expq3 (problem 3.4.2)

Average Error: 58.7 → 4.0

Time: 36.3s

Precision: 64

• could not determine a ground truth for program body

  1. a = 1.6206549579252368e+91
  2. b = 8.267413230461951e+54
  3. eps = 7.285710797787771e-16

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le 1.0362962652190794 \cdot 10^{+163}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(\varepsilon \cdot \left(b + a\right)\right)\right) \cdot \frac{\frac{\varepsilon}{\sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}}{\sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r3243865 = eps;
        double r3243866 = a;
        double r3243867 = b;
        double r3243868 = r3243866 + r3243867;
        double r3243869 = r3243868 * r3243865;
        double r3243870 = exp(r3243869);
        double r3243871 = 1.0;
        double r3243872 = r3243870 - r3243871;
        double r3243873 = r3243865 * r3243872;
        double r3243874 = r3243866 * r3243865;
        double r3243875 = exp(r3243874);
        double r3243876 = r3243875 - r3243871;
        double r3243877 = r3243867 * r3243865;
        double r3243878 = exp(r3243877);
        double r3243879 = r3243878 - r3243871;
        double r3243880 = r3243876 * r3243879;
        double r3243881 = r3243873 / r3243880;
        return r3243881;
}

↓

double f(double a, double b, double eps) {
        double r3243882 = a;
        double r3243883 = 1.0362962652190794e+163;
        bool r3243884 = r3243882 <= r3243883;
        double r3243885 = 1.0;
        double r3243886 = b;
        double r3243887 = r3243885 / r3243886;
        double r3243888 = r3243885 / r3243882;
        double r3243889 = r3243887 + r3243888;
        double r3243890 = eps;
        double r3243891 = r3243886 + r3243882;
        double r3243892 = r3243890 * r3243891;
        double r3243893 = expm1(r3243892);
        double r3243894 = r3243890 * r3243886;
        double r3243895 = expm1(r3243894);
        double r3243896 = cbrt(r3243895);
        double r3243897 = r3243896 * r3243896;
        double r3243898 = r3243890 / r3243897;
        double r3243899 = r3243898 / r3243896;
        double r3243900 = r3243893 * r3243899;
        double r3243901 = r3243890 * r3243882;
        double r3243902 = expm1(r3243901);
        double r3243903 = r3243900 / r3243902;
        double r3243904 = r3243884 ? r3243889 : r3243903;
        return r3243904;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.7
Target	13.8
Herbie	4.0

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

Derivation

1. Split input into 2 regimes

2. if a < 1.0362962652190794e+163

   1. Initial program 59.5

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

   2. Simplified 27.9

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}}\]

   3. Taylor expanded around 0 2.5

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]

   if 1.0362962652190794e+163 < a

   1. Initial program 50.9

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

   2. Simplified 17.1

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 17.7

      \[\leadsto \frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\left(\sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}}}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}\]

   5. Applied associate-/r* 17.7

      \[\leadsto \frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \color{blue}{\frac{\frac{\varepsilon}{\sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}}{\sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}}}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 1.0362962652190794 \cdot 10^{+163}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(\varepsilon \cdot \left(b + a\right)\right)\right) \cdot \frac{\frac{\varepsilon}{\sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}}{\sqrt[3]{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))