Quotient of products

Average Error: 11.3 → 2.4

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.12917095165229 \cdot 10^{-304}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a2}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.4409601181936536 \cdot 10^{+266}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) (- INFINITY))
   (* (/ a1 b1) (/ a2 b2))
   (if (<= (/ (* a1 a2) (* b1 b2)) -1.12917095165229e-304)
     (/ (* 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)) 4.4409601181936536e+266)
         (/ (* a1 a2) (* b1 b2))
         (* (/ a2 b1) (/ a1 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)) <= -((double) INFINITY)) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (((a1 * a2) / (b1 * b2)) <= -1.12917095165229e-304) {
		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)) <= 4.4409601181936536e+266) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = (a2 / b1) * (a1 / 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	11.2
Herbie	2.4

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied times-frac_binary64_5540 9.9

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.12917095165229e-304 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 4.4409601181936536e266

   1. Initial program 0.8

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

   if -1.12917095165229e-304 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.7

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

   3. Applied associate-/r*_binary64_5478 6.5

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

   4. Simplified 3.4

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

   6. Applied add-cube-cbrt_binary64_5569 3.6

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

   7. Applied *-un-lft-identity_binary64_5534 3.6

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

   8. Applied times-frac_binary64_5540 3.6

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

   9. Applied associate-*r*_binary64_5474 3.1

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

   10. Simplified 3.1

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

   if 4.4409601181936536e266 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 54.8

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

   3. Applied associate-/l*_binary64_5479 42.6

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

   4. Simplified 15.1

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

   6. Applied associate-/r/_binary64_5480 14.5

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

   7. Applied *-un-lft-identity_binary64_5534 14.5

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

   8. Applied times-frac_binary64_5540 8.9

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

   9. Simplified 8.8

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.12917095165229 \cdot 10^{-304}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a2}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.4409601181936536 \cdot 10^{+266}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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