Average Error: 39.8 → 0.4
Time: 11.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.6704963527402913 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left({\left(e^{x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(\mathsf{fma}\left(1, 1 + e^{x}, e^{x + x}\right) \cdot \mathsf{fma}\left({1}^{3}, {\left(e^{x}\right)}^{3} + {1}^{3}, {\left(e^{x}\right)}^{6}\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right)}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.6704963527402913 \cdot 10^{-4}:\\
\;\;\;\;\frac{{\left({\left(e^{x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(\mathsf{fma}\left(1, 1 + e^{x}, e^{x + x}\right) \cdot \mathsf{fma}\left({1}^{3}, {\left(e^{x}\right)}^{3} + {1}^{3}, {\left(e^{x}\right)}^{6}\right)\right) \cdot x}\\

\mathbf{else}:\\
\;\;\;\;e^{\mathsf{log1p}\left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right)}\\

\end{array}
double f(double x) {
        double r45864 = x;
        double r45865 = exp(r45864);
        double r45866 = 1.0;
        double r45867 = r45865 - r45866;
        double r45868 = r45867 / r45864;
        return r45868;
}
↓
double f(double x) {
        double r45869 = x;
        double r45870 = -0.00016704963527402913;
        bool r45871 = r45869 <= r45870;
        double r45872 = exp(r45869);
        double r45873 = 3.0;
        double r45874 = pow(r45872, r45873);
        double r45875 = pow(r45874, r45873);
        double r45876 = 1.0;
        double r45877 = pow(r45876, r45873);
        double r45878 = pow(r45877, r45873);
        double r45879 = r45875 - r45878;
        double r45880 = r45876 + r45872;
        double r45881 = r45869 + r45869;
        double r45882 = exp(r45881);
        double r45883 = fma(r45876, r45880, r45882);
        double r45884 = r45874 + r45877;
        double r45885 = 6.0;
        double r45886 = pow(r45872, r45885);
        double r45887 = fma(r45877, r45884, r45886);
        double r45888 = r45883 * r45887;
        double r45889 = r45888 * r45869;
        double r45890 = r45879 / r45889;
        double r45891 = 0.16666666666666666;
        double r45892 = 0.5;
        double r45893 = fma(r45869, r45891, r45892);
        double r45894 = r45869 * r45893;
        double r45895 = log1p(r45894);
        double r45896 = exp(r45895);
        double r45897 = r45871 ? r45890 : r45896;
        return r45897;
}

Error

Bits error versus x

Target

Original39.8
Target40.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00016704963527402913

    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. Applied associate-/l/0.1

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

    5. Simplified0.1

      \[\leadsto \frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\color{blue}{x \cdot \mathsf{fma}\left(1, 1 + e^{x}, e^{x + x}\right)}}\]
    6. Using strategy rm

    7. Applied flip3--0.1

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

    8. Applied associate-/l/0.1

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

    9. Simplified0.1

      \[\leadsto \frac{{\left({\left(e^{x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(1, 1 + e^{x}, e^{x + x}\right) \cdot \mathsf{fma}\left({1}^{3}, {\left(e^{x}\right)}^{3} + {1}^{3}, {\left(e^{x}\right)}^{6}\right)\right) \cdot x}}\]

    if -0.00016704963527402913 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{fma}\left(\frac{1}{6}, {x}^{2}, 1\right)\right)}\]
    4. Using strategy rm

    5. Applied add-exp-log0.5

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{1}{2}, x, \mathsf{fma}\left(\frac{1}{6}, {x}^{2}, 1\right)\right)\right)}}\]

    6. Simplified0.5

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6704963527402913 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left({\left(e^{x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(\mathsf{fma}\left(1, 1 + e^{x}, e^{x + x}\right) \cdot \mathsf{fma}\left({1}^{3}, {\left(e^{x}\right)}^{3} + {1}^{3}, {\left(e^{x}\right)}^{6}\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))