Average Error: 39.5 → 0.0
Time: 14.9s
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 r1584243 = x;
        double r1584244 = exp(r1584243);
        double r1584245 = 1.0;
        double r1584246 = r1584244 - r1584245;
        double r1584247 = r1584246 / r1584243;
        return r1584247;
}
↓
double f(double x) {
        double r1584248 = 1.0;
        double r1584249 = x;
        double r1584250 = expm1(r1584249);
        double r1584251 = r1584249 / r1584250;
        double r1584252 = r1584248 / r1584251;
        return r1584252;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.5
Target38.6
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 39.5

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

  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 2019135 +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))