Quotient of products

Average Error: 11.3 → 2.6

Time: 11.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.331097236289201929838111566058261923975 \cdot 10^{-312} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 2.759064889840746628621735483350290847538 \cdot 10^{303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r185570 = a1;
        double r185571 = a2;
        double r185572 = r185570 * r185571;
        double r185573 = b1;
        double r185574 = b2;
        double r185575 = r185573 * r185574;
        double r185576 = r185572 / r185575;
        return r185576;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r185577 = a1;
        double r185578 = a2;
        double r185579 = r185577 * r185578;
        double r185580 = b1;
        double r185581 = b2;
        double r185582 = r185580 * r185581;
        double r185583 = r185579 / r185582;
        double r185584 = -inf.0;
        bool r185585 = r185583 <= r185584;
        double r185586 = r185579 / r185580;
        double r185587 = r185586 / r185581;
        double r185588 = -4.3310972362892e-312;
        bool r185589 = r185583 <= r185588;
        double r185590 = 0.0;
        bool r185591 = r185583 <= r185590;
        double r185592 = !r185591;
        double r185593 = 2.7590648898407466e+303;
        bool r185594 = r185583 <= r185593;
        bool r185595 = r185592 && r185594;
        bool r185596 = r185589 || r185595;
        double r185597 = r185577 / r185580;
        double r185598 = r185578 / r185581;
        double r185599 = r185597 * r185598;
        double r185600 = r185596 ? r185583 : r185599;
        double r185601 = r185585 ? r185587 : r185600;
        return r185601;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.3
Herbie	2.6

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/r* 28.8

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -4.3310972362892e-312 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 2.7590648898407466e+303

   1. Initial program 3.4

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

   if -4.3310972362892e-312 < (/ (* a1 a2) (* b1 b2)) < 0.0 or 2.7590648898407466e+303 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 29.3

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

   3. Applied times-frac 3.6

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.331097236289201929838111566058261923975 \cdot 10^{-312} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \le 2.759064889840746628621735483350290847538 \cdot 10^{303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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