Quotient of products

Average Error: 11.4 → 5.2

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty \lor \neg \left(a1 \cdot a2 \le -4.0878819804843486 \cdot 10^{-254} \lor \neg \left(a1 \cdot a2 \le 1.7944304405375 \cdot 10^{-310} \lor \neg \left(a1 \cdot a2 \le 3.0403037297418217 \cdot 10^{246}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r133634 = a1;
        double r133635 = a2;
        double r133636 = r133634 * r133635;
        double r133637 = b1;
        double r133638 = b2;
        double r133639 = r133637 * r133638;
        double r133640 = r133636 / r133639;
        return r133640;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r133641 = a1;
        double r133642 = a2;
        double r133643 = r133641 * r133642;
        double r133644 = -inf.0;
        bool r133645 = r133643 <= r133644;
        double r133646 = -4.0878819804843486e-254;
        bool r133647 = r133643 <= r133646;
        double r133648 = 1.7944304405375e-310;
        bool r133649 = r133643 <= r133648;
        double r133650 = 3.0403037297418217e+246;
        bool r133651 = r133643 <= r133650;
        double r133652 = !r133651;
        bool r133653 = r133649 || r133652;
        double r133654 = !r133653;
        bool r133655 = r133647 || r133654;
        double r133656 = !r133655;
        bool r133657 = r133645 || r133656;
        double r133658 = b1;
        double r133659 = r133641 / r133658;
        double r133660 = b2;
        double r133661 = r133642 / r133660;
        double r133662 = r133659 * r133661;
        double r133663 = 1.0;
        double r133664 = r133663 / r133658;
        double r133665 = r133664 / r133660;
        double r133666 = r133643 * r133665;
        double r133667 = r133657 ? r133662 : r133666;
        return r133667;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.0
Herbie	5.2

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

Derivation

1. Split input into 2 regimes

2. if (* a1 a2) < -inf.0 or -4.0878819804843486e-254 < (* a1 a2) < 1.7944304405375e-310 or 3.0403037297418217e+246 < (* a1 a2)

   1. Initial program 27.6

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

   3. Applied times-frac 4.3

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

   if -inf.0 < (* a1 a2) < -4.0878819804843486e-254 or 1.7944304405375e-310 < (* a1 a2) < 3.0403037297418217e+246

   1. Initial program 5.5

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

   3. Applied div-inv 5.8

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

   5. Applied associate-/r* 5.5

      \[\leadsto \left(a1 \cdot a2\right) \cdot \color{blue}{\frac{\frac{1}{b1}}{b2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 = -\infty \lor \neg \left(a1 \cdot a2 \le -4.0878819804843486 \cdot 10^{-254} \lor \neg \left(a1 \cdot a2 \le 1.7944304405375 \cdot 10^{-310} \lor \neg \left(a1 \cdot a2 \le 3.0403037297418217 \cdot 10^{246}\right)\right)\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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