Average Error: 39.8 → 0.3
Time: 14.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.245218611780339091916341986987504242279 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{1 + e^{x}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{1 + e^{x}}} \cdot \sqrt[3]{\sqrt[3]{1 + e^{x}}}\right) \cdot \sqrt[3]{\sqrt[3]{1 + e^{x}}}\right)} \cdot \frac{e^{x + x} - 1 \cdot 1}{\sqrt[3]{1 + e^{x}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.245218611780339091916341986987504242279 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{1 + e^{x}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{1 + e^{x}}} \cdot \sqrt[3]{\sqrt[3]{1 + e^{x}}}\right) \cdot \sqrt[3]{\sqrt[3]{1 + e^{x}}}\right)} \cdot \frac{e^{x + x} - 1 \cdot 1}{\sqrt[3]{1 + e^{x}}}}{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 r105395 = x;
        double r105396 = exp(r105395);
        double r105397 = 1.0;
        double r105398 = r105396 - r105397;
        double r105399 = r105398 / r105395;
        return r105399;
}
↓
double f(double x) {
        double r105400 = x;
        double r105401 = -0.0001245218611780339;
        bool r105402 = r105400 <= r105401;
        double r105403 = 1.0;
        double r105404 = 1.0;
        double r105405 = exp(r105400);
        double r105406 = r105404 + r105405;
        double r105407 = cbrt(r105406);
        double r105408 = cbrt(r105407);
        double r105409 = r105408 * r105408;
        double r105410 = r105409 * r105408;
        double r105411 = r105407 * r105410;
        double r105412 = r105403 / r105411;
        double r105413 = r105400 + r105400;
        double r105414 = exp(r105413);
        double r105415 = r105404 * r105404;
        double r105416 = r105414 - r105415;
        double r105417 = r105416 / r105407;
        double r105418 = r105412 * r105417;
        double r105419 = r105418 / r105400;
        double r105420 = 0.16666666666666666;
        double r105421 = 0.5;
        double r105422 = fma(r105420, r105400, r105421);
        double r105423 = fma(r105400, r105422, r105403);
        double r105424 = r105402 ? r105419 : r105423;
        return r105424;
}

Error

Bits error versus x

Target

Original39.8
Target40.2
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 < -0.0001245218611780339

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

      \[\leadsto \frac{\frac{e^{x + x} - 1 \cdot 1}{\color{blue}{1 + e^{x}}}}{x}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.1

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

    8. Applied *-un-lft-identity0.1

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

    9. Applied times-frac0.0

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}} \cdot \frac{e^{x + x} - 1 \cdot 1}{\sqrt[3]{1 + e^{x}}}}}{x}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt0.1

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

    if -0.0001245218611780339 < 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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.245218611780339091916341986987504242279 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{1 + e^{x}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{1 + e^{x}}} \cdot \sqrt[3]{\sqrt[3]{1 + e^{x}}}\right) \cdot \sqrt[3]{\sqrt[3]{1 + e^{x}}}\right)} \cdot \frac{e^{x + x} - 1 \cdot 1}{\sqrt[3]{1 + e^{x}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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))