expax (section 3.5)

Average Error: 29.3 → 0.2

Time: 35.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.997847041171635 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{e^{\left(a \cdot x\right) \cdot 9} + -1}{1 - \left(-1 - e^{3 \cdot \left(a \cdot x\right)}\right) \cdot e^{3 \cdot \left(a \cdot x\right)}}}{e^{a \cdot x} + \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6}\right) + a \cdot x\\ \end{array}\]

C

double f(double a, double x) {
        double r3843691 = a;
        double r3843692 = x;
        double r3843693 = r3843691 * r3843692;
        double r3843694 = exp(r3843693);
        double r3843695 = 1.0;
        double r3843696 = r3843694 - r3843695;
        return r3843696;
}

↓

double f(double a, double x) {
        double r3843697 = a;
        double r3843698 = x;
        double r3843699 = r3843697 * r3843698;
        double r3843700 = -8.997847041171635e-05;
        bool r3843701 = r3843699 <= r3843700;
        double r3843702 = 9.0;
        double r3843703 = r3843699 * r3843702;
        double r3843704 = exp(r3843703);
        double r3843705 = -1.0;
        double r3843706 = r3843704 + r3843705;
        double r3843707 = 1.0;
        double r3843708 = 3.0;
        double r3843709 = r3843708 * r3843699;
        double r3843710 = exp(r3843709);
        double r3843711 = r3843705 - r3843710;
        double r3843712 = r3843711 * r3843710;
        double r3843713 = r3843707 - r3843712;
        double r3843714 = r3843706 / r3843713;
        double r3843715 = exp(r3843699);
        double r3843716 = r3843715 * r3843715;
        double r3843717 = r3843716 + r3843707;
        double r3843718 = r3843715 + r3843717;
        double r3843719 = r3843714 / r3843718;
        double r3843720 = 0.5;
        double r3843721 = r3843699 * r3843720;
        double r3843722 = r3843699 * r3843721;
        double r3843723 = r3843699 * r3843699;
        double r3843724 = r3843723 * r3843699;
        double r3843725 = 0.16666666666666666;
        double r3843726 = r3843724 * r3843725;
        double r3843727 = r3843722 + r3843726;
        double r3843728 = r3843727 + r3843699;
        double r3843729 = r3843701 ? r3843719 : r3843728;
        return r3843729;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -8.997847041171635e-05

   1. Initial program 0.0

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

   3. Applied flip3-- 0.1

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

   4. Simplified 0.0

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

   5. Simplified 0.0

      \[\leadsto \frac{e^{\left(a \cdot x\right) \cdot 3} + -1}{\color{blue}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right) + e^{a \cdot x}}}\]
   6. Using strategy rm

   7. Applied flip3-+ 0.0

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

   8. Simplified 0.0

      \[\leadsto \frac{\frac{\color{blue}{e^{\left(a \cdot x\right) \cdot 9} + -1}}{e^{\left(a \cdot x\right) \cdot 3} \cdot e^{\left(a \cdot x\right) \cdot 3} + \left(-1 \cdot -1 - e^{\left(a \cdot x\right) \cdot 3} \cdot -1\right)}}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right) + e^{a \cdot x}}\]

   9. Simplified 0.0

      \[\leadsto \frac{\frac{e^{\left(a \cdot x\right) \cdot 9} + -1}{\color{blue}{1 - e^{\left(a \cdot x\right) \cdot 3} \cdot \left(-1 - e^{\left(a \cdot x\right) \cdot 3}\right)}}}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right) + e^{a \cdot x}}\]

   if -8.997847041171635e-05 < (* 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

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.997847041171635 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{e^{\left(a \cdot x\right) \cdot 9} + -1}{1 - \left(-1 - e^{3 \cdot \left(a \cdot x\right)}\right) \cdot e^{3 \cdot \left(a \cdot x\right)}}}{e^{a \cdot x} + \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6}\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :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))