expax (section 3.5)

Average Error: 29.0 → 8.7

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.97221861134718174 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}} \cdot \sqrt[3]{\sqrt[3]{{\left(e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}\right)}^{3}}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}} \cdot \frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\\ \mathbf{elif}\;a \cdot x \le 6.8409562196091337 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}} \cdot \sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}} \cdot \frac{\sqrt[3]{\left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}} - {\left(\sqrt{1}\right)}^{3}\right)}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\\ \end{array}\]

C

double f(double a, double x) {
        double r108857 = a;
        double r108858 = x;
        double r108859 = r108857 * r108858;
        double r108860 = exp(r108859);
        double r108861 = 1.0;
        double r108862 = r108860 - r108861;
        return r108862;
}

↓

double f(double a, double x) {
        double r108863 = a;
        double r108864 = x;
        double r108865 = r108863 * r108864;
        double r108866 = -1.9722186113471817e-05;
        bool r108867 = r108865 <= r108866;
        double r108868 = 3.0;
        double r108869 = r108865 * r108868;
        double r108870 = exp(r108869);
        double r108871 = 1.0;
        double r108872 = pow(r108871, r108868);
        double r108873 = r108870 - r108872;
        double r108874 = cbrt(r108873);
        double r108875 = pow(r108873, r108868);
        double r108876 = cbrt(r108875);
        double r108877 = cbrt(r108876);
        double r108878 = r108874 * r108877;
        double r108879 = exp(r108865);
        double r108880 = r108879 + r108871;
        double r108881 = r108879 * r108880;
        double r108882 = r108871 * r108871;
        double r108883 = r108881 + r108882;
        double r108884 = cbrt(r108883);
        double r108885 = r108884 * r108884;
        double r108886 = r108878 / r108885;
        double r108887 = r108874 / r108884;
        double r108888 = r108886 * r108887;
        double r108889 = 6.840956219609134e-13;
        bool r108890 = r108865 <= r108889;
        double r108891 = 0.5;
        double r108892 = 2.0;
        double r108893 = pow(r108863, r108892);
        double r108894 = r108891 * r108893;
        double r108895 = r108894 * r108864;
        double r108896 = r108863 + r108895;
        double r108897 = r108864 * r108896;
        double r108898 = 0.16666666666666666;
        double r108899 = pow(r108863, r108868);
        double r108900 = pow(r108864, r108868);
        double r108901 = r108899 * r108900;
        double r108902 = r108898 * r108901;
        double r108903 = r108897 + r108902;
        double r108904 = r108874 * r108874;
        double r108905 = r108904 / r108885;
        double r108906 = sqrt(r108870);
        double r108907 = sqrt(r108871);
        double r108908 = pow(r108907, r108868);
        double r108909 = r108906 + r108908;
        double r108910 = r108906 - r108908;
        double r108911 = r108909 * r108910;
        double r108912 = cbrt(r108911);
        double r108913 = r108912 / r108884;
        double r108914 = r108905 * r108913;
        double r108915 = r108890 ? r108903 : r108914;
        double r108916 = r108867 ? r108888 : r108915;
        return r108916;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.0
Target	0.2
Herbie	8.7

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -1.9722186113471817e-05

   1. Initial program 0.1

      \[e^{a \cdot x} - 1\]
   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 0.1

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

   6. Applied pow-exp 0.1

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

   8. Applied add-cube-cbrt 0.1

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

   9. Applied add-cube-cbrt 0.1

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

   10. Applied times-frac 0.1

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

   12. Applied add-cbrt-cube 0.1

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

   13. Simplified 0.1

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

   if -1.9722186113471817e-05 < (* a x) < 6.840956219609134e-13

   1. Initial program 44.8

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

   2. Taylor expanded around 0 13.1

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

   3. Simplified 13.1

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

   if 6.840956219609134e-13 < (* a x)

   1. Initial program 14.9

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

   3. Applied flip3-- 15.8

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

   4. Simplified 15.8

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

   6. Applied pow-exp 14.9

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

   8. Applied add-cube-cbrt 15.0

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

   9. Applied add-cube-cbrt 15.0

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

   10. Applied times-frac 15.0

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

   12. Applied add-sqr-sqrt 15.0

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

   13. Applied unpow-prod-down 15.0

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

   14. Applied add-sqr-sqrt 15.1

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

   15. Applied difference-of-squares 15.2

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.97221861134718174 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}} \cdot \sqrt[3]{\sqrt[3]{{\left(e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}\right)}^{3}}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}} \cdot \frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\\ \mathbf{elif}\;a \cdot x \le 6.8409562196091337 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}} \cdot \sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}} \cdot \frac{\sqrt[3]{\left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}} - {\left(\sqrt{1}\right)}^{3}\right)}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))