expax (section 3.5)

Average Error: 28.9 → 7.2

Time: 20.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.875657620123969931839940451833229073664 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, {\left(e^{2}\right)}^{\left(a \cdot x\right)}\right)}}{1 + e^{a \cdot x}}}\\ \mathbf{elif}\;a \cdot x \le 8.25458109042831203908792331233248718025 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{{\left(e^{a \cdot x} + 1\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\\ \end{array}\]

C

double f(double a, double x) {
        double r84856 = a;
        double r84857 = x;
        double r84858 = r84856 * r84857;
        double r84859 = exp(r84858);
        double r84860 = 1.0;
        double r84861 = r84859 - r84860;
        return r84861;
}

↓

double f(double a, double x) {
        double r84862 = a;
        double r84863 = x;
        double r84864 = r84862 * r84863;
        double r84865 = -6.87565762012397e-09;
        bool r84866 = r84864 <= r84865;
        double r84867 = exp(r84864);
        double r84868 = 1.0;
        double r84869 = r84867 - r84868;
        double r84870 = cbrt(r84869);
        double r84871 = r84870 * r84870;
        double r84872 = 4.0;
        double r84873 = r84864 * r84872;
        double r84874 = exp(r84873);
        double r84875 = pow(r84868, r84872);
        double r84876 = r84874 - r84875;
        double r84877 = 2.0;
        double r84878 = exp(r84877);
        double r84879 = pow(r84878, r84864);
        double r84880 = fma(r84868, r84868, r84879);
        double r84881 = r84876 / r84880;
        double r84882 = r84868 + r84867;
        double r84883 = r84881 / r84882;
        double r84884 = cbrt(r84883);
        double r84885 = r84871 * r84884;
        double r84886 = 8.254581090428312e-23;
        bool r84887 = r84864 <= r84886;
        double r84888 = pow(r84863, r84877);
        double r84889 = 0.16666666666666666;
        double r84890 = 3.0;
        double r84891 = pow(r84862, r84890);
        double r84892 = r84889 * r84891;
        double r84893 = 0.5;
        double r84894 = pow(r84862, r84877);
        double r84895 = r84893 * r84894;
        double r84896 = fma(r84892, r84863, r84895);
        double r84897 = fma(r84888, r84896, r84864);
        double r84898 = r84877 * r84864;
        double r84899 = exp(r84898);
        double r84900 = r84868 * r84868;
        double r84901 = r84899 - r84900;
        double r84902 = cbrt(r84901);
        double r84903 = r84867 + r84868;
        double r84904 = 0.3333333333333333;
        double r84905 = pow(r84903, r84904);
        double r84906 = r84902 / r84905;
        double r84907 = r84870 * r84906;
        double r84908 = r84901 / r84882;
        double r84909 = cbrt(r84908);
        double r84910 = r84907 * r84909;
        double r84911 = r84887 ? r84897 : r84910;
        double r84912 = r84866 ? r84885 : r84911;
        return r84912;
}

Error

Bits error versus a

Bits error versus x

Target

Original	28.9
Target	0.2
Herbie	7.2

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -6.87565762012397e-09

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt 0.3

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

   5. Applied flip-- 0.3

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

   6. Simplified 0.3

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

   7. Simplified 0.3

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

   9. Applied flip-- 0.3

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

   10. Simplified 0.3

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

   11. Simplified 0.3

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

   if -6.87565762012397e-09 < (* a x) < 8.254581090428312e-23

   1. Initial program 44.5

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

   2. Taylor expanded around 0 13.0

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

   3. Simplified 9.9

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

   if 8.254581090428312e-23 < (* a x)

   1. Initial program 31.6

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

   3. Applied add-cube-cbrt 31.6

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

   5. Applied flip-- 32.5

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

   6. Simplified 32.2

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

   7. Simplified 32.2

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

   9. Applied flip-- 32.3

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

   10. Applied cbrt-div 32.3

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

   11. Simplified 31.7

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

   13. Applied pow1/3 31.7

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.875657620123969931839940451833229073664 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{\frac{\frac{e^{\left(a \cdot x\right) \cdot 4} - {1}^{4}}{\mathsf{fma}\left(1, 1, {\left(e^{2}\right)}^{\left(a \cdot x\right)}\right)}}{1 + e^{a \cdot x}}}\\ \mathbf{elif}\;a \cdot x \le 8.25458109042831203908792331233248718025 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{6} \cdot {a}^{3}, x, \frac{1}{2} \cdot {a}^{2}\right), a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{{\left(e^{a \cdot x} + 1\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{1 + e^{a \cdot x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))