Quotient of products

Average Error: 11.8 → 5.9

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -6.389965488148195 \cdot 10^{+119}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -3.313835999822937 \cdot 10^{-202}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 5.581682978566765 \cdot 10^{-265}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 4.762500496650808 \cdot 10^{+291}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* a1 a2) -6.389965488148195e+119)
   (* (/ a2 b1) (/ a1 b2))
   (if (<= (* a1 a2) -3.313835999822937e-202)
     (* (* a1 a2) (/ 1.0 (* b1 b2)))
     (if (<= (* a1 a2) 5.581682978566765e-265)
       (* a2 (/ (/ a1 b1) b2))
       (if (<= (* a1 a2) 4.762500496650808e+291)
         (* (* a1 a2) (/ 1.0 (* b1 b2)))
         (* (/ a2 b1) (/ a1 b2)))))))

C

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

↓

double code(double a1, double a2, double b1, double b2) {
	double VAR;
	if ((((double) (a1 * a2)) <= -6.389965488148195e+119)) {
		VAR = ((double) ((a2 / b1) * (a1 / b2)));
	} else {
		double VAR_1;
		if ((((double) (a1 * a2)) <= -3.313835999822937e-202)) {
			VAR_1 = ((double) (((double) (a1 * a2)) * (1.0 / ((double) (b1 * b2)))));
		} else {
			double VAR_2;
			if ((((double) (a1 * a2)) <= 5.581682978566765e-265)) {
				VAR_2 = ((double) (a2 * ((a1 / b1) / b2)));
			} else {
				double VAR_3;
				if ((((double) (a1 * a2)) <= 4.762500496650808e+291)) {
					VAR_3 = ((double) (((double) (a1 * a2)) * (1.0 / ((double) (b1 * b2)))));
				} else {
					VAR_3 = ((double) ((a2 / b1) * (a1 / b2)));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.8
Target	11.1
Herbie	5.9

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

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -6.3899654881481949e119 or 4.7625004966508079e291 < (* a1 a2)

   1. Initial program 33.2

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

   2. Taylor expanded around 0 33.2

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

   3. Simplified 17.7

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

   5. Applied *-un-lft-identity 17.7

      \[\leadsto a2 \cdot \frac{\color{blue}{1 \cdot a1}}{b1 \cdot b2}\]

   6. Applied times-frac 10.2

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

   7. Applied associate-*r* 11.5

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

   8. Simplified 11.4

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

   if -6.3899654881481949e119 < (* a1 a2) < -3.3138359998229371e-202 or 5.5816829785667653e-265 < (* a1 a2) < 4.7625004966508079e291

   1. Initial program 4.9

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

   3. Applied div-inv 5.2

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

   if -3.3138359998229371e-202 < (* a1 a2) < 5.5816829785667653e-265

   1. Initial program 16.4

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

   2. Taylor expanded around 0 16.4

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

   3. Simplified 8.1

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

   5. Applied associate-/r* 4.2

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -6.389965488148195 \cdot 10^{+119}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -3.313835999822937 \cdot 10^{-202}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 5.581682978566765 \cdot 10^{-265}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 4.762500496650808 \cdot 10^{+291}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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