Average Error: 40.4 → 0.2
Time: 13.2s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)\]
\frac{e^{x} - 1}{x}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)
double f(double x) {
        double r3385625 = x;
        double r3385626 = exp(r3385625);
        double r3385627 = 1.0;
        double r3385628 = r3385626 - r3385627;
        double r3385629 = r3385628 / r3385625;
        return r3385629;
}
↓
double f(double x) {
        double r3385630 = x;
        double r3385631 = expm1(r3385630);
        double r3385632 = r3385631 / r3385630;
        double r3385633 = expm1(r3385632);
        double r3385634 = log1p(r3385633);
        return r3385634;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.4
Target39.6
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 40.4

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x\right)}{x}}\]
  3. Using strategy rm

  4. Applied log1p-expm1-u0.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)}\]

  5. Final simplification0.2

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{expm1}\left(x\right)}{x}\right)\right)\]

Reproduce

herbie shell --seed 2019134 +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))