Average Error: 39.7 → 0.0
Time: 44.8s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - 1}{x}\]
↓
\[\frac{\frac{1}{x}}{\frac{1}{(e^{x} - 1)^*}}\]

Error

Bits error versus x

Target

Original39.7
Target38.9
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 39.7

    \[\frac{e^{x} - 1}{x}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{(e^{x} - 1)^*}{x}}\]
  3. Using strategy rm

  4. Applied clear-num0.0

    \[\leadsto \color{blue}{\frac{1}{\frac{x}{(e^{x} - 1)^*}}}\]
  5. Using strategy rm

  6. Applied div-inv0.1

    \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{(e^{x} - 1)^*}}}\]

  7. Applied associate-/r*0.0

    \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{1}{(e^{x} - 1)^*}}}\]

  8. Final simplification0.0

    \[\leadsto \frac{\frac{1}{x}}{\frac{1}{(e^{x} - 1)^*}}\]

Reproduce

herbie shell --seed 2019090 +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))