Average Error: 40.2 → 0.6

Time: 1.5m

Precision: 64

Internal Precision: 1344

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9983021854475828:\\ \;\;\;\;\sqrt[3]{{\left(\frac{e^{x}}{e^{x} - 1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus x

The X axis uses an exponential scale — Bits error versus x

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	40.2
Target	39.8
Herbie	0.6

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

Derivation

Split input into 2 regimes
if (exp x) < 0.9983021854475828
1. Initial program 0.0
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied add-cbrt-cube0.0
  \[\leadsto \frac{e^{x}}{\color{blue}{\sqrt[3]{\left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right) \cdot \left(e^{x} - 1\right)}}}\]
4. Applied add-cbrt-cube0.1
  \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(e^{x} \cdot e^{x}\right) \cdot e^{x}}}}{\sqrt[3]{\left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right) \cdot \left(e^{x} - 1\right)}}\]
5. Applied cbrt-undiv0.1
  \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(e^{x} \cdot e^{x}\right) \cdot e^{x}}{\left(\left(e^{x} - 1\right) \cdot \left(e^{x} - 1\right)\right) \cdot \left(e^{x} - 1\right)}}}\]
6. Applied simplify0.1
  \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{e^{x}}{e^{x} - 1}\right)}^{3}}}\]
if 0.9983021854475828 < (exp x)
1. Initial program 60.4
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Taylor expanded around 0 0.8
  \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed 2018198 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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