Average Error: 40.2 → 0.3
Time: 3.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.4843984217390087 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\sqrt[3]{e^{x} + 1}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.4843984217390087 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\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 <= -0.00014843984217390087)) {
		VAR = ((cbrt((fma(-1.0, 1.0, exp((x + x))) / (exp(x) + 1.0))) * cbrt((fma(-1.0, 1.0, exp((x + x))) / (exp(x) + 1.0)))) * ((cbrt(((cbrt(fma(-1.0, 1.0, exp((x + x)))) * cbrt(fma(-1.0, 1.0, exp((x + x))))) / (cbrt((exp(x) + 1.0)) * cbrt((exp(x) + 1.0))))) * cbrt((cbrt(fma(-1.0, 1.0, exp((x + x)))) / 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

Original40.2
Target40.7
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 < -0.00014843984217390087

    1. Initial program 0.0

      \[\frac{e^{x} - 1}{x}\]
    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. Simplified0.0

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

    6. Applied *-un-lft-identity0.0

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

    7. Applied add-cube-cbrt0.0

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

    8. Applied times-frac0.0

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

    9. Simplified0.0

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

    11. Applied add-cube-cbrt0.0

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

    12. Applied add-cube-cbrt0.0

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

    13. Applied times-frac0.0

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

    14. Applied cbrt-prod0.0

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

    if -0.00014843984217390087 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4843984217390087 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\sqrt[3]{e^{x} + 1}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +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))