Quotient of products

Average Error: 11.4 → 2.3

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.7094671346107 \cdot 10^{-321}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.9859388062377627 \cdot 10^{+301}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

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 (<= (/ (* a1 a2) (* b1 b2)) -1.7094671346107e-321)
     (* (* a1 a2) (/ (/ 1.0 b1) b2))
     (if (or (<= (/ (* a1 a2) (* b1 b2)) 0.0)
             (not (<= (/ (* a1 a2) (* b1 b2)) 3.9859388062377627e+301)))
       (* (/ a1 b1) (/ a2 b2))
       (/ (* a1 a2) (* b1 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)) <= -1.7094671346107e-321) {
		tmp = (a1 * a2) * ((1.0 / b1) / b2);
	} else if ((((a1 * a2) / (b1 * b2)) <= 0.0) || !(((a1 * a2) / (b1 * b2)) <= 3.9859388062377627e+301)) {
		tmp = (a1 / b1) * (a2 / b2);
	} else {
		tmp = (a1 * a2) / (b1 * b2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.5
Herbie	2.3

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

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 associate-/r*_binary64_6160 35.1

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

   4. Simplified 17.4

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.70947e-321

   1. Initial program 0.7

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

   3. Applied div-inv_binary64_6213 0.9

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}}\]
   4. Using strategy rm

   5. Applied associate-/r*_binary64_6160 0.9

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

   if -1.70947e-321 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0 or 3.9859388062377627e301 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 22.8

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

   3. Applied times-frac_binary64_6222 3.0

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 3.9859388062377627e301

   1. Initial program 0.9

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.7094671346107 \cdot 10^{-321}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.9859388062377627 \cdot 10^{+301}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020357 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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