Average Error: 39.6 → 0.3
Time: 3.4s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.03925223585416463 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)}}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}}{\frac{x \cdot \sqrt[3]{e^{x} + 1}}{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.03925223585416463 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)}}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}}{\frac{x \cdot \sqrt[3]{e^{x} + 1}}{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}}\\

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

\end{array}
double f(double x) {
        double r113073 = x;
        double r113074 = exp(r113073);
        double r113075 = 1.0;
        double r113076 = r113074 - r113075;
        double r113077 = r113076 / r113073;
        return r113077;
}
↓
double f(double x) {
        double r113078 = x;
        double r113079 = -0.00010392522358541646;
        bool r113080 = r113078 <= r113079;
        double r113081 = 1.0;
        double r113082 = -r113081;
        double r113083 = r113078 + r113078;
        double r113084 = exp(r113083);
        double r113085 = fma(r113082, r113081, r113084);
        double r113086 = exp(r113085);
        double r113087 = log(r113086);
        double r113088 = cbrt(r113087);
        double r113089 = r113088 * r113088;
        double r113090 = exp(r113078);
        double r113091 = r113090 + r113081;
        double r113092 = cbrt(r113091);
        double r113093 = r113092 * r113092;
        double r113094 = r113089 / r113093;
        double r113095 = r113078 * r113092;
        double r113096 = cbrt(r113085);
        double r113097 = r113095 / r113096;
        double r113098 = r113094 / r113097;
        double r113099 = 0.16666666666666666;
        double r113100 = 2.0;
        double r113101 = pow(r113078, r113100);
        double r113102 = 0.5;
        double r113103 = 1.0;
        double r113104 = fma(r113102, r113078, r113103);
        double r113105 = fma(r113099, r113101, r113104);
        double r113106 = r113080 ? r113098 : r113105;
        return r113106;
}

Error

Bits error versus x

Target

Original39.6
Target40.1
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 < -0.00010392522358541646

    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. Simplified0.0

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

    6. Applied add-log-exp0.1

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

    8. Applied add-cube-cbrt0.1

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

    9. Applied add-cube-cbrt0.1

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

    10. Applied times-frac0.1

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

    11. Applied associate-/l*0.1

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

    12. Simplified0.1

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

    if -0.00010392522358541646 < x

    1. Initial program 60.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.03925223585416463 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)} \cdot \sqrt[3]{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)}}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}}{\frac{x \cdot \sqrt[3]{e^{x} + 1}}{\sqrt[3]{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))