Quotient of products

Average Error: 11.2 → 2.7

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.13537030152012078 \cdot 10^{239}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.1780918079753167 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.26933436524312406 \cdot 10^{289}:\\ \;\;\;\;\sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

C

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

↓

double code(double a1, double a2, double b1, double b2) {
	double VAR;
	if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -3.135370301520121e+239)) {
		VAR = ((double) (((double) (a1 / b1)) * ((double) (a2 / b2))));
	} else {
		double VAR_1;
		if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= -2.1780918079753167e-308)) {
			VAR_1 = ((double) (1.0 / ((double) (((double) (b1 * b2)) / ((double) (a1 * a2))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 0.0)) {
				VAR_2 = ((double) (((double) (((double) (((double) cbrt(a2)) * ((double) cbrt(a2)))) / b1)) * ((double) (a1 / ((double) (b2 / ((double) cbrt(a2))))))));
			} else {
				double VAR_3;
				if ((((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))) <= 4.269334365243124e+289)) {
					VAR_3 = ((double) (((double) sqrt(((double) (((double) (a1 * a2)) / ((double) (b1 * b2)))))) * ((double) sqrt(((double) (((double) (a1 * a2)) / ((double) (b1 * b2))))))));
				} else {
					VAR_3 = ((double) (((double) (a1 / b1)) / ((double) (b2 / a2))));
				}
				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	11.2
Target	11.6
Herbie	2.7

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

Derivation

1. Split input into 5 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -3.13537030152012078e239

   1. Initial program 39.1

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

   3. Applied times-frac 18.8

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

   if -3.13537030152012078e239 < (/ (* a1 a2) (* b1 b2)) < -2.1780918079753167e-308

   1. Initial program 0.8

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

   3. Applied clear-num 0.9

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

   if -2.1780918079753167e-308 < (/ (* a1 a2) (* b1 b2)) < 0.0

   1. Initial program 13.7

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

   3. Applied associate-/l* 7.6

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

   5. Applied add-cube-cbrt 7.7

      \[\leadsto \frac{a1}{\frac{b1 \cdot b2}{\color{blue}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}}}\]

   6. Applied times-frac 4.0

      \[\leadsto \frac{a1}{\color{blue}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{b2}{\sqrt[3]{a2}}}}\]

   7. Applied *-un-lft-identity 4.0

      \[\leadsto \frac{\color{blue}{1 \cdot a1}}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{b2}{\sqrt[3]{a2}}}\]

   8. Applied times-frac 2.2

      \[\leadsto \color{blue}{\frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}}\]

   9. Simplified 2.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\]

   if 0.0 < (/ (* a1 a2) (* b1 b2)) < 4.26933436524312406e289

   1. Initial program 0.8

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

   3. Applied add-sqr-sqrt 1.1

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

   if 4.26933436524312406e289 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 58.8

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

   3. Applied associate-/l* 45.3

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

   5. Applied *-un-lft-identity 45.3

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

   6. Applied times-frac 12.0

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

   7. Applied associate-/r* 7.4

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

   8. Simplified 7.4

      \[\leadsto \frac{\color{blue}{\frac{a1}{b1}}}{\frac{b2}{a2}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.13537030152012078 \cdot 10^{239}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.1780918079753167 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.26933436524312406 \cdot 10^{289}:\\ \;\;\;\;\sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}} \cdot \sqrt{\frac{a1 \cdot a2}{b1 \cdot b2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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