Average Error: 40.3 → 0.3
Time: 8.6s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.681655298641986193575875718764223165635 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{{\left(e^{x}\right)}^{2} - 1 \cdot 1}\right)}{x \cdot \mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.681655298641986193575875718764223165635 \cdot 10^{-4}:\\
\;\;\;\;\frac{\log \left(e^{{\left(e^{x}\right)}^{2} - 1 \cdot 1}\right)}{x \cdot \mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, 1\right)}\\

\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 r64946 = x;
        double r64947 = exp(r64946);
        double r64948 = 1.0;
        double r64949 = r64947 - r64948;
        double r64950 = r64949 / r64946;
        return r64950;
}
↓
double f(double x) {
        double r64951 = x;
        double r64952 = -0.00016816552986419862;
        bool r64953 = r64951 <= r64952;
        double r64954 = exp(r64951);
        double r64955 = 2.0;
        double r64956 = pow(r64954, r64955);
        double r64957 = 1.0;
        double r64958 = r64957 * r64957;
        double r64959 = r64956 - r64958;
        double r64960 = exp(r64959);
        double r64961 = log(r64960);
        double r64962 = cbrt(r64954);
        double r64963 = r64962 * r64962;
        double r64964 = fma(r64963, r64962, r64957);
        double r64965 = r64951 * r64964;
        double r64966 = r64961 / r64965;
        double r64967 = 0.16666666666666666;
        double r64968 = 0.5;
        double r64969 = fma(r64967, r64951, r64968);
        double r64970 = 1.0;
        double r64971 = fma(r64951, r64969, r64970);
        double r64972 = r64953 ? r64966 : r64971;
        return r64972;
}

Error

Bits error versus x

Target

Original40.3
Target40.8
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 < -0.00016816552986419862

    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. Applied associate-/l/0.1

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

    6. Applied add-log-exp0.1

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

    7. Applied add-log-exp0.1

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

    8. Applied diff-log0.1

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

    9. Simplified0.1

      \[\leadsto \frac{\log \color{blue}{\left(e^{{\left(e^{x}\right)}^{2} - 1 \cdot 1}\right)}}{x \cdot \left(e^{x} + 1\right)}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt0.1

      \[\leadsto \frac{\log \left(e^{{\left(e^{x}\right)}^{2} - 1 \cdot 1}\right)}{x \cdot \left(\color{blue}{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}} + 1\right)}\]

    12. Applied fma-def0.1

      \[\leadsto \frac{\log \left(e^{{\left(e^{x}\right)}^{2} - 1 \cdot 1}\right)}{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, 1\right)}}\]

    if -0.00016816552986419862 < x

    1. Initial program 60.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.681655298641986193575875718764223165635 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{{\left(e^{x}\right)}^{2} - 1 \cdot 1}\right)}{x \cdot \mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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))