Average Error: 40.1 → 0.4
Time: 11.4s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{1}{(e^{x} - 1)^* \cdot e^{-x}}\]

Error

Bits error versus x

Target

Original40.1
Target39.6
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.1

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

  2. Applied simplify0.4

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

  4. Applied clear-num0.5

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

  6. Applied div-inv0.5

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

  7. Applied simplify0.4

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

Runtime

Time bar (total: 11.4s)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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