Average Error: 40.7 → 0.3
Time: 2.4s
Precision: binary64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.00016710770978154628:\\ \;\;\;\;\frac{\sqrt[3]{{\left({\left(e^{x}\right)}^{2} + -1\right)}^{3}}}{x \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.00016710770978154628:\\
\;\;\;\;\frac{\sqrt[3]{{\left({\left(e^{x}\right)}^{2} + -1\right)}^{3}}}{x \cdot \left(e^{x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\

\end{array}
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.00016710770978154628)
   (/ (cbrt (pow (+ (pow (exp x) 2.0) -1.0) 3.0)) (* x (+ (exp x) 1.0)))
   (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))
double code(double x) {
	return (exp(x) - 1.0) / x;
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.00016710770978154628) {
		tmp = cbrt(pow((pow(exp(x), 2.0) + -1.0), 3.0)) / (x * (exp(x) + 1.0));
	} else {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.7
Target41.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.67107709781546279e-4

    1. Initial program 0.0

      \[\frac{e^{x} - 1}{x}\]
    2. Using strategy rm

    3. Applied flip--_binary640.1

      \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}{x}\]

    4. Applied associate-/l/_binary640.1

      \[\leadsto \color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{x \cdot \left(e^{x} + 1\right)}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube_binary640.1

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

    7. Simplified0.1

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left({\left(e^{x}\right)}^{2} + -1\right)}^{3}}}}{x \cdot \left(e^{x} + 1\right)}\]

    if -1.67107709781546279e-4 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{0.5 \cdot x + \left(0.16666666666666666 \cdot {x}^{2} + 1\right)}\]

    3. Simplified0.4

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00016710770978154628:\\ \;\;\;\;\frac{\sqrt[3]{{\left({\left(e^{x}\right)}^{2} + -1\right)}^{3}}}{x \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))