Average Error: 40.3 → 1.2
Time: 33.1s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\left(\sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}} \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}\right) \cdot \sqrt[3]{\frac{e^{x}}{(e^{x} - 1)^*}}\]

Error

Bits error versus x

Target

Original40.3
Target40.0
Herbie1.2
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.3

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

  2. Applied simplify0.4

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

  4. Applied add-cube-cbrt1.2

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

Runtime

Time bar (total: 33.1s)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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