Quotient of products

Average Error: 11.6 → 5.0

Time: 8.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.4931232845681356 \cdot 10^{168}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.0866604942007535 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.01600999395086732 \cdot 10^{-257}:\\ \;\;\;\;\frac{\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.59608712487395094 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \end{array}\]

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) <= -1.4931232845681356e+168)) {
		temp = ((a1 / b1) * (a2 / b2));
	} else {
		double temp_1;
		if (((a1 * a2) <= -2.0866604942007535e-124)) {
			temp_1 = (1.0 / (b2 / ((a1 * a2) / b1)));
		} else {
			double temp_2;
			if (((a1 * a2) <= 1.0160099939508673e-257)) {
				temp_2 = (((((a1 / (cbrt(b1) * cbrt(b1))) / (cbrt(b2) * cbrt(b2))) * (cbrt(a2) * cbrt(a2))) / cbrt((cbrt(b2) * cbrt(b2)))) * ((cbrt(a2) / cbrt(b1)) / cbrt(cbrt(b2))));
			} else {
				double temp_3;
				if (((a1 * a2) <= 1.596087124873951e+126)) {
					temp_3 = (1.0 / (b2 / ((a1 * a2) / b1)));
				} else {
					temp_3 = (((a1 / (cbrt(b1) * cbrt(b1))) / (cbrt(b2) * cbrt(b2))) * ((a2 / cbrt(b1)) / cbrt(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	11.6
Target	11.3
Herbie	5.0

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -1.4931232845681356e+168

   1. Initial program 30.4

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

   3. Applied times-frac 12.5

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

   if -1.4931232845681356e+168 < (* a1 a2) < -2.0866604942007535e-124 or 1.0160099939508673e-257 < (* a1 a2) < 1.596087124873951e+126

   1. Initial program 4.0

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

   3. Applied associate-/r* 3.7

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

   5. Applied clear-num 4.1

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

   if -2.0866604942007535e-124 < (* a1 a2) < 1.0160099939508673e-257

   1. Initial program 14.7

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

   3. Applied associate-/r* 14.7

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

   5. Applied add-cube-cbrt 15.0

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

   6. Applied add-cube-cbrt 15.1

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

   7. Applied times-frac 8.4

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

   8. Applied times-frac 3.5

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

   10. Applied add-cube-cbrt 3.5

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

   11. Applied cbrt-prod 3.6

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

   12. Applied *-un-lft-identity 3.6

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

   13. Applied add-cube-cbrt 3.6

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

   14. Applied times-frac 3.6

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

   15. Applied times-frac 3.0

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

   16. Applied associate-*r* 3.0

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

   17. Simplified 3.0

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

   if 1.596087124873951e+126 < (* a1 a2)

   1. Initial program 25.3

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

   3. Applied associate-/r* 27.1

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

   5. Applied add-cube-cbrt 27.7

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

   6. Applied add-cube-cbrt 27.8

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

   7. Applied times-frac 17.4

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

   8. Applied times-frac 9.4

      \[\leadsto \color{blue}{\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.4931232845681356 \cdot 10^{168}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.0866604942007535 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.01600999395086732 \cdot 10^{-257}:\\ \;\;\;\;\frac{\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.59608712487395094 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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