Average Error: 40.1 → 0.3
Time: 3.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.0413827379038481 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{\left(\sqrt[3]{e^{x} + 1} \cdot \left(\sqrt[3]{\sqrt{e^{x} + 1}} \cdot \sqrt[3]{\sqrt{e^{x} + 1}}\right)\right) \cdot \sqrt[3]{e^{x} + 1}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.0413827379038481 \cdot 10^{-5}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{\left(\sqrt[3]{e^{x} + 1} \cdot \left(\sqrt[3]{\sqrt{e^{x} + 1}} \cdot \sqrt[3]{\sqrt{e^{x} + 1}}\right)\right) \cdot \sqrt[3]{e^{x} + 1}}}{x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\

\end{array}
double code(double x) {
	return ((exp(x) - 1.0) / x);
}
↓
double code(double x) {
	double VAR;
	if ((x <= -9.041382737903848e-05)) {
		VAR = ((fma(-1.0, 1.0, exp((x + x))) / ((cbrt((exp(x) + 1.0)) * (cbrt(sqrt((exp(x) + 1.0))) * cbrt(sqrt((exp(x) + 1.0))))) * cbrt((exp(x) + 1.0)))) / x);
	} else {
		VAR = fma(0.16666666666666666, pow(x, 2.0), fma(0.5, x, 1.0));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target40.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -9.041382737903848e-05

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Simplified0.1

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

    6. Applied add-cube-cbrt0.1

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{\color{blue}{\left(\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}\right) \cdot \sqrt[3]{e^{x} + 1}}}}{x}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt0.1

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

    9. Applied cbrt-prod0.1

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

    if -9.041382737903848e-05 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]

    3. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.0413827379038481 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{\left(\sqrt[3]{e^{x} + 1} \cdot \left(\sqrt[3]{\sqrt{e^{x} + 1}} \cdot \sqrt[3]{\sqrt{e^{x} + 1}}\right)\right) \cdot \sqrt[3]{e^{x} + 1}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

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

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