Quotient of products

Average Error: 11.4 → 3.4

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.68280036335944034 \cdot 10^{301}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.5204447990899316 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.51733405101081216 \cdot 10^{288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \end{array}\]

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 temp;
	if ((((a1 * a2) / (b1 * b2)) <= -1.6828003633594403e+301)) {
		temp = ((a1 * (a2 / b1)) * (1.0 / b2));
	} else {
		double temp_1;
		if ((((a1 * a2) / (b1 * b2)) <= -4.520444799089932e-307)) {
			temp_1 = ((a1 * a2) / (b1 * b2));
		} else {
			double temp_2;
			if ((((a1 * a2) / (b1 * b2)) <= 0.0)) {
				temp_2 = (a1 / (b1 / (a2 / b2)));
			} else {
				double temp_3;
				if ((((a1 * a2) / (b1 * b2)) <= 4.517334051010812e+288)) {
					temp_3 = ((a1 * a2) / (b1 * b2));
				} else {
					temp_3 = ((a1 * (a2 / b1)) * (1.0 / b2));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.5
Herbie	3.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -1.6828003633594403e+301 or 4.517334051010812e+288 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 59.3

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

   3. Applied associate-/l* 39.8

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

   5. Applied div-inv 39.9

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

   6. Simplified 39.8

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

   8. Applied associate-/r* 15.2

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

   10. Applied div-inv 15.3

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

   11. Applied associate-*r* 14.9

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

   if -1.6828003633594403e+301 < (/ (* a1 a2) (* b1 b2)) < -4.520444799089932e-307 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 4.517334051010812e+288

   1. Initial program 0.8

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

   3. Applied associate-/l* 7.3

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

   5. Applied div-inv 7.6

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

   6. Simplified 7.3

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

   8. Applied associate-*r/ 0.8

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

   if -4.520444799089932e-307 < (/ (* a1 a2) (* b1 b2)) < 0.0

   1. Initial program 13.1

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

   3. Applied associate-/l* 7.1

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

   5. Applied associate-/l* 3.8

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.68280036335944034 \cdot 10^{301}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.5204447990899316 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.51733405101081216 \cdot 10^{288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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