expax (section 3.5)

Average Error: 29.7 → 0.4

Time: 22.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.574663701892631815539014272786744186305 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{e^{3 \cdot \left(a \cdot x\right)}} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \frac{1}{6}\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r11859448 = a;
        double r11859449 = x;
        double r11859450 = r11859448 * r11859449;
        double r11859451 = exp(r11859450);
        double r11859452 = 1.0;
        double r11859453 = r11859451 - r11859452;
        return r11859453;
}

↓

double f(double a, double x) {
        double r11859454 = a;
        double r11859455 = x;
        double r11859456 = r11859454 * r11859455;
        double r11859457 = -0.00015746637018926318;
        bool r11859458 = r11859456 <= r11859457;
        double r11859459 = 3.0;
        double r11859460 = r11859459 * r11859456;
        double r11859461 = exp(r11859460);
        double r11859462 = cbrt(r11859461);
        double r11859463 = 1.0;
        double r11859464 = r11859462 - r11859463;
        double r11859465 = r11859456 * r11859456;
        double r11859466 = 0.5;
        double r11859467 = 0.16666666666666666;
        double r11859468 = r11859456 * r11859467;
        double r11859469 = r11859466 + r11859468;
        double r11859470 = r11859465 * r11859469;
        double r11859471 = r11859456 + r11859470;
        double r11859472 = r11859458 ? r11859464 : r11859471;
        return r11859472;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.7
Target	0.1
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.00015746637018926318

   1. Initial program 0.0

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

   3. Applied add-cbrt-cube 0.0

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

   4. Simplified 0.0

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

   if -0.00015746637018926318 < (* a x)

   1. Initial program 44.0

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

   2. Taylor expanded around 0 15.1

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

   3. Simplified 0.6

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

   4. Taylor expanded around inf 15.1

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

   5. Simplified 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.574663701892631815539014272786744186305 \cdot 10^{-4}:\\ \;\;\;\;\sqrt[3]{e^{3 \cdot \left(a \cdot x\right)}} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \frac{1}{6}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(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))