Average Error: 40.5 → 0.4
Time: 20.1s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\sqrt{e^{x}}}{\frac{(e^{x} - 1)^*}{\sqrt{e^{x}}}}\]

Error

Bits error versus x

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Results

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Target

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

Derivation

  1. Initial program 40.5

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

  2. Initial simplification0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Runtime

Time bar (total: 20.1s)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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