expax (section 3.5)

Average Error: 29.7 → 9.1

Time: 16.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.92826377784041773 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\frac{\sqrt[3]{{1}^{4} - {\left(e^{4}\right)}^{\left(a \cdot x\right)}}}{\sqrt[3]{1 \cdot 1 - e^{2 \cdot \left(a \cdot x\right)}}} \cdot \sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{\sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + 0.5 \cdot \left(a \cdot x\right)\right) + 0.16666666666666663 \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r60674 = a;
        double r60675 = x;
        double r60676 = r60674 * r60675;
        double r60677 = exp(r60676);
        double r60678 = 1.0;
        double r60679 = r60677 - r60678;
        return r60679;
}

↓

double f(double a, double x) {
        double r60680 = a;
        double r60681 = x;
        double r60682 = r60680 * r60681;
        double r60683 = -2.9282637778404177e-16;
        bool r60684 = r60682 <= r60683;
        double r60685 = 4.0;
        double r60686 = r60682 * r60685;
        double r60687 = exp(r60686);
        double r60688 = 1.0;
        double r60689 = pow(r60688, r60685);
        double r60690 = r60687 - r60689;
        double r60691 = exp(r60685);
        double r60692 = pow(r60691, r60682);
        double r60693 = r60689 - r60692;
        double r60694 = cbrt(r60693);
        double r60695 = r60688 * r60688;
        double r60696 = 2.0;
        double r60697 = r60696 * r60682;
        double r60698 = exp(r60697);
        double r60699 = r60695 - r60698;
        double r60700 = cbrt(r60699);
        double r60701 = r60694 / r60700;
        double r60702 = r60695 + r60698;
        double r60703 = cbrt(r60702);
        double r60704 = r60701 * r60703;
        double r60705 = r60690 / r60704;
        double r60706 = r60705 / r60703;
        double r60707 = exp(r60682);
        double r60708 = r60707 + r60688;
        double r60709 = r60706 / r60708;
        double r60710 = 0.5;
        double r60711 = r60710 * r60682;
        double r60712 = r60688 + r60711;
        double r60713 = r60682 * r60712;
        double r60714 = 0.16666666666666663;
        double r60715 = 3.0;
        double r60716 = pow(r60680, r60715);
        double r60717 = pow(r60681, r60715);
        double r60718 = r60716 * r60717;
        double r60719 = r60714 * r60718;
        double r60720 = r60713 + r60719;
        double r60721 = r60684 ? r60709 : r60720;
        return r60721;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.7
Target	0.2
Herbie	9.1

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

Derivation

1. Split input into 2 regimes

2. if (* a x) < -2.9282637778404177e-16

   1. Initial program 1.0

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

   3. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

   4. Simplified 1.0

      \[\leadsto \frac{\color{blue}{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{e^{a \cdot x} + 1}\]
   5. Using strategy rm

   6. Applied flip-- 1.0

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

   7. Simplified 1.0

      \[\leadsto \frac{\frac{\color{blue}{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}}{e^{2 \cdot \left(a \cdot x\right)} + 1 \cdot 1}}{e^{a \cdot x} + 1}\]

   8. Simplified 1.0

      \[\leadsto \frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\color{blue}{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{e^{a \cdot x} + 1}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 1.0

       \[\leadsto \frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\color{blue}{\left(\sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}} \cdot \sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}\right) \cdot \sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}}{e^{a \cdot x} + 1}\]

   11. Applied associate-/r* 1.0

       \[\leadsto \frac{\color{blue}{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}} \cdot \sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{\sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}}{e^{a \cdot x} + 1}\]
   12. Using strategy rm

   13. Applied flip-+ 1.0

       \[\leadsto \frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\sqrt[3]{\color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - e^{2 \cdot \left(a \cdot x\right)} \cdot e^{2 \cdot \left(a \cdot x\right)}}{1 \cdot 1 - e^{2 \cdot \left(a \cdot x\right)}}}} \cdot \sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{\sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{e^{a \cdot x} + 1}\]

   14. Applied cbrt-div 1.0

       \[\leadsto \frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\color{blue}{\frac{\sqrt[3]{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - e^{2 \cdot \left(a \cdot x\right)} \cdot e^{2 \cdot \left(a \cdot x\right)}}}{\sqrt[3]{1 \cdot 1 - e^{2 \cdot \left(a \cdot x\right)}}}} \cdot \sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{\sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{e^{a \cdot x} + 1}\]

   15. Simplified 0.9

       \[\leadsto \frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\frac{\color{blue}{\sqrt[3]{{1}^{4} - {\left(e^{4}\right)}^{\left(a \cdot x\right)}}}}{\sqrt[3]{1 \cdot 1 - e^{2 \cdot \left(a \cdot x\right)}}} \cdot \sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{\sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{e^{a \cdot x} + 1}\]

   if -2.9282637778404177e-16 < (* a x)

   1. Initial program 44.6

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

   3. Applied flip-- 44.6

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

   4. Simplified 44.6

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

   5. Taylor expanded around 0 13.3

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666663 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 1 \cdot \left(a \cdot x\right)\right)}\]

   6. Simplified 13.3

      \[\leadsto \color{blue}{\left(a \cdot x\right) \cdot \left(1 + 0.5 \cdot \left(a \cdot x\right)\right) + 0.16666666666666663 \cdot \left({a}^{3} \cdot {x}^{3}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.92826377784041773 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\frac{\sqrt[3]{{1}^{4} - {\left(e^{4}\right)}^{\left(a \cdot x\right)}}}{\sqrt[3]{1 \cdot 1 - e^{2 \cdot \left(a \cdot x\right)}}} \cdot \sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{\sqrt[3]{1 \cdot 1 + e^{2 \cdot \left(a \cdot x\right)}}}}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + 0.5 \cdot \left(a \cdot x\right)\right) + 0.16666666666666663 \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(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))