Quotient of products

Average Error: 11.0 → 2.1

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.6303120180766 \cdot 10^{-313}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.9080829902179166 \cdot 10^{-292}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.6301067955831595 \cdot 10^{+297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r8780582 = a1;
        double r8780583 = a2;
        double r8780584 = r8780582 * r8780583;
        double r8780585 = b1;
        double r8780586 = b2;
        double r8780587 = r8780585 * r8780586;
        double r8780588 = r8780584 / r8780587;
        return r8780588;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r8780589 = a1;
        double r8780590 = a2;
        double r8780591 = r8780589 * r8780590;
        double r8780592 = b1;
        double r8780593 = b2;
        double r8780594 = r8780592 * r8780593;
        double r8780595 = r8780591 / r8780594;
        double r8780596 = -inf.0;
        bool r8780597 = r8780595 <= r8780596;
        double r8780598 = r8780589 / r8780593;
        double r8780599 = r8780590 / r8780592;
        double r8780600 = r8780598 * r8780599;
        double r8780601 = -2.6303120180766e-313;
        bool r8780602 = r8780595 <= r8780601;
        double r8780603 = 3.9080829902179166e-292;
        bool r8780604 = r8780595 <= r8780603;
        double r8780605 = 1.6301067955831595e+297;
        bool r8780606 = r8780595 <= r8780605;
        double r8780607 = r8780606 ? r8780595 : r8780600;
        double r8780608 = r8780604 ? r8780600 : r8780607;
        double r8780609 = r8780602 ? r8780595 : r8780608;
        double r8780610 = r8780597 ? r8780600 : r8780609;
        return r8780610;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.0
Target	10.8
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -2.6303120180766e-313 < (/ (* a1 a2) (* b1 b2)) < 3.9080829902179166e-292 or 1.6301067955831595e+297 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 24.8

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

   3. Applied *-commutative 24.8

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

   4. Applied times-frac 3.8

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -2.6303120180766e-313 or 3.9080829902179166e-292 < (/ (* a1 a2) (* b1 b2)) < 1.6301067955831595e+297

   1. Initial program 0.8

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.6303120180766 \cdot 10^{-313}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 3.9080829902179166 \cdot 10^{-292}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.6301067955831595 \cdot 10^{+297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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