Quotient of products

Average Error: 11.3 → 3.4

Time: 2.9s

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 r145120 = a1;
        double r145121 = a2;
        double r145122 = r145120 * r145121;
        double r145123 = b1;
        double r145124 = b2;
        double r145125 = r145123 * r145124;
        double r145126 = r145122 / r145125;
        return r145126;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r145127 = a1;
        double r145128 = a2;
        double r145129 = r145127 * r145128;
        double r145130 = b1;
        double r145131 = b2;
        double r145132 = r145130 * r145131;
        double r145133 = r145129 / r145132;
        double r145134 = -5.6299209626543135e+278;
        bool r145135 = r145133 <= r145134;
        double r145136 = -1.8763565820874656e-287;
        bool r145137 = r145133 <= r145136;
        double r145138 = 1.1310223173930025e-246;
        bool r145139 = r145133 <= r145138;
        double r145140 = 5.499100526970725e+259;
        bool r145141 = r145133 <= r145140;
        double r145142 = !r145141;
        bool r145143 = r145139 || r145142;
        double r145144 = !r145143;
        bool r145145 = r145137 || r145144;
        double r145146 = !r145145;
        bool r145147 = r145135 || r145146;
        double r145148 = r145127 / r145130;
        double r145149 = r145131 / r145128;
        double r145150 = r145148 / r145149;
        double r145151 = r145147 ? r145150 : r145133;
        return r145151;
}

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)))