Average Error: 39.5 → 0.3
Time: 3.6s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.709877439748755422518206037452159762324 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{x \cdot \mathsf{fma}\left(1, e^{x} + 1, e^{x + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.709877439748755422518206037452159762324 \cdot 10^{-4}:\\
\;\;\;\;\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{x \cdot \mathsf{fma}\left(1, e^{x} + 1, e^{x + x}\right)}\\

\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 r83070 = x;
        double r83071 = exp(r83070);
        double r83072 = 1.0;
        double r83073 = r83071 - r83072;
        double r83074 = r83073 / r83070;
        return r83074;
}
↓
double f(double x) {
        double r83075 = x;
        double r83076 = -0.00017098774397487554;
        bool r83077 = r83075 <= r83076;
        double r83078 = exp(r83075);
        double r83079 = 3.0;
        double r83080 = pow(r83078, r83079);
        double r83081 = 1.0;
        double r83082 = pow(r83081, r83079);
        double r83083 = r83080 - r83082;
        double r83084 = r83078 + r83081;
        double r83085 = r83075 + r83075;
        double r83086 = exp(r83085);
        double r83087 = fma(r83081, r83084, r83086);
        double r83088 = r83075 * r83087;
        double r83089 = r83083 / r83088;
        double r83090 = 0.5;
        double r83091 = 2.0;
        double r83092 = pow(r83075, r83091);
        double r83093 = 0.16666666666666666;
        double r83094 = pow(r83075, r83079);
        double r83095 = fma(r83093, r83094, r83075);
        double r83096 = fma(r83090, r83092, r83095);
        double r83097 = r83096 / r83075;
        double r83098 = r83077 ? r83089 : r83097;
        return r83098;
}

Error

Bits error versus x

Target

Original39.5
Target40.0
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 < -0.00017098774397487554

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Applied associate-/l/0.0

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

    5. Simplified0.0

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

    if -0.00017098774397487554 < x

    1. Initial program 60.2

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.709877439748755422518206037452159762324 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{x \cdot \mathsf{fma}\left(1, e^{x} + 1, e^{x + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))