Average Error: 38.9 → 0.3
Time: 3.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.405557861940245 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\left(\sqrt[3]{\sqrt[3]{e^{x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.405557861940245 \cdot 10^{-4}:\\
\;\;\;\;\frac{\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)}}{\left(\sqrt[3]{\sqrt[3]{e^{x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}{x}\\

\end{array}
double f(double x) {
        double r54641 = x;
        double r54642 = exp(r54641);
        double r54643 = 1.0;
        double r54644 = r54642 - r54643;
        double r54645 = r54644 / r54641;
        return r54645;
}
↓
double f(double x) {
        double r54646 = x;
        double r54647 = -0.00014055578619402449;
        bool r54648 = r54646 <= r54647;
        double r54649 = 1.0;
        double r54650 = -r54649;
        double r54651 = r54646 + r54646;
        double r54652 = exp(r54651);
        double r54653 = fma(r54650, r54649, r54652);
        double r54654 = cbrt(r54653);
        double r54655 = r54654 * r54654;
        double r54656 = exp(r54646);
        double r54657 = r54656 + r54649;
        double r54658 = cbrt(r54657);
        double r54659 = r54658 * r54658;
        double r54660 = r54655 / r54659;
        double r54661 = cbrt(r54658);
        double r54662 = r54661 * r54661;
        double r54663 = r54662 * r54661;
        double r54664 = r54654 / r54663;
        double r54665 = r54660 * r54664;
        double r54666 = r54665 / r54646;
        double r54667 = 0.5;
        double r54668 = 2.0;
        double r54669 = pow(r54646, r54668);
        double r54670 = 0.16666666666666666;
        double r54671 = 3.0;
        double r54672 = pow(r54646, r54671);
        double r54673 = fma(r54670, r54672, r54646);
        double r54674 = fma(r54667, r54669, r54673);
        double r54675 = r54674 / r54646;
        double r54676 = r54648 ? r54666 : r54675;
        return r54676;
}

Error

Bits error versus x

Target

Original38.9
Target39.3
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 < -0.00014055578619402449

    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. Applied add-cube-cbrt0.1

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

    8. Applied times-frac0.1

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

    10. Applied add-cube-cbrt0.1

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

    if -0.00014055578619402449 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}{x}\]

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.405557861940245 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{\left(\sqrt[3]{\sqrt[3]{e^{x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))