Average Error: 40.3 → 0.0
Time: 7.4s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}\]
\frac{e^{x} - 1}{x}
↓
\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}
double f(double x) {
        double r1169152 = x;
        double r1169153 = exp(r1169152);
        double r1169154 = 1.0;
        double r1169155 = r1169153 - r1169154;
        double r1169156 = r1169155 / r1169152;
        return r1169156;
}
↓
double f(double x) {
        double r1169157 = 1.0;
        double r1169158 = x;
        double r1169159 = expm1(r1169158);
        double r1169160 = r1169158 / r1169159;
        double r1169161 = r1169157 / r1169160;
        return r1169161;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
Target39.5
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 40.3

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

  2. Simplified0.0

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

  4. Applied clear-num0.0

    \[\leadsto \color{blue}{\frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}}\]

  5. Final simplification0.0

    \[\leadsto \frac{1}{\frac{x}{\mathsf{expm1}\left(x\right)}}\]

Reproduce

herbie shell --seed 2019156 +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))