expax (section 3.5)

Average Error: 29.4 → 0.3

Time: 10.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0001742103476160202:\\ \;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\\ \end{array}\]

C

double f(double a, double x) {
        double r1573897 = a;
        double r1573898 = x;
        double r1573899 = r1573897 * r1573898;
        double r1573900 = exp(r1573899);
        double r1573901 = 1.0;
        double r1573902 = r1573900 - r1573901;
        return r1573902;
}

↓

double f(double a, double x) {
        double r1573903 = a;
        double r1573904 = x;
        double r1573905 = r1573903 * r1573904;
        double r1573906 = -0.0001742103476160202;
        bool r1573907 = r1573905 <= r1573906;
        double r1573908 = exp(r1573905);
        double r1573909 = 1.0;
        double r1573910 = r1573908 - r1573909;
        double r1573911 = r1573910 * r1573910;
        double r1573912 = r1573910 * r1573911;
        double r1573913 = cbrt(r1573912);
        double r1573914 = r1573905 * r1573905;
        double r1573915 = r1573905 * r1573914;
        double r1573916 = 0.16666666666666666;
        double r1573917 = r1573915 * r1573916;
        double r1573918 = 0.5;
        double r1573919 = r1573918 * r1573905;
        double r1573920 = r1573905 * r1573919;
        double r1573921 = r1573917 + r1573920;
        double r1573922 = r1573921 + r1573905;
        double r1573923 = r1573907 ? r1573913 : r1573922;
        return r1573923;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.4
Target	0.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.0001742103476160202

   1. Initial program 0.0

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

   3. Applied add-cbrt-cube 0.0

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

   if -0.0001742103476160202 < (* a x)

   1. Initial program 44.1

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

   2. Taylor expanded around 0 14.2

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

   3. Simplified 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.0001742103476160202:\\ \;\;\;\;\sqrt[3]{\left(e^{a \cdot x} - 1\right) \cdot \left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) \cdot \frac{1}{6} + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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))