Average Error: 40.1 → 0.0
Time: 13.0s
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 r9822407 = x;
        double r9822408 = exp(r9822407);
        double r9822409 = 1.0;
        double r9822410 = r9822408 - r9822409;
        double r9822411 = r9822410 / r9822407;
        return r9822411;
}
↓
double f(double x) {
        double r9822412 = x;
        double r9822413 = expm1(r9822412);
        double r9822414 = r9822413 / r9822412;
        return r9822414;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target39.3
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 40.1

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

  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 2019128 +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))