Quotient of products

Average Error: 10.8 → 5.2

Time: 12.2s

Precision: 64

Math

\[\frac{a1 \cdot a2}{b1 \cdot b2}\]

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.975410788026315 \cdot 10^{+305}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -5.1555799853760176 \cdot 10^{-213}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.1084891218651217 \cdot 10^{-234}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.666768526518194 \cdot 10^{+83}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r2733908 = a1;
        double r2733909 = a2;
        double r2733910 = r2733908 * r2733909;
        double r2733911 = b1;
        double r2733912 = b2;
        double r2733913 = r2733911 * r2733912;
        double r2733914 = r2733910 / r2733913;
        return r2733914;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r2733915 = a1;
        double r2733916 = a2;
        double r2733917 = r2733915 * r2733916;
        double r2733918 = -3.975410788026315e+305;
        bool r2733919 = r2733917 <= r2733918;
        double r2733920 = b1;
        double r2733921 = r2733915 / r2733920;
        double r2733922 = b2;
        double r2733923 = r2733916 / r2733922;
        double r2733924 = r2733921 * r2733923;
        double r2733925 = -5.1555799853760176e-213;
        bool r2733926 = r2733917 <= r2733925;
        double r2733927 = r2733922 * r2733920;
        double r2733928 = r2733917 / r2733927;
        double r2733929 = 1.1084891218651217e-234;
        bool r2733930 = r2733917 <= r2733929;
        double r2733931 = r2733916 / r2733920;
        double r2733932 = r2733931 / r2733922;
        double r2733933 = r2733915 * r2733932;
        double r2733934 = 5.666768526518194e+83;
        bool r2733935 = r2733917 <= r2733934;
        double r2733936 = r2733935 ? r2733928 : r2733924;
        double r2733937 = r2733930 ? r2733933 : r2733936;
        double r2733938 = r2733926 ? r2733928 : r2733937;
        double r2733939 = r2733919 ? r2733924 : r2733938;
        return r2733939;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.8
Target	11.0
Herbie	5.2

\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -3.975410788026315e+305 or 5.666768526518194e+83 < (* a1 a2)

   1. Initial program 28.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied times-frac 10.8

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]

   if -3.975410788026315e+305 < (* a1 a2) < -5.1555799853760176e-213 or 1.1084891218651217e-234 < (* a1 a2) < 5.666768526518194e+83

   1. Initial program 3.9

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]

   if -5.1555799853760176e-213 < (* a1 a2) < 1.1084891218651217e-234

   1. Initial program 15.4

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   2. Using strategy rm

   3. Applied times-frac 4.5

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]
   4. Using strategy rm

   5. Applied div-inv 4.5

      \[\leadsto \color{blue}{\left(a1 \cdot \frac{1}{b1}\right)} \cdot \frac{a2}{b2}\]

   6. Applied associate-*l* 4.6

      \[\leadsto \color{blue}{a1 \cdot \left(\frac{1}{b1} \cdot \frac{a2}{b2}\right)}\]

   7. Simplified 4.4

      \[\leadsto a1 \cdot \color{blue}{\frac{\frac{a2}{b1}}{b2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -3.975410788026315 \cdot 10^{+305}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -5.1555799853760176 \cdot 10^{-213}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.1084891218651217 \cdot 10^{-234}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 5.666768526518194 \cdot 10^{+83}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))