expq3 (problem 3.4.2)

Average Error: 60.5 → 52.1

Time: 35.5s

Precision: 64

• could not determine a ground truth for program body

  1. a = -4.378245219510849e+106
  2. b = -1.404089089984425e+261
  3. eps = -4.518429747333079e-151

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -5.308175442042669859643036141092548933227 \cdot 10^{128} \lor \neg \left(b \le 1.811738455791938600211664020093860840226 \cdot 10^{53}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(e^{b \cdot \varepsilon} - 1\right) \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \left(a \cdot a\right), \varepsilon, a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left({b}^{3}, \frac{1}{6} \cdot {\varepsilon}^{3}, b \cdot \left(\varepsilon + \left(\frac{1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot b\right)\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r72842 = eps;
        double r72843 = a;
        double r72844 = b;
        double r72845 = r72843 + r72844;
        double r72846 = r72845 * r72842;
        double r72847 = exp(r72846);
        double r72848 = 1.0;
        double r72849 = r72847 - r72848;
        double r72850 = r72842 * r72849;
        double r72851 = r72843 * r72842;
        double r72852 = exp(r72851);
        double r72853 = r72852 - r72848;
        double r72854 = r72844 * r72842;
        double r72855 = exp(r72854);
        double r72856 = r72855 - r72848;
        double r72857 = r72853 * r72856;
        double r72858 = r72850 / r72857;
        return r72858;
}

↓

double f(double a, double b, double eps) {
        double r72859 = b;
        double r72860 = -5.30817544204267e+128;
        bool r72861 = r72859 <= r72860;
        double r72862 = 1.8117384557919386e+53;
        bool r72863 = r72859 <= r72862;
        double r72864 = !r72863;
        bool r72865 = r72861 || r72864;
        double r72866 = eps;
        double r72867 = a;
        double r72868 = r72867 + r72859;
        double r72869 = r72868 * r72866;
        double r72870 = exp(r72869);
        double r72871 = 1.0;
        double r72872 = r72870 - r72871;
        double r72873 = r72866 * r72872;
        double r72874 = r72859 * r72866;
        double r72875 = exp(r72874);
        double r72876 = r72875 - r72871;
        double r72877 = r72876 * r72866;
        double r72878 = 0.5;
        double r72879 = r72867 * r72867;
        double r72880 = r72878 * r72879;
        double r72881 = fma(r72880, r72866, r72867);
        double r72882 = r72877 * r72881;
        double r72883 = expm1(r72882);
        double r72884 = log1p(r72883);
        double r72885 = r72873 / r72884;
        double r72886 = r72867 * r72866;
        double r72887 = exp(r72886);
        double r72888 = r72887 - r72871;
        double r72889 = 3.0;
        double r72890 = pow(r72859, r72889);
        double r72891 = 0.16666666666666666;
        double r72892 = pow(r72866, r72889);
        double r72893 = r72891 * r72892;
        double r72894 = r72866 * r72866;
        double r72895 = r72878 * r72894;
        double r72896 = r72895 * r72859;
        double r72897 = r72866 + r72896;
        double r72898 = r72859 * r72897;
        double r72899 = fma(r72890, r72893, r72898);
        double r72900 = r72888 * r72899;
        double r72901 = r72873 / r72900;
        double r72902 = r72865 ? r72885 : r72901;
        return r72902;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.5
Target	14.7
Herbie	52.1

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

Derivation

1. Split input into 2 regimes

2. if b < -5.30817544204267e+128 or 1.8117384557919386e+53 < b

   1. Initial program 53.7

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

   2. Taylor expanded around 0 45.9

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

   3. Simplified 45.9

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

   4. Taylor expanded around 0 43.0

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

   5. Simplified 42.4

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \varepsilon \cdot \left(a \cdot a\right), a\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 42.7

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

   8. Applied associate-*l* 42.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(\sqrt[3]{\varepsilon} \cdot \mathsf{fma}\left(\frac{1}{2}, \varepsilon \cdot \left(a \cdot a\right), a\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
   9. Using strategy rm

   10. Applied log1p-expm1-u 41.8

       \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right) \cdot \left(\sqrt[3]{\varepsilon} \cdot \mathsf{fma}\left(\frac{1}{2}, \varepsilon \cdot \left(a \cdot a\right), a\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)\right)}}\]

   11. Simplified 41.4

       \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\left(\left(e^{b \cdot \varepsilon} - 1\right) \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \left(a \cdot a\right), \varepsilon, a\right)\right)}\right)}\]

   if -5.30817544204267e+128 < b < 1.8117384557919386e+53

   1. Initial program 63.3

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

   2. Taylor expanded around 0 56.6

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

   3. Simplified 56.6

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

4. Final simplification 52.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.308175442042669859643036141092548933227 \cdot 10^{128} \lor \neg \left(b \le 1.811738455791938600211664020093860840226 \cdot 10^{53}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(e^{b \cdot \varepsilon} - 1\right) \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \left(a \cdot a\right), \varepsilon, a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left({b}^{3}, \frac{1}{6} \cdot {\varepsilon}^{3}, b \cdot \left(\varepsilon + \left(\frac{1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot b\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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))))