Average Error: 39.9 → 0.4
Time: 19.7s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{(e^{x} - 1)^*}\]

Error

Bits error versus x

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Results

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Target

Original39.9
Target39.5
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.9

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

  2. Initial simplification0.4

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

  3. Taylor expanded around -inf 39.9

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Runtime

Time bar (total: 19.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.40.40.40.10%
herbie shell --seed 2018351 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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