expq3 (problem 3.4.2)

Average Error: 60.5 → 52.1

Time: 35.3s

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 r72888 = eps;
        double r72889 = a;
        double r72890 = b;
        double r72891 = r72889 + r72890;
        double r72892 = r72891 * r72888;
        double r72893 = exp(r72892);
        double r72894 = 1.0;
        double r72895 = r72893 - r72894;
        double r72896 = r72888 * r72895;
        double r72897 = r72889 * r72888;
        double r72898 = exp(r72897);
        double r72899 = r72898 - r72894;
        double r72900 = r72890 * r72888;
        double r72901 = exp(r72900);
        double r72902 = r72901 - r72894;
        double r72903 = r72899 * r72902;
        double r72904 = r72896 / r72903;
        return r72904;
}

↓

double f(double a, double b, double eps) {
        double r72905 = b;
        double r72906 = -5.30817544204267e+128;
        bool r72907 = r72905 <= r72906;
        double r72908 = 1.8117384557919386e+53;
        bool r72909 = r72905 <= r72908;
        double r72910 = !r72909;
        bool r72911 = r72907 || r72910;
        double r72912 = eps;
        double r72913 = a;
        double r72914 = r72913 + r72905;
        double r72915 = r72914 * r72912;
        double r72916 = exp(r72915);
        double r72917 = 1.0;
        double r72918 = r72916 - r72917;
        double r72919 = r72912 * r72918;
        double r72920 = r72905 * r72912;
        double r72921 = exp(r72920);
        double r72922 = r72921 - r72917;
        double r72923 = r72922 * r72912;
        double r72924 = 0.5;
        double r72925 = r72913 * r72913;
        double r72926 = r72924 * r72925;
        double r72927 = fma(r72926, r72912, r72913);
        double r72928 = r72923 * r72927;
        double r72929 = expm1(r72928);
        double r72930 = log1p(r72929);
        double r72931 = r72919 / r72930;
        double r72932 = r72913 * r72912;
        double r72933 = exp(r72932);
        double r72934 = r72933 - r72917;
        double r72935 = 3.0;
        double r72936 = pow(r72905, r72935);
        double r72937 = 0.16666666666666666;
        double r72938 = pow(r72912, r72935);
        double r72939 = r72937 * r72938;
        double r72940 = r72912 * r72912;
        double r72941 = r72924 * r72940;
        double r72942 = r72941 * r72905;
        double r72943 = r72912 + r72942;
        double r72944 = r72905 * r72943;
        double r72945 = fma(r72936, r72939, r72944);
        double r72946 = r72934 * r72945;
        double r72947 = r72919 / r72946;
        double r72948 = r72911 ? r72931 : r72947;
        return r72948;
}

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))))