Quotient of products

Average Error: 10.9 → 5.2

Time: 2.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -5.562117873525825 \cdot 10^{+190}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1.6952518629488907 \cdot 10^{-183}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 8.602609044046132 \cdot 10^{-236} \lor \neg \left(b1 \cdot b2 \leq 1.3186271336181104 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* b1 b2) -5.562117873525825e+190)
   (* (/ a1 b1) (/ a2 b2))
   (if (<= (* b1 b2) -1.6952518629488907e-183)
     (/ (* a1 a2) (* b1 b2))
     (if (or (<= (* b1 b2) 8.602609044046132e-236)
             (not (<= (* b1 b2) 1.3186271336181104e+160)))
       (/ (* (/ 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 ((b1 * b2) <= -5.562117873525825e+190) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if ((b1 * b2) <= -1.6952518629488907e-183) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((b1 * b2) <= 8.602609044046132e-236) || !((b1 * b2) <= 1.3186271336181104e+160)) {
		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	10.9
Target	11.7
Herbie	5.2

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 b1 b2) < -5.5621178735258251e190

   1. Initial program 14.8

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

   3. Applied times-frac_binary64_3847 4.4

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

   if -5.5621178735258251e190 < (*.f64 b1 b2) < -1.6952518629488907e-183

   1. Initial program 3.9

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

   if -1.6952518629488907e-183 < (*.f64 b1 b2) < 8.6026090440461317e-236 or 1.3186271336181104e160 < (*.f64 b1 b2)

   1. Initial program 21.8

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

   3. Applied associate-/r*_binary64_3785 11.5

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

   4. Simplified 7.9

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

   if 8.6026090440461317e-236 < (*.f64 b1 b2) < 1.3186271336181104e160

   1. Initial program 4.6

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

   3. Applied div-inv_binary64_3838 4.6

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

   5. Applied associate-*l*_binary64_3782 4.1

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

   6. Simplified 4.0

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

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -5.562117873525825 \cdot 10^{+190}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1.6952518629488907 \cdot 10^{-183}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 8.602609044046132 \cdot 10^{-236} \lor \neg \left(b1 \cdot b2 \leq 1.3186271336181104 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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