expq3 (problem 3.4.2)

Average Error: 59.4 → 58.7

Time: 11.5s

Precision: 64

\[-1 \lt \varepsilon \land \varepsilon \lt 1\]

Math

\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le -3.6563033602620086 \cdot 10^{36}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt{e^{a \cdot \varepsilon}} + \sqrt{1}\right) \cdot \left(\left(\left(\sqrt{\sqrt{e^{a \cdot \varepsilon}}} + \sqrt{\sqrt{1}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}} \cdot \sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}}\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)}\\ \mathbf{elif}\;a \le 2.575614997139679 \cdot 10^{-24}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r91835 = eps;
        double r91836 = a;
        double r91837 = b;
        double r91838 = r91836 + r91837;
        double r91839 = r91838 * r91835;
        double r91840 = exp(r91839);
        double r91841 = 1.0;
        double r91842 = r91840 - r91841;
        double r91843 = r91835 * r91842;
        double r91844 = r91836 * r91835;
        double r91845 = exp(r91844);
        double r91846 = r91845 - r91841;
        double r91847 = r91837 * r91835;
        double r91848 = exp(r91847);
        double r91849 = r91848 - r91841;
        double r91850 = r91846 * r91849;
        double r91851 = r91843 / r91850;
        return r91851;
}

↓

double f(double a, double b, double eps) {
        double r91852 = a;
        double r91853 = -3.6563033602620086e+36;
        bool r91854 = r91852 <= r91853;
        double r91855 = eps;
        double r91856 = b;
        double r91857 = r91852 + r91856;
        double r91858 = r91857 * r91855;
        double r91859 = exp(r91858);
        double r91860 = 1.0;
        double r91861 = r91859 - r91860;
        double r91862 = r91855 * r91861;
        double r91863 = r91852 * r91855;
        double r91864 = exp(r91863);
        double r91865 = sqrt(r91864);
        double r91866 = sqrt(r91860);
        double r91867 = r91865 + r91866;
        double r91868 = sqrt(r91865);
        double r91869 = sqrt(r91866);
        double r91870 = r91868 + r91869;
        double r91871 = r91868 - r91869;
        double r91872 = cbrt(r91871);
        double r91873 = r91872 * r91872;
        double r91874 = r91873 * r91872;
        double r91875 = cbrt(r91874);
        double r91876 = r91872 * r91875;
        double r91877 = r91876 * r91872;
        double r91878 = r91870 * r91877;
        double r91879 = r91856 * r91855;
        double r91880 = exp(r91879);
        double r91881 = r91880 - r91860;
        double r91882 = r91878 * r91881;
        double r91883 = r91867 * r91882;
        double r91884 = r91862 / r91883;
        double r91885 = 2.5756149971396786e-24;
        bool r91886 = r91852 <= r91885;
        double r91887 = 0.16666666666666666;
        double r91888 = 3.0;
        double r91889 = pow(r91852, r91888);
        double r91890 = pow(r91855, r91888);
        double r91891 = r91889 * r91890;
        double r91892 = r91887 * r91891;
        double r91893 = 0.5;
        double r91894 = 2.0;
        double r91895 = pow(r91852, r91894);
        double r91896 = pow(r91855, r91894);
        double r91897 = r91895 * r91896;
        double r91898 = r91893 * r91897;
        double r91899 = r91898 + r91863;
        double r91900 = r91892 + r91899;
        double r91901 = r91900 * r91881;
        double r91902 = r91862 / r91901;
        double r91903 = r91864 - r91860;
        double r91904 = exp(r91903);
        double r91905 = log(r91904);
        double r91906 = r91905 * r91881;
        double r91907 = r91862 / r91906;
        double r91908 = r91886 ? r91902 : r91907;
        double r91909 = r91854 ? r91884 : r91908;
        return r91909;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	59.4
Target	17.0
Herbie	58.7

\[\frac{a + b}{a \cdot b}\]

Derivation

1. Split input into 3 regimes

2. if a < -3.6563033602620086e+36

   1. Initial program 51.8

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 51.8

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

   4. Applied add-sqr-sqrt 51.8

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

   5. Applied difference-of-squares 51.8

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

   6. Applied associate-*l* 51.8

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

   8. Applied add-sqr-sqrt 51.8

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

   9. Applied sqrt-prod 51.8

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

   10. Applied add-sqr-sqrt 51.8

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

   11. Applied sqrt-prod 51.9

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

   12. Applied difference-of-squares 51.9

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

   14. Applied add-cube-cbrt 51.9

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

   16. Applied add-cube-cbrt 51.9

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

   if -3.6563033602620086e+36 < a < 2.5756149971396786e-24

   1. Initial program 63.9

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   2. Taylor expanded around 0 62.6

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

   if 2.5756149971396786e-24 < a

   1. Initial program 53.3

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
   2. Using strategy rm

   3. Applied add-log-exp 53.3

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

   4. Applied add-log-exp 53.4

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

   5. Applied diff-log 53.4

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

   6. Simplified 53.4

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

4. Final simplification 58.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.6563033602620086 \cdot 10^{36}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt{e^{a \cdot \varepsilon}} + \sqrt{1}\right) \cdot \left(\left(\left(\sqrt{\sqrt{e^{a \cdot \varepsilon}}} + \sqrt{\sqrt{1}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}} \cdot \sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{e^{a \cdot \varepsilon}}} - \sqrt{\sqrt{1}}}\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)}\\ \mathbf{elif}\;a \le 2.575614997139679 \cdot 10^{-24}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))