Average Error: 40.8 → 0.4
Time: 8.1s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{(e^{x} - 1)^*}\]

Error

Bits error versus x

Target

Original40.8
Target40.5
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.8

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

  3. Applied expm1-def0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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