Average Error: 29.9 → 0.0
Time: 19.1s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\mathsf{expm1}\left(a \cdot x\right)\]
e^{a \cdot x} - 1
↓
\mathsf{expm1}\left(a \cdot x\right)
double f(double a, double x) {
        double r5256120 = a;
        double r5256121 = x;
        double r5256122 = r5256120 * r5256121;
        double r5256123 = exp(r5256122);
        double r5256124 = 1.0;
        double r5256125 = r5256123 - r5256124;
        return r5256125;
}
↓
double f(double a, double x) {
        double r5256126 = a;
        double r5256127 = x;
        double r5256128 = r5256126 * r5256127;
        double r5256129 = expm1(r5256128);
        return r5256129;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Initial program 29.9

    \[e^{a \cdot x} - 1\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{expm1}\left(a \cdot x\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))