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 r157954 = a1;
        double r157955 = a2;
        double r157956 = r157954 * r157955;
        double r157957 = b1;
        double r157958 = b2;
        double r157959 = r157957 * r157958;
        double r157960 = r157956 / r157959;
        return r157960;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r157961 = a1;
        double r157962 = a2;
        double r157963 = r157961 * r157962;
        double r157964 = b1;
        double r157965 = b2;
        double r157966 = r157964 * r157965;
        double r157967 = r157963 / r157966;
        double r157968 = -5.6299209626543135e+278;
        bool r157969 = r157967 <= r157968;
        double r157970 = -1.8763565820874656e-287;
        bool r157971 = r157967 <= r157970;
        double r157972 = 1.1310223173930025e-246;
        bool r157973 = r157967 <= r157972;
        double r157974 = 5.499100526970725e+259;
        bool r157975 = r157967 <= r157974;
        double r157976 = !r157975;
        bool r157977 = r157973 || r157976;
        double r157978 = !r157977;
        bool r157979 = r157971 || r157978;
        double r157980 = !r157979;
        bool r157981 = r157969 || r157980;
        double r157982 = r157961 / r157964;
        double r157983 = r157965 / r157962;
        double r157984 = r157982 / r157983;
        double r157985 = r157981 ? r157984 : r157967;
        return r157985;
}

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 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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