Average Error: 39.7 → 0.3
Time: 16.4s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.392605479495764375999500295222333079437 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, e^{x} + 1, e^{x + x}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.392605479495764375999500295222333079437 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, e^{x} + 1, e^{x + x}\right)}}{x}\\

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

\end{array}
double f(double x) {
        double r62517 = x;
        double r62518 = exp(r62517);
        double r62519 = 1.0;
        double r62520 = r62518 - r62519;
        double r62521 = r62520 / r62517;
        return r62521;
}
↓
double f(double x) {
        double r62522 = x;
        double r62523 = -0.00013926054794957644;
        bool r62524 = r62522 <= r62523;
        double r62525 = exp(r62522);
        double r62526 = 3.0;
        double r62527 = pow(r62525, r62526);
        double r62528 = 1.0;
        double r62529 = pow(r62528, r62526);
        double r62530 = r62527 - r62529;
        double r62531 = r62525 + r62528;
        double r62532 = r62522 + r62522;
        double r62533 = exp(r62532);
        double r62534 = fma(r62528, r62531, r62533);
        double r62535 = r62530 / r62534;
        double r62536 = r62535 / r62522;
        double r62537 = 0.16666666666666666;
        double r62538 = 0.5;
        double r62539 = fma(r62537, r62522, r62538);
        double r62540 = 1.0;
        double r62541 = fma(r62522, r62539, r62540);
        double r62542 = r62524 ? r62536 : r62541;
        return r62542;
}

Error

Bits error versus x

Target

Original39.7
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 < -0.00013926054794957644

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    if -0.00013926054794957644 < x

    1. Initial program 59.9

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.392605479495764375999500295222333079437 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, e^{x} + 1, e^{x + x}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))