Average Error: 39.2 → 0.3
Time: 15.4s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{(e^{x} - 1)^*}\]

Error

Bits error versus x

Target

Original39.2
Target38.9
Herbie0.3
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.2

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

  3. Applied add-exp-log39.2

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

  4. Applied expm1-def39.2

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

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