Average Error: 29.2 → 0.0
Time: 20.3s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\mathsf{expm1}\left(\left(a \cdot x\right)\right)\]
e^{a \cdot x} - 1
↓
\mathsf{expm1}\left(\left(a \cdot x\right)\right)
double f(double a, double x) {
        double r10049934 = a;
        double r10049935 = x;
        double r10049936 = r10049934 * r10049935;
        double r10049937 = exp(r10049936);
        double r10049938 = 1.0;
        double r10049939 = r10049937 - r10049938;
        return r10049939;
}
↓
double f(double a, double x) {
        double r10049940 = a;
        double r10049941 = x;
        double r10049942 = r10049940 * r10049941;
        double r10049943 = expm1(r10049942);
        return r10049943;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.2
Target0.2
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 29.2

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"

  :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))