expax (section 3.5)

Average Error: 29.3 → 0.6

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.090734393683841924:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, a \cdot x\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r105555 = a;
        double r105556 = x;
        double r105557 = r105555 * r105556;
        double r105558 = exp(r105557);
        double r105559 = 1.0;
        double r105560 = r105558 - r105559;
        return r105560;
}

↓

double f(double a, double x) {
        double r105561 = a;
        double r105562 = x;
        double r105563 = r105561 * r105562;
        double r105564 = -0.09073439368384192;
        bool r105565 = r105563 <= r105564;
        double r105566 = exp(r105563);
        double r105567 = 1.0;
        double r105568 = r105566 - r105567;
        double r105569 = cbrt(r105568);
        double r105570 = r105569 * r105569;
        double r105571 = r105570 * r105569;
        double r105572 = 0.5;
        double r105573 = r105562 * r105561;
        double r105574 = fabs(r105573);
        double r105575 = r105574 * r105574;
        double r105576 = fma(r105572, r105575, r105563);
        double r105577 = r105565 ? r105571 : r105576;
        return r105577;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.09073439368384192

   1. Initial program 0.0

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

   3. Applied add-cube-cbrt 0.0

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

   if -0.09073439368384192 < (* a x)

   1. Initial program 44.1

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

   2. Taylor expanded around 0 14.5

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

   3. Simplified 14.5

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

   4. Taylor expanded around 0 8.6

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

   5. Simplified 8.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, a \cdot x\right)}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 8.6

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{{a}^{2} \cdot {x}^{2}} \cdot \sqrt{{a}^{2} \cdot {x}^{2}}}, a \cdot x\right)\]

   8. Simplified 8.6

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left|x \cdot a\right|} \cdot \sqrt{{a}^{2} \cdot {x}^{2}}, a \cdot x\right)\]

   9. Simplified 0.8

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.090734393683841924:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))