Average Error: 40.6 → 0.0
Time: 33.1s
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 r2094043 = x;
        double r2094044 = exp(r2094043);
        double r2094045 = 1.0;
        double r2094046 = r2094044 - r2094045;
        double r2094047 = r2094046 / r2094043;
        return r2094047;
}
↓
double f(double x) {
        double r2094048 = x;
        double r2094049 = expm1(r2094048);
        double r2094050 = r2094049 / r2094048;
        return r2094050;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.6
Target39.8
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 40.6

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019142 +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))