Average Error: 40.0 → 0.3
Time: 6.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.02234544586829302 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt{1} + \sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.02234544586829302 \cdot 10^{-4}:\\
\;\;\;\;\frac{\left(\sqrt{1} + \sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}{x}\\

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

\end{array}
double f(double x) {
        double r99080 = x;
        double r99081 = exp(r99080);
        double r99082 = 1.0;
        double r99083 = r99081 - r99082;
        double r99084 = r99083 / r99080;
        return r99084;
}
↓
double f(double x) {
        double r99085 = x;
        double r99086 = -0.0002022345445868293;
        bool r99087 = r99085 <= r99086;
        double r99088 = 1.0;
        double r99089 = sqrt(r99088);
        double r99090 = exp(r99085);
        double r99091 = sqrt(r99090);
        double r99092 = r99089 + r99091;
        double r99093 = r99091 - r99089;
        double r99094 = r99092 * r99093;
        double r99095 = r99094 / r99085;
        double r99096 = 0.16666666666666666;
        double r99097 = r99085 * r99085;
        double r99098 = 0.5;
        double r99099 = 1.0;
        double r99100 = fma(r99098, r99085, r99099);
        double r99101 = fma(r99096, r99097, r99100);
        double r99102 = r99087 ? r99095 : r99101;
        return r99102;
}

Error

Bits error versus x

Target

Original40.0
Target40.5
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 < -0.0002022345445868293

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{e^{x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{x}\]

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - \sqrt{1} \cdot \sqrt{1}}{x}\]

    5. Applied difference-of-squares0.1

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

    6. Simplified0.1

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

    if -0.0002022345445868293 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.02234544586829302 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt{1} + \sqrt{e^{x}}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))