Quotient of products

Average Error: 11.2 → 2.1

Time: 1.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.6712351622532716 \cdot 10^{-305}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.9400705823713053 \cdot 10^{287}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \end{array}\]

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 VAR;
	if ((((a1 * a2) / (b1 * b2)) <= -inf.0)) {
		VAR = ((a1 / b1) * (a2 / b2));
	} else {
		double VAR_1;
		if ((((a1 * a2) / (b1 * b2)) <= -2.6712351622532716e-305)) {
			VAR_1 = ((a1 * a2) / (b1 * b2));
		} else {
			double VAR_2;
			if ((((a1 * a2) / (b1 * b2)) <= -0.0)) {
				VAR_2 = ((a2 / b1) / (b2 / a1));
			} else {
				double VAR_3;
				if ((((a1 * a2) / (b1 * b2)) <= 2.9400705823713053e+287)) {
					VAR_3 = ((a1 * a2) / (b1 * b2));
				} else {
					VAR_3 = ((a2 / b1) / (b2 / a1));
				}
				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	11.7
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied times-frac 10.0

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -2.6712351622532716e-305 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 2.9400705823713053e+287

   1. Initial program 0.9

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

   if -2.6712351622532716e-305 < (/ (* a1 a2) (* b1 b2)) < -0.0 or 2.9400705823713053e+287 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 22.5

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

   3. Applied *-commutative 22.5

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

   4. Applied associate-/l* 14.7

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

   6. Applied div-inv 14.7

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

   7. Simplified 6.1

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

   9. Applied div-inv 6.1

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

   11. Applied clear-num 6.2

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

   12. Applied associate-*l/ 6.2

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

   13. Applied associate-*r/ 3.4

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

   14. Simplified 3.3

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.6712351622532716 \cdot 10^{-305}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.9400705823713053 \cdot 10^{287}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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