Average Error: 29.9 → 0.0
Time: 18.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[(e^{a \cdot x} - 1)^*\]
double f(double a, double x) {
        double r7262297 = a;
        double r7262298 = x;
        double r7262299 = r7262297 * r7262298;
        double r7262300 = exp(r7262299);
        double r7262301 = 1.0;
        double r7262302 = r7262300 - r7262301;
        return r7262302;
}
↓
double f(double a, double x) {
        double r7262303 = a;
        double r7262304 = x;
        double r7262305 = r7262303 * r7262304;
        double r7262306 = expm1(r7262305);
        return r7262306;
}


e^{a \cdot x} - 1
↓
(e^{a \cdot x} - 1)^*

Error

Bits error versus a

Bits error versus x

Target

Original29.9
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 29.9

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

  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 2019101 +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))