Average Error: 39.8 → 0.0
Time: 16.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\frac{\mathsf{expm1}\left(x\right)}{x}\]
\frac{e^{x} - 1}{x}
↓
\frac{\mathsf{expm1}\left(x\right)}{x}
double f(double x) {
        double r1933928 = x;
        double r1933929 = exp(r1933928);
        double r1933930 = 1.0;
        double r1933931 = r1933929 - r1933930;
        double r1933932 = r1933931 / r1933928;
        return r1933932;
}
↓
double f(double x) {
        double r1933933 = x;
        double r1933934 = expm1(r1933933);
        double r1933935 = r1933934 / r1933933;
        return r1933935;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target38.9
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 39.8

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}}\]

  3. Taylor expanded around inf 39.8

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

  4. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}}\]

  5. Final simplification0.0

    \[\leadsto \frac{\mathsf{expm1}\left(x\right)}{x}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))