Average Error: 39.8 → 1.4
Time: 15.7s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\sqrt{e^{x}}}{\sqrt[3]{(e^{x} - 1)^*} \cdot \sqrt[3]{(e^{x} - 1)^*}} \cdot \frac{\sqrt{e^{x}}}{\sqrt[3]{(e^{x} - 1)^*}}\]

Error

Bits error versus x

Target

Original39.8
Target39.4
Herbie1.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.8

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

  2. Applied simplify0.4

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

  4. Applied 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 add-sqr-sqrt1.3

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

  6. Applied times-frac1.4

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

Runtime

Time bar (total: 15.7s)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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