expax (section 3.5)

Average Error: 29.3 → 9.1

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.33595634735350902 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{\left(\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}\right) \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

C

double f(double a, double x) {
        double r106997 = a;
        double r106998 = x;
        double r106999 = r106997 * r106998;
        double r107000 = exp(r106999);
        double r107001 = 1.0;
        double r107002 = r107000 - r107001;
        return r107002;
}

↓

double f(double a, double x) {
        double r107003 = a;
        double r107004 = x;
        double r107005 = r107003 * r107004;
        double r107006 = -6.335956347353509e-09;
        bool r107007 = r107005 <= r107006;
        double r107008 = exp(r107005);
        double r107009 = 1.0;
        double r107010 = r107008 - r107009;
        double r107011 = cbrt(r107010);
        double r107012 = r107008 * r107008;
        double r107013 = r107009 * r107009;
        double r107014 = r107012 - r107013;
        double r107015 = cbrt(r107014);
        double r107016 = r107015 * r107015;
        double r107017 = r107016 * r107015;
        double r107018 = cbrt(r107017);
        double r107019 = r107018 * r107015;
        double r107020 = r107008 + r107009;
        double r107021 = cbrt(r107020);
        double r107022 = r107021 * r107021;
        double r107023 = r107019 / r107022;
        double r107024 = r107011 * r107023;
        double r107025 = 0.5;
        double r107026 = 2.0;
        double r107027 = pow(r107003, r107026);
        double r107028 = pow(r107004, r107026);
        double r107029 = r107027 * r107028;
        double r107030 = 0.16666666666666666;
        double r107031 = 3.0;
        double r107032 = pow(r107003, r107031);
        double r107033 = pow(r107004, r107031);
        double r107034 = r107032 * r107033;
        double r107035 = fma(r107030, r107034, r107005);
        double r107036 = fma(r107025, r107029, r107035);
        double r107037 = r107007 ? r107024 : r107036;
        return r107037;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
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) < -6.335956347353509e-09

   1. Initial program 0.3

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

   3. Applied add-log-exp 0.3

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

   4. Applied add-log-exp 0.3

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

   5. Applied diff-log 0.3

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

   6. Simplified 0.3

      \[\leadsto \log \color{blue}{\left(e^{e^{a \cdot x} - 1}\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 0.3

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

   9. Applied exp-prod 0.3

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

   10. Applied log-pow 0.3

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

   11. Simplified 0.3

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

   13. Applied flip-- 0.3

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

   14. Applied cbrt-div 0.3

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

   15. Applied flip-- 0.3

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

   16. Applied cbrt-div 0.3

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

   17. Applied frac-times 0.3

       \[\leadsto \sqrt[3]{e^{a \cdot x} - 1} \cdot \color{blue}{\frac{\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}}}\]
   18. Using strategy rm

   19. Applied add-cube-cbrt 0.3

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

   if -6.335956347353509e-09 < (* a x)

   1. Initial program 44.6

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

   2. Taylor expanded around 0 13.7

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

   3. Simplified 13.7

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.33595634735350902 \cdot 10^{-9}:\\ \;\;\;\;\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{\left(\sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}\right) \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}} \cdot \sqrt[3]{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}}{\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))