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 r129735 = a1;
        double r129736 = a2;
        double r129737 = r129735 * r129736;
        double r129738 = b1;
        double r129739 = b2;
        double r129740 = r129738 * r129739;
        double r129741 = r129737 / r129740;
        return r129741;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r129742 = a1;
        double r129743 = a2;
        double r129744 = r129742 * r129743;
        double r129745 = b1;
        double r129746 = b2;
        double r129747 = r129745 * r129746;
        double r129748 = r129744 / r129747;
        double r129749 = -5.6299209626543135e+278;
        bool r129750 = r129748 <= r129749;
        double r129751 = -1.8763565820874656e-287;
        bool r129752 = r129748 <= r129751;
        double r129753 = 1.1310223173930025e-246;
        bool r129754 = r129748 <= r129753;
        double r129755 = 5.499100526970725e+259;
        bool r129756 = r129748 <= r129755;
        double r129757 = !r129756;
        bool r129758 = r129754 || r129757;
        double r129759 = !r129758;
        bool r129760 = r129752 || r129759;
        double r129761 = !r129760;
        bool r129762 = r129750 || r129761;
        double r129763 = r129742 / r129745;
        double r129764 = r129746 / r129743;
        double r129765 = r129763 / r129764;
        double r129766 = r129762 ? r129765 : r129748;
        return r129766;
}

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)))