Quotient of products

Average Error: 11.6 → 2.4

Time: 4.7s

Precision: binary64

\[[a1, a2]=\mathsf{sort}([a1, a2])\]

\[[b1, b2]=\mathsf{sort}([b1, b2])\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -4.9179948584692854 \cdot 10^{+302}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b1}}{b2}\\ \mathbf{elif}\;t_0 \leq -4.626874562349005 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a1}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;t_0 \leq 1.8486007324360962 \cdot 10^{+306}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -4.9179948584692854e+302)
     (* a2 (/ (/ a1 b1) b2))
     (if (<= t_0 -4.626874562349005e-299)
       t_0
       (if (<= t_0 0.0)
         (* (/ a2 (* (cbrt b2) (cbrt b2))) (/ (/ a1 b1) (cbrt b2)))
         (if (<= t_0 1.8486007324360962e+306) t_0 (* (/ 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 t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -4.9179948584692854e+302) {
		tmp = a2 * ((a1 / b1) / b2);
	} else if (t_0 <= -4.626874562349005e-299) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a2 / (cbrt(b2) * cbrt(b2))) * ((a1 / b1) / cbrt(b2));
	} else if (t_0 <= 1.8486007324360962e+306) {
		tmp = t_0;
	} 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	11.6
Target	11.6
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)) < -4.9179948584692854e302

   1. Initial program 62.4

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

   2. Applied associate-/r*_binary64 32.6

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

   3. Simplified 15.9

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

   4. Applied *-un-lft-identity_binary64 15.9

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

   5. Applied times-frac_binary64 16.7

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

   if -4.9179948584692854e302 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.62687456234900487e-299 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.84860073243609623e306

   1. Initial program 0.8

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

   if -4.62687456234900487e-299 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.5

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

   2. Applied associate-/r*_binary64 6.2

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

   3. Simplified 3.9

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

   4. Applied add-cube-cbrt_binary64 4.0

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

   5. Applied times-frac_binary64 2.8

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

   if 1.84860073243609623e306 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 63.6

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

   2. Applied times-frac_binary64 5.9

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.9179948584692854 \cdot 10^{+302}:\\ \;\;\;\;a2 \cdot \frac{\frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.626874562349005 \cdot 10^{-299}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a1}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.8486007324360962 \cdot 10^{+306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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