Quotient of products

Average Error: 11.3 → 3.1

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.282096742419842 \cdot 10^{+229}:\\ \;\;\;\;\frac{a2}{\frac{b2}{\frac{a1}{b1}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.478366537771996 \cdot 10^{-297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{\frac{a1}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.988148331797139 \cdot 10^{+259}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) -1.282096742419842e+229)
   (/ a2 (/ b2 (/ a1 b1)))
   (if (<= (/ (* a1 a2) (* b1 b2)) -3.478366537771996e-297)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (* (/ a2 (* (cbrt b1) (cbrt b1))) (/ (/ a1 (cbrt b1)) b2))
       (if (<= (/ (* a1 a2) (* b1 b2)) 1.988148331797139e+259)
         (/ (* a1 a2) (* b1 b2))
         (* (/ a1 b1) (/ a2 b2)))))))

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 tmp;
	if (((a1 * a2) / (b1 * b2)) <= -1.282096742419842e+229) {
		tmp = a2 / (b2 / (a1 / b1));
	} else if (((a1 * a2) / (b1 * b2)) <= -3.478366537771996e-297) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 0.0) {
		tmp = (a2 / (cbrt(b1) * cbrt(b1))) * ((a1 / cbrt(b1)) / b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 1.988148331797139e+259) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = (a1 / b1) * (a2 / b2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.0
Herbie	3.1

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.2820967424198419e229

   1. Initial program 37.5

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

   3. Applied associate-/r*_binary64_6501 24.8

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

   4. Simplified 17.0

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

   6. Applied associate-/l*_binary64_6502 17.8

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

   if -1.2820967424198419e229 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.4783665377719959e-297 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.9881483317971391e259

   1. Initial program 0.9

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

   if -3.4783665377719959e-297 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.0

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

   3. Applied associate-/r*_binary64_6501 6.5

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

   4. Simplified 3.9

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

   6. Applied *-un-lft-identity_binary64_6557 3.9

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

   7. Applied times-frac_binary64_6563 4.0

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

   8. Simplified 4.0

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

   10. Applied *-un-lft-identity_binary64_6557 4.0

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

   11. Applied add-cube-cbrt_binary64_6592 4.1

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

   12. Applied *-un-lft-identity_binary64_6557 4.1

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

   13. Applied times-frac_binary64_6563 4.1

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

   14. Applied times-frac_binary64_6563 3.9

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

   15. Applied associate-*r*_binary64_6497 2.3

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

   16. Simplified 2.3

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

   if 1.9881483317971391e259 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 53.7

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

   3. Applied times-frac_binary64_6563 10.2

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.282096742419842 \cdot 10^{+229}:\\ \;\;\;\;\frac{a2}{\frac{b2}{\frac{a1}{b1}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.478366537771996 \cdot 10^{-297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{\frac{a1}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.988148331797139 \cdot 10^{+259}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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