Average Error: 40.2 → 0.4
Time: 7.9s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[e^{x} \cdot \frac{1}{(e^{x} - 1)^*}\]

Error

Bits error versus x

Target

Original40.2
Target39.8
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.2

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

  2. Applied simplify0.4

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

  4. Applied div-inv0.4

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

Runtime

Time bar (total: 7.9s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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