Average Error: 39.8 → 0.3
Time: 2.8s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.4842288339664386 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.4842288339664386 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)}{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 f(double x) {
        double r48950 = x;
        double r48951 = exp(r48950);
        double r48952 = 1.0;
        double r48953 = r48951 - r48952;
        double r48954 = r48953 / r48950;
        return r48954;
}
↓
double f(double x) {
        double r48955 = x;
        double r48956 = -0.00014842288339664386;
        bool r48957 = r48955 <= r48956;
        double r48958 = 1.0;
        double r48959 = -r48958;
        double r48960 = r48955 + r48955;
        double r48961 = exp(r48960);
        double r48962 = fma(r48959, r48958, r48961);
        double r48963 = exp(r48962);
        double r48964 = log(r48963);
        double r48965 = exp(r48955);
        double r48966 = r48965 + r48958;
        double r48967 = r48964 / r48966;
        double r48968 = r48967 / r48955;
        double r48969 = 0.16666666666666666;
        double r48970 = 2.0;
        double r48971 = pow(r48955, r48970);
        double r48972 = 0.5;
        double r48973 = 1.0;
        double r48974 = fma(r48972, r48955, r48973);
        double r48975 = fma(r48969, r48971, r48974);
        double r48976 = r48957 ? r48968 : r48975;
        return r48976;
}

Error

Bits error versus x

Target

Original39.8
Target40.3
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 < -0.00014842288339664386

    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-log-exp0.1

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

    if -0.00014842288339664386 < x

    1. Initial program 60.2

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4842288339664386 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\log \left(e^{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))