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consecutive bump nodes with image textures give nan in normal pass
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System Information
All OS tested

Blender Version
Broken: 2.79.6
Worked: 2.79

In our materials we use textures to add scratches and dirt to the normals via bump nodes (textures go to the height input). For some time now we have broken normal passes. Instead of sending you our shitty shaders I isolated the problematic parts:

If you use an image texture as height input for the bump node and then connect the normal output to another bump node also with an image texture for the height you will end with nan pixels in the normal pass. This "works" even if no image is selected in the image texture node. See blend file for reproduction.

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Event Timeline

While updating our shaders to workaround this bug we noticed that the added bump node could use any texture for height input and the resulting normal pass after adding this to another bump node would be wrong.

So for instance a noise texture bump node added via the normal input to an image texture bump node would give nan in the normal pass.