Average Error: 29.3 → 0.0
Time: 24.5s
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 r3717224 = a;
        double r3717225 = x;
        double r3717226 = r3717224 * r3717225;
        double r3717227 = exp(r3717226);
        double r3717228 = 1.0;
        double r3717229 = r3717227 - r3717228;
        return r3717229;
}
↓
double f(double a, double x) {
        double r3717230 = a;
        double r3717231 = x;
        double r3717232 = r3717230 * r3717231;
        double r3717233 = expm1(r3717232);
        return r3717233;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
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 29.3

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

  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 2019144 +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))