Error

Target

Derivation

1. Initial program 40.8

   \[\frac{e^{x} - 1}{x}\]
2. Using strategy rm

3. Applied add-sqr-sqrt40.8

   \[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - 1}{x}\]

4. Applied fma-neg40.8

   \[\leadsto \frac{\color{blue}{(\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}}\right) + \left(-1\right))_*}}{x}\]
5. Using strategy rm

6. Applied add-sqr-sqrt40.8

   \[\leadsto \frac{(\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}}\right) + \left(-1\right))_*}{x}\]

7. Applied sqrt-prod40.8

   \[\leadsto \frac{(\left(\sqrt{e^{x}}\right) \cdot \color{blue}{\left(\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}\right)} + \left(-1\right))_*}{x}\]
8. Using strategy rm

9. Applied add-cube-cbrt40.8

   \[\leadsto \frac{(\left(\sqrt{e^{x}}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}} \cdot \sqrt[3]{\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}}\right)} + \left(-1\right))_*}{x}\]

10. Final simplification40.8

    \[\leadsto \frac{(\left(\sqrt{e^{x}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}} \cdot \sqrt[3]{\sqrt{\sqrt{e^{x}}} \cdot \sqrt{\sqrt{e^{x}}}}\right)\right) + -1)_*}{x}\]

Runtime

Time bar (total: 10.7s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))