expax (section 3.5)

Average Error: 29.6 → 0.7

Time: 55.0s

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 r6362485 = a;
        double r6362486 = x;
        double r6362487 = r6362485 * r6362486;
        double r6362488 = exp(r6362487);
        double r6362489 = 1.0;
        double r6362490 = r6362488 - r6362489;
        return r6362490;
}

↓

double f(double a, double x) {
        double r6362491 = x;
        double r6362492 = a;
        double r6362493 = r6362491 * r6362492;
        double r6362494 = -649331.2251280493;
        bool r6362495 = r6362493 <= r6362494;
        double r6362496 = exp(r6362493);
        double r6362497 = 1.0;
        double r6362498 = r6362496 - r6362497;
        double r6362499 = cbrt(r6362498);
        double r6362500 = r6362499 * r6362499;
        double r6362501 = r6362500 * r6362499;
        double r6362502 = r6362493 * r6362493;
        double r6362503 = r6362493 * r6362502;
        double r6362504 = 0.16666666666666666;
        double r6362505 = 0.5;
        double r6362506 = fma(r6362502, r6362505, r6362493);
        double r6362507 = fma(r6362503, r6362504, r6362506);
        double r6362508 = r6362495 ? r6362501 : r6362507;
        return r6362508;
}

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))