Quotient of products

Average Error: 11.5 → 3.8

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.2602034138658774 \cdot 10^{-231} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.10047474805390751 \cdot 10^{-168} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.70834244534979421 \cdot 10^{294}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r178845 = a1;
        double r178846 = a2;
        double r178847 = r178845 * r178846;
        double r178848 = b1;
        double r178849 = b2;
        double r178850 = r178848 * r178849;
        double r178851 = r178847 / r178850;
        return r178851;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r178852 = a1;
        double r178853 = a2;
        double r178854 = r178852 * r178853;
        double r178855 = b1;
        double r178856 = b2;
        double r178857 = r178855 * r178856;
        double r178858 = r178854 / r178857;
        double r178859 = -inf.0;
        bool r178860 = r178858 <= r178859;
        double r178861 = -4.2602034138658774e-231;
        bool r178862 = r178858 <= r178861;
        double r178863 = 7.1004747480539075e-168;
        bool r178864 = r178858 <= r178863;
        double r178865 = 3.708342445349794e+294;
        bool r178866 = r178858 <= r178865;
        double r178867 = !r178866;
        bool r178868 = r178864 || r178867;
        double r178869 = !r178868;
        bool r178870 = r178862 || r178869;
        double r178871 = !r178870;
        bool r178872 = r178860 || r178871;
        double r178873 = r178852 / r178855;
        double r178874 = r178853 / r178856;
        double r178875 = r178873 * r178874;
        double r178876 = r178872 ? r178875 : r178858;
        return r178876;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.5
Herbie	3.8

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -4.2602034138658774e-231 < (/ (* a1 a2) (* b1 b2)) < 7.1004747480539075e-168 or 3.708342445349794e+294 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 21.8

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

   3. Applied times-frac 6.7

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -4.2602034138658774e-231 or 7.1004747480539075e-168 < (/ (* a1 a2) (* b1 b2)) < 3.708342445349794e+294

   1. Initial program 0.8

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.2602034138658774 \cdot 10^{-231} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 7.10047474805390751 \cdot 10^{-168} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.70834244534979421 \cdot 10^{294}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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