Quotient of products

Average Error: 11.3 → 3.1

Time: 12.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.843324609247471005376211875053257441102 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.464542161086205916851616380334111321007 \cdot 10^{288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r127592 = a1;
        double r127593 = a2;
        double r127594 = r127592 * r127593;
        double r127595 = b1;
        double r127596 = b2;
        double r127597 = r127595 * r127596;
        double r127598 = r127594 / r127597;
        return r127598;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r127599 = a1;
        double r127600 = a2;
        double r127601 = r127599 * r127600;
        double r127602 = b1;
        double r127603 = b2;
        double r127604 = r127602 * r127603;
        double r127605 = r127601 / r127604;
        double r127606 = -inf.0;
        bool r127607 = r127605 <= r127606;
        double r127608 = r127600 / r127603;
        double r127609 = r127599 * r127608;
        double r127610 = r127609 / r127602;
        double r127611 = -8.843324609247471e-307;
        bool r127612 = r127605 <= r127611;
        double r127613 = -0.0;
        bool r127614 = r127605 <= r127613;
        double r127615 = 1.464542161086206e+288;
        bool r127616 = r127605 <= r127615;
        double r127617 = r127599 / r127602;
        double r127618 = r127617 * r127600;
        double r127619 = r127618 / r127603;
        double r127620 = r127616 ? r127605 : r127619;
        double r127621 = r127614 ? r127610 : r127620;
        double r127622 = r127612 ? r127605 : r127621;
        double r127623 = r127607 ? r127610 : r127622;
        return r127623;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	10.9
Herbie	3.1

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -8.843324609247471e-307 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 18.3

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

   3. Applied times-frac 3.3

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

   5. Applied associate-*l/ 4.7

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -8.843324609247471e-307 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 1.464542161086206e+288

   1. Initial program 0.8

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

   3. Applied times-frac 16.0

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

   5. Applied div-inv 16.1

      \[\leadsto \frac{a1}{b1} \cdot \color{blue}{\left(a2 \cdot \frac{1}{b2}\right)}\]

   6. Applied associate-*r* 14.6

      \[\leadsto \color{blue}{\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}}\]
   7. Using strategy rm

   8. Applied un-div-inv 14.6

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

   10. Applied associate-*l/ 8.1

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

   11. Applied associate-/l/ 0.8

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

   12. Simplified 0.8

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

   if 1.464542161086206e+288 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 59.6

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

   3. Applied times-frac 7.2

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

   5. Applied div-inv 7.3

      \[\leadsto \frac{a1}{b1} \cdot \color{blue}{\left(a2 \cdot \frac{1}{b2}\right)}\]

   6. Applied associate-*r* 13.1

      \[\leadsto \color{blue}{\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}}\]
   7. Using strategy rm

   8. Applied un-div-inv 13.1

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.843324609247471005376211875053257441102 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.464542161086205916851616380334111321007 \cdot 10^{288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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