expax (section 3.5)

Average Error: 29.6 → 0.7

Time: 51.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot a \le -649331.22512804926373064517974853515625:\\ \;\;\;\;\left(\sqrt[3]{e^{x \cdot a} - 1} \cdot \sqrt[3]{e^{x \cdot a} - 1}\right) \cdot \sqrt[3]{e^{x \cdot a} - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot a\right) \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right), \frac{1}{6}, \mathsf{fma}\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right), \frac{1}{2}, x \cdot a\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4226749 = a;
        double r4226750 = x;
        double r4226751 = r4226749 * r4226750;
        double r4226752 = exp(r4226751);
        double r4226753 = 1.0;
        double r4226754 = r4226752 - r4226753;
        return r4226754;
}

↓

double f(double a, double x) {
        double r4226755 = x;
        double r4226756 = a;
        double r4226757 = r4226755 * r4226756;
        double r4226758 = -649331.2251280493;
        bool r4226759 = r4226757 <= r4226758;
        double r4226760 = exp(r4226757);
        double r4226761 = 1.0;
        double r4226762 = r4226760 - r4226761;
        double r4226763 = cbrt(r4226762);
        double r4226764 = r4226763 * r4226763;
        double r4226765 = r4226764 * r4226763;
        double r4226766 = r4226757 * r4226757;
        double r4226767 = r4226757 * r4226766;
        double r4226768 = 0.16666666666666666;
        double r4226769 = 0.5;
        double r4226770 = fma(r4226766, r4226769, r4226757);
        double r4226771 = fma(r4226767, r4226768, r4226770);
        double r4226772 = r4226759 ? r4226765 : r4226771;
        return r4226772;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.6
Target	0.2
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -649331.2251280493

   1. Initial program 0

      \[e^{a \cdot x} - 1\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}}\]

   if -649331.2251280493 < (* a x)

   1. Initial program 43.8

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

   2. Taylor expanded around 0 14.9

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

   3. Simplified 1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right), \frac{1}{6}, \mathsf{fma}\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right), \frac{1}{2}, x \cdot a\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot a \le -649331.22512804926373064517974853515625:\\ \;\;\;\;\left(\sqrt[3]{e^{x \cdot a} - 1} \cdot \sqrt[3]{e^{x \cdot a} - 1}\right) \cdot \sqrt[3]{e^{x \cdot a} - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot a\right) \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right), \frac{1}{6}, \mathsf{fma}\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right), \frac{1}{2}, x \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))