Average Error: 39.6 → 0.0
Time: 9.5s
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 r860568 = x;
        double r860569 = exp(r860568);
        double r860570 = 1.0;
        double r860571 = r860569 - r860570;
        double r860572 = r860571 / r860568;
        return r860572;
}
↓
double f(double x) {
        double r860573 = x;
        double r860574 = expm1(r860573);
        double r860575 = r860574 / r860573;
        return r860575;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.6
Target38.8
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 39.6

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

  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 2019155 +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))