Quotient of products

Average Error: 11.5 → 5.9

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -8.065931767366266 \cdot 10^{+116}:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b1}{a2} \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -2.5844249635960764 \cdot 10^{-115}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 8.259474310150762 \cdot 10^{-200}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 6.232999595325203 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* a1 a2) -8.065931767366266e+116)
   (* a1 (/ 1.0 (* (/ b1 a2) b2)))
   (if (<= (* a1 a2) -2.5844249635960764e-115)
     (* (* a1 a2) (/ 1.0 (* b1 b2)))
     (if (<= (* a1 a2) 8.259474310150762e-200)
       (* (/ a1 b1) (/ a2 b2))
       (if (<= (* a1 a2) 6.232999595325203e+152)
         (/ (/ (* a1 a2) b2) b1)
         (/ a1 (* b1 (/ b2 a2))))))))

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 tmp;
	if ((((double) (a1 * a2)) <= -8.065931767366266e+116)) {
		tmp = ((double) (a1 * (1.0 / ((double) ((b1 / a2) * b2)))));
	} else {
		double tmp_1;
		if ((((double) (a1 * a2)) <= -2.5844249635960764e-115)) {
			tmp_1 = ((double) (((double) (a1 * a2)) * (1.0 / ((double) (b1 * b2)))));
		} else {
			double tmp_2;
			if ((((double) (a1 * a2)) <= 8.259474310150762e-200)) {
				tmp_2 = ((double) ((a1 / b1) * (a2 / b2)));
			} else {
				double tmp_3;
				if ((((double) (a1 * a2)) <= 6.232999595325203e+152)) {
					tmp_3 = ((((double) (a1 * a2)) / b2) / b1);
				} else {
					tmp_3 = (a1 / ((double) (b1 * (b2 / a2))));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.4
Herbie	5.9

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

Derivation

1. Split input into 5 regimes

2. if (* a1 a2) < -8.06593176736626629e116

   1. Initial program Error: 24.4 bits

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

   2. Simplified Error: 15.6 bits

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

   4. Applied clear-num Error: 15.8 bits

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

   5. Simplified Error: 10.8 bits

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

   if -8.06593176736626629e116 < (* a1 a2) < -2.58442496359607641e-115

   1. Initial program Error: 3.1 bits

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

   2. Simplified Error: 13.2 bits

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

   4. Applied div-inv Error: 13.3 bits

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

   5. Applied associate-*r* Error: 3.5 bits

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

   6. Simplified Error: 3.5 bits

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

   if -2.58442496359607641e-115 < (* a1 a2) < 8.25947431015076204e-200

   1. Initial program Error: 13.2 bits

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

   2. Simplified Error: 8.1 bits

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

   4. Applied *-un-lft-identity Error: 8.1 bits

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

   5. Applied times-frac Error: 5.5 bits

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

   6. Applied associate-*r* Error: 5.4 bits

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

   7. Simplified Error: 5.4 bits

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

   if 8.25947431015076204e-200 < (* a1 a2) < 6.2329995953252028e152

   1. Initial program Error: 4.0 bits

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

   2. Simplified Error: 12.0 bits

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

   4. Applied add-cube-cbrt Error: 12.7 bits

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

   5. Applied times-frac Error: 14.7 bits

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

   6. Applied associate-*r* Error: 11.7 bits

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

   7. Simplified Error: 11.7 bits

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

   9. Applied associate-*l/ Error: 11.7 bits

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

   10. Applied associate-*l/ Error: 6.7 bits

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

   11. Applied associate-*l/ Error: 6.3 bits

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

   12. Simplified Error: 4.2 bits

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

   if 6.2329995953252028e152 < (* a1 a2)

   1. Initial program Error: 30.3 bits

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

   2. Simplified Error: 18.2 bits

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

   4. Applied add-cube-cbrt Error: 18.8 bits

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

   5. Applied times-frac Error: 10.6 bits

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

   6. Applied associate-*r* Error: 13.8 bits

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

   7. Simplified Error: 13.8 bits

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

   9. Applied associate-*r/ Error: 15.7 bits

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

   10. Simplified Error: 15.4 bits

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

   12. Applied associate-/l* Error: 11.9 bits

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

   13. Simplified Error: 12.4 bits

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

4. Final simplification Error: 5.9 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -8.065931767366266 \cdot 10^{+116}:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b1}{a2} \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -2.5844249635960764 \cdot 10^{-115}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 8.259474310150762 \cdot 10^{-200}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq 6.232999595325203 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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