Quotient of products

Average Error: 11.4 → 2.9

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -8.778186171616543 \cdot 10^{-268}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.677872322695465 \cdot 10^{+299}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array}\]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) (- INFINITY))
   (/ (* a1 (/ a2 b2)) b1)
   (if (<= (/ (* a1 a2) (* b1 b2)) -8.778186171616543e-268)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (/ (* a1 (/ a2 b2)) b1)
       (if (<= (/ (* a1 a2) (* b1 b2)) 4.677872322695465e+299)
         (/ (* a1 a2) (* b1 b2))
         (* (/ a2 b2) (/ a1 b1)))))))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

↓

double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (((a1 * a2) / (b1 * b2)) <= -((double) INFINITY)) {
		tmp = (a1 * (a2 / b2)) / b1;
	} else if (((a1 * a2) / (b1 * b2)) <= -8.778186171616543e-268) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 0.0) {
		tmp = (a1 * (a2 / b2)) / b1;
	} else if (((a1 * a2) / (b1 * b2)) <= 4.677872322695465e+299) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = (a2 / b2) * (a1 / b1);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.8
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or -8.7781861716165429e-268 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 18.2

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

   3. Applied times-frac_binary64_2105 4.2

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

   5. Applied associate-*l/_binary64_2044 5.7

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -8.7781861716165429e-268 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 4.6778723226954651e299

   1. Initial program 0.8

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

   if 4.6778723226954651e299 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 61.6

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

   3. Applied times-frac_binary64_2105 6.7

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -8.778186171616543 \cdot 10^{-268}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.677872322695465 \cdot 10^{+299}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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