Average Error: 40.7 → 1.3
Time: 42.3s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{1}{\sqrt[3]{(e^{x} - 1)^*} \cdot \sqrt[3]{(e^{x} - 1)^*}} \cdot \frac{e^{x}}{\sqrt[3]{(e^{x} - 1)^*}}\]

Error

Bits error versus x

Target

Original40.7
Target40.4
Herbie1.3
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.7

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

  2. Applied simplify0.3

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

  4. Applied add-cube-cbrt1.3

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

  5. Applied *-un-lft-identity1.3

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

  6. Applied times-frac1.3

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

Runtime

Time bar (total: 42.3s)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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