Average Error: 29.4 → 0.0
Time: 16.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[(e^{a \cdot x} - 1)^*\]
e^{a \cdot x} - 1
↓
(e^{a \cdot x} - 1)^*
double f(double a, double x) {
        double r8091349 = a;
        double r8091350 = x;
        double r8091351 = r8091349 * r8091350;
        double r8091352 = exp(r8091351);
        double r8091353 = 1.0;
        double r8091354 = r8091352 - r8091353;
        return r8091354;
}
↓
double f(double a, double x) {
        double r8091355 = a;
        double r8091356 = x;
        double r8091357 = r8091355 * r8091356;
        double r8091358 = expm1(r8091357);
        return r8091358;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
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 29.4

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

  2. Simplified0.0

    \[\leadsto \color{blue}{(e^{a \cdot x} - 1)^*}\]

  3. Final simplification0.0

    \[\leadsto (e^{a \cdot x} - 1)^*\]

Reproduce

herbie shell --seed 2019107 +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))