Average Error: 42.1 → 41.4
Time: 15.8s
Precision: 64
Internal Precision: 128
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}} \cdot \sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}}\right)\]

Error

Bits error versus x

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Results

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Target

Original42.1
Target42.1
Herbie41.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 42.1

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

  2. Initial simplification41.4

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

  4. Applied add-cube-cbrt41.4

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

  6. Applied add-cube-cbrt41.4

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}} \cdot \sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}}\right)} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}\]

  7. Final simplification41.4

    \[\leadsto \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}} \cdot \sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}}\right)\]

Runtime

Time bar (total: 15.8s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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