Is there any real difference between the math functions performed by numpy and tensorflow. For example, exponential function, or the max function?

The only difference I noticed is that tensorflow takes input of tensors, and not numpy arrays. Is this the only difference, and no difference in the results of the function, by value?

As has been mentioned, there is the performance difference. TensorFlow has the advantage that it has been designed to work on both CPUs or GPUs, so if you have a CUDA-enabled GPU, chances are TensorFlow is going to be much faster. You can find several benchmarks on the web with different comparisons, and also with other packages such as Numba or Theano.

However, I think that you are talking about whether NumPy and TensorFlow operations are exactly equivalent. The answer is basically yes, that is, the meaning of the operations is the same. However, since they are completely separate libraries with different implementations for everything, you will find small differences in the results. Take this code, for example (TensorFlow 1.2.0, NumPy 1.13.1):

# Force TensorFlow to run on CPU only
import os
os.environ["CUDA_VISIBLE_DEVICES"] = "-1"import numpy as np
import tensorflow as tf# float32 NumPy array
a = np.arange(100, dtype=np.float32)
# The same array with the same dtype in TensorFlow
a_tf = tf.constant(a, dtype=tf.float32)
# Square root with NumPy
sqrt = np.sqrt(a)
# Square root with TensorFlow
with tf.Session() as sess:sqrt_tf = sess.run(tf.sqrt(a_tf))

You would expect to get pretty much the same output from both, I mean, a square root doesn't sound like an extremely complex operation after all. However, printing these arrays in my computer I get:

print(sqrt)
>>> array([ 0.        ,  1.        ,  1.41421354,  1.73205078,  2.        ,2.23606801,  2.44948983,  2.64575124,  2.82842708,  3.        ,3.1622777 ,  3.31662488,  3.46410155,  3.60555124,  3.7416575 ,3.87298346,  4.        ,  4.12310553,  4.2426405 ,  4.35889912,4.47213602,  4.5825758 ,  4.69041586,  4.79583168,  4.89897966,5.        ,  5.09901953,  5.19615221,  5.29150248,  5.38516474,5.47722578,  5.56776428,  5.65685415,  5.74456263,  5.83095169,5.91608   ,  6.        ,  6.08276272,  6.16441393,  6.24499798,6.3245554 ,  6.40312433,  6.48074055,  6.55743837,  6.63324976,6.70820379,  6.78233004,  6.85565472,  6.92820311,  7.        ,7.07106781,  7.14142847,  7.21110249,  7.28010988,  7.34846926,7.41619825,  7.48331499,  7.54983425,  7.6157732 ,  7.68114567,7.74596691,  7.81024981,  7.8740077 ,  7.93725395,  8.        ,8.06225777,  8.1240387 ,  8.18535233,  8.24621105,  8.30662346,8.36660004,  8.42614937,  8.48528099,  8.54400349,  8.60232544,8.66025448,  8.71779823,  8.77496433,  8.83176041,  8.88819408,8.94427204,  9.        ,  9.05538559,  9.11043358,  9.1651516 ,9.21954441,  9.2736187 ,  9.32737923,  9.38083172,  9.43398094,9.48683262,  9.53939247,  9.59166336,  9.64365101,  9.69536018,9.7467947 ,  9.79795933,  9.84885788,  9.89949512,  9.94987392], dtype=float32)print(sqrt_tf)
>>> array([ 0.        ,  0.99999994,  1.41421342,  1.73205078,  1.99999988,2.23606801,  2.44948959,  2.64575124,  2.82842684,  2.99999976,3.1622777 ,  3.31662488,  3.46410155,  3.60555077,  3.74165726,3.87298322,  3.99999976,  4.12310553,  4.2426405 ,  4.35889864,4.47213602,  4.58257532,  4.69041538,  4.79583073,  4.89897919,5.        ,  5.09901857,  5.19615221,  5.29150248,  5.38516474,5.47722483,  5.56776428,  5.65685368,  5.74456215,  5.83095121,5.91607952,  5.99999952,  6.08276224,  6.16441393,  6.24499846,6.3245554 ,  6.40312433,  6.48074055,  6.5574379 ,  6.63324976,6.70820427,  6.78233004,  6.85565472,  6.92820311,  6.99999952,7.07106733,  7.14142799,  7.21110153,  7.28010893,  7.34846973,7.41619825,  7.48331451,  7.54983425,  7.61577368,  7.68114567,7.74596643,  7.81025028,  7.8740077 ,  7.93725395,  7.99999952,8.06225681,  8.12403774,  8.18535233,  8.24621105,  8.30662346,8.36660004,  8.42614937,  8.48528099,  8.54400253,  8.60232449,8.66025352,  8.71779728,  8.77496433,  8.83176041,  8.88819408,8.94427204,  8.99999905,  9.05538464,  9.11043262,  9.16515064,9.21954441,  9.27361774,  9.32737923,  9.38083076,  9.43398094,9.48683357,  9.53939152,  9.59166145,  9.64365005,  9.69535923,9.7467947 ,  9.79795837,  9.84885788,  9.89949417,  9.94987392], dtype=float32)

So, okay, it's similar, but there are obvious differences. TensorFlow couldn't even get right the square roots of 1, 4 or 9, for example. And you would probably get yet a different result if you run it on a GPU (due to the GPU kernels being different from the CPU kernels and the dependence on CUDA routines implemented by NVIDIA, another player in the field).

My impression (although I may be wrong) is that TensorFlow is more willing to sacrifice a bit of precision in exchange of performance (which would make sense considering its typical use case). I have even seen some more complicated operations to produce (very slightly) different results just running it twice (on the same hardware), probably due to unspecified order in aggregation and averaging operations causing rounding errors (I generally use float32, so that's a factor too I guess).