Average Error: 39.8 → 0.3
Time: 2.7s
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 r76660 = x;
        double r76661 = exp(r76660);
        double r76662 = 1.0;
        double r76663 = r76661 - r76662;
        double r76664 = r76663 / r76660;
        return r76664;
}
↓
double f(double x) {
        double r76665 = x;
        double r76666 = -0.00014842288339664386;
        bool r76667 = r76665 <= r76666;
        double r76668 = 1.0;
        double r76669 = -r76668;
        double r76670 = r76665 + r76665;
        double r76671 = exp(r76670);
        double r76672 = fma(r76669, r76668, r76671);
        double r76673 = exp(r76672);
        double r76674 = log(r76673);
        double r76675 = exp(r76665);
        double r76676 = r76675 + r76668;
        double r76677 = r76674 / r76676;
        double r76678 = r76677 / r76665;
        double r76679 = 0.16666666666666666;
        double r76680 = 2.0;
        double r76681 = pow(r76665, r76680);
        double r76682 = 0.5;
        double r76683 = 1.0;
        double r76684 = fma(r76682, r76665, r76683);
        double r76685 = fma(r76679, r76681, r76684);
        double r76686 = r76667 ? r76678 : r76685;
        return r76686;
}

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))