Quotient of products

Average Error: 11.7 → 2.7

Time: 5.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.7805680563426 \cdot 10^{-322} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.3879797649998844 \cdot 10^{-268}\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.041374688326402 \cdot 10^{+308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

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))
   (/ (* a2 (/ a1 b1)) b2)
   (if (or (<= (/ (* a1 a2) (* b1 b2)) -5.7805680563426e-322)
           (and (not (<= (/ (* a1 a2) (* b1 b2)) 1.3879797649998844e-268))
                (<= (/ (* a1 a2) (* b1 b2)) 1.041374688326402e+308)))
     (/ (* 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 = (a2 * (a1 / b1)) / b2;
	} else if ((((a1 * a2) / (b1 * b2)) <= -5.7805680563426e-322) || (!(((a1 * a2) / (b1 * b2)) <= 1.3879797649998844e-268) && (((a1 * a2) / (b1 * b2)) <= 1.041374688326402e+308))) {
		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	11.7
Target	11.2
Herbie	2.7

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

Derivation

1. Split input into 3 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 associate-/r*_binary64_2719 37.2

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

   4. Simplified 19.0

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.78057e-322 or 1.3879797649998844e-268 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.04137468832640207e308

   1. Initial program 0.9

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

   if -5.78057e-322 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.3879797649998844e-268 or 1.04137468832640207e308 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 22.2

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

   3. Applied associate-/l*_binary64_2720 14.7

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

   4. Simplified 5.4

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

   6. Applied associate-/r/_binary64_2721 5.9

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

   7. Applied *-un-lft-identity_binary64_2773 5.9

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

   8. Applied times-frac_binary64_2779 3.8

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

   9. Simplified 3.7

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.7805680563426 \cdot 10^{-322} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.3879797649998844 \cdot 10^{-268}\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.041374688326402 \cdot 10^{+308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020303 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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