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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.0:\\ \;\;\;\;\sqrt[3]{\left(\left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) + \frac{1}{2} \cdot x\right) \cdot \left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) + \frac{1}{2} \cdot x\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) + \frac{1}{2} \cdot x\right)}\\ \mathbf{elif}\;\frac{e^{x} - 1}{x} \le 0.9877278429083478:\\ \;\;\;\;\frac{-1 + e^{x}}{x}\\ \mathbf{elif}\;\frac{e^{x} - 1}{x} \le 1.753990269224867:\\ \;\;\;\;\left({x}^{2} \cdot \frac{1}{6} + 1\right) + \frac{1}{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + e^{x}}{x}\\ \end{array}\]

Original	40.2
Target	39.4
Herbie	0.3

Derivation

Split input into 3 regimes
if (/ (- (exp x) 1) x) < 0.0
1. Initial program 62.0
  \[\frac{e^{x} - 1}{x}\]
2. Taylor expanded around 0 0
  \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]
3. Using strategy rm
4. Applied add-cbrt-cube0
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right) \cdot \left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)\right) \cdot \left(\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)\right)}}\]
if 0.0 < (/ (- (exp x) 1) x) < 0.9877278429083478 or 1.753990269224867 < (/ (- (exp x) 1) x)
1. Initial program 0.3
  \[\frac{e^{x} - 1}{x}\]
2. Taylor expanded around -inf 0.3
  \[\leadsto \color{blue}{-1 \cdot \frac{1 - e^{x}}{x}}\]
3. Simplified0.3
  \[\leadsto \color{blue}{\frac{e^{x} + -1}{x}}\]
if 0.9877278429083478 < (/ (- (exp x) 1) x) < 1.753990269224867
1. Initial program 23.3
  \[\frac{e^{x} - 1}{x}\]
2. Taylor expanded around 0 6.4
  \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]
Recombined 3 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \le 0.0:\\ \;\;\;\;\sqrt[3]{\left(\left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) + \frac{1}{2} \cdot x\right) \cdot \left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) + \frac{1}{2} \cdot x\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{1}{6} + 1\right) + \frac{1}{2} \cdot x\right)}\\ \mathbf{elif}\;\frac{e^{x} - 1}{x} \le 0.9877278429083478:\\ \;\;\;\;\frac{-1 + e^{x}}{x}\\ \mathbf{elif}\;\frac{e^{x} - 1}{x} \le 1.753990269224867:\\ \;\;\;\;\left({x}^{2} \cdot \frac{1}{6} + 1\right) + \frac{1}{2} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + e^{x}}{x}\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	20.7	0.3	0.0	20.7	98.9%

herbie shell --seed 2018274 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

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

Error

Try it out

Target

Derivation

`if (/ (- (exp x) 1) x) < 0.0`

`if 0.0 < (/ (- (exp x) 1) x) < 0.9877278429083478 or 1.753990269224867 < (/ (- (exp x) 1) x)`

`if 0.9877278429083478 < (/ (- (exp x) 1) x) < 1.753990269224867`

Runtime

In
Out