Quotient of products

Average Error: 11.3 → 3.4

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -5.629920962654313471604012923529225270914 \cdot 10^{278} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.876356582087465582822541150969070437304 \cdot 10^{-287} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.131022317393002517657401368626620541691 \cdot 10^{-246} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.499100526970725014397291959470552919566 \cdot 10^{259}\right)\right)\right):\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r159902 = a1;
        double r159903 = a2;
        double r159904 = r159902 * r159903;
        double r159905 = b1;
        double r159906 = b2;
        double r159907 = r159905 * r159906;
        double r159908 = r159904 / r159907;
        return r159908;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r159909 = a1;
        double r159910 = a2;
        double r159911 = r159909 * r159910;
        double r159912 = b1;
        double r159913 = b2;
        double r159914 = r159912 * r159913;
        double r159915 = r159911 / r159914;
        double r159916 = -5.6299209626543135e+278;
        bool r159917 = r159915 <= r159916;
        double r159918 = -1.8763565820874656e-287;
        bool r159919 = r159915 <= r159918;
        double r159920 = 1.1310223173930025e-246;
        bool r159921 = r159915 <= r159920;
        double r159922 = 5.499100526970725e+259;
        bool r159923 = r159915 <= r159922;
        double r159924 = !r159923;
        bool r159925 = r159921 || r159924;
        double r159926 = !r159925;
        bool r159927 = r159919 || r159926;
        double r159928 = !r159927;
        bool r159929 = r159917 || r159928;
        double r159930 = r159909 / r159912;
        double r159931 = r159913 / r159910;
        double r159932 = r159930 / r159931;
        double r159933 = r159929 ? r159932 : r159915;
        return r159933;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.5
Herbie	3.4

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -5.6299209626543135e+278 or -1.8763565820874656e-287 < (/ (* a1 a2) (* b1 b2)) < 1.1310223173930025e-246 or 5.499100526970725e+259 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 23.1

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

   3. Applied associate-/r* 16.0

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

   5. Applied associate-/l* 8.0

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

   7. Applied associate-/r/ 8.3

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

   8. Applied associate-/l* 6.4

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

   if -5.6299209626543135e+278 < (/ (* a1 a2) (* b1 b2)) < -1.8763565820874656e-287 or 1.1310223173930025e-246 < (/ (* a1 a2) (* b1 b2)) < 5.499100526970725e+259

   1. Initial program 0.7

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -5.629920962654313471604012923529225270914 \cdot 10^{278} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.876356582087465582822541150969070437304 \cdot 10^{-287} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.131022317393002517657401368626620541691 \cdot 10^{-246} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 5.499100526970725014397291959470552919566 \cdot 10^{259}\right)\right)\right):\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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