expax (section 3.5)

Average Error: 29.8 → 0.4

Time: 16.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01289790256637027461572575504078486119397:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(9 \cdot a\right) \cdot x + \left(\left(9 \cdot a\right) \cdot x + \left(9 \cdot a\right) \cdot x\right)} - \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right)}{\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot e^{\left(9 \cdot a\right) \cdot x} + e^{\left(9 \cdot a\right) \cdot x + \left(9 \cdot a\right) \cdot x}\right)}}{\left(e^{3 \cdot \left(a \cdot x\right)} \cdot \left(1 \cdot \left(1 \cdot 1\right)\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right)\right) + e^{3 \cdot \left(a \cdot x\right)} \cdot e^{3 \cdot \left(a \cdot x\right)}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{6}\right) + a\right) \cdot x + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r4850658 = a;
        double r4850659 = x;
        double r4850660 = r4850658 * r4850659;
        double r4850661 = exp(r4850660);
        double r4850662 = 1.0;
        double r4850663 = r4850661 - r4850662;
        return r4850663;
}

↓

double f(double a, double x) {
        double r4850664 = a;
        double r4850665 = x;
        double r4850666 = r4850664 * r4850665;
        double r4850667 = -0.012897902566370275;
        bool r4850668 = r4850666 <= r4850667;
        double r4850669 = 9.0;
        double r4850670 = r4850669 * r4850664;
        double r4850671 = r4850670 * r4850665;
        double r4850672 = r4850671 + r4850671;
        double r4850673 = r4850671 + r4850672;
        double r4850674 = exp(r4850673);
        double r4850675 = 1.0;
        double r4850676 = r4850675 * r4850675;
        double r4850677 = r4850676 * r4850676;
        double r4850678 = r4850677 * r4850677;
        double r4850679 = r4850678 * r4850675;
        double r4850680 = r4850679 * r4850679;
        double r4850681 = r4850679 * r4850680;
        double r4850682 = r4850674 - r4850681;
        double r4850683 = exp(r4850671);
        double r4850684 = r4850679 * r4850683;
        double r4850685 = exp(r4850672);
        double r4850686 = r4850684 + r4850685;
        double r4850687 = r4850680 + r4850686;
        double r4850688 = r4850682 / r4850687;
        double r4850689 = 3.0;
        double r4850690 = r4850689 * r4850666;
        double r4850691 = exp(r4850690);
        double r4850692 = r4850675 * r4850676;
        double r4850693 = r4850691 * r4850692;
        double r4850694 = r4850692 * r4850692;
        double r4850695 = r4850693 + r4850694;
        double r4850696 = r4850691 * r4850691;
        double r4850697 = r4850695 + r4850696;
        double r4850698 = r4850688 / r4850697;
        double r4850699 = exp(r4850666);
        double r4850700 = r4850699 + r4850675;
        double r4850701 = r4850699 * r4850700;
        double r4850702 = r4850701 + r4850676;
        double r4850703 = r4850698 / r4850702;
        double r4850704 = r4850666 * r4850666;
        double r4850705 = 0.16666666666666666;
        double r4850706 = r4850664 * r4850705;
        double r4850707 = r4850704 * r4850706;
        double r4850708 = r4850707 + r4850664;
        double r4850709 = r4850708 * r4850665;
        double r4850710 = 0.5;
        double r4850711 = r4850710 * r4850704;
        double r4850712 = r4850709 + r4850711;
        double r4850713 = r4850668 ? r4850703 : r4850712;
        return r4850713;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.8
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) < -0.012897902566370275

   1. Initial program 0.0

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

   3. Applied flip3-- 0.0

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

   4. Simplified 0.0

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

   5. Simplified 0.0

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

   7. Applied flip3-- 0.0

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

   8. Simplified 0.0

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

   10. Applied flip3-- 0.0

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

   11. Simplified 0.0

       \[\leadsto \frac{\frac{\frac{\color{blue}{e^{x \cdot \left(a \cdot 9\right) + \left(x \cdot \left(a \cdot 9\right) + x \cdot \left(a \cdot 9\right)\right)} - \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right)}}{e^{\left(a \cdot x\right) \cdot \left(6 + 3\right)} \cdot e^{\left(a \cdot x\right) \cdot \left(6 + 3\right)} + \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) + e^{\left(a \cdot x\right) \cdot \left(6 + 3\right)} \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right)\right)}}{e^{3 \cdot \left(x \cdot a\right)} \cdot e^{3 \cdot \left(x \cdot a\right)} + \left(\left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right) + e^{3 \cdot \left(x \cdot a\right)} \cdot \left(\left(1 \cdot 1\right) \cdot 1\right)\right)}}{1 \cdot 1 + e^{x \cdot a} \cdot \left(e^{x \cdot a} + 1\right)}\]

   12. Simplified 0.0

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

   if -0.012897902566370275 < (* a x)

   1. Initial program 44.7

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

   2. Taylor expanded around 0 15.5

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

   3. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.01289790256637027461572575504078486119397:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(9 \cdot a\right) \cdot x + \left(\left(9 \cdot a\right) \cdot x + \left(9 \cdot a\right) \cdot x\right)} - \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right)\right)}{\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot \left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) + \left(\left(\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1\right) \cdot e^{\left(9 \cdot a\right) \cdot x} + e^{\left(9 \cdot a\right) \cdot x + \left(9 \cdot a\right) \cdot x}\right)}}{\left(e^{3 \cdot \left(a \cdot x\right)} \cdot \left(1 \cdot \left(1 \cdot 1\right)\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right)\right) + e^{3 \cdot \left(a \cdot x\right)} \cdot e^{3 \cdot \left(a \cdot x\right)}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{6}\right) + a\right) \cdot x + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))