Quotient of products

Average Error: 11.3 → 3.3

Time: 3.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.0886227990026322 \cdot 10^{+265}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.9315844250235 \cdot 10^{-317}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.673913532993364 \cdot 10^{+303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b2 \cdot \frac{b1}{a1}}{a2}}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) -3.0886227990026322e+265)
   (/ a2 (* b1 (/ b2 a1)))
   (if (<= (/ (* a1 a2) (* b1 b2)) -4.9315844250235e-317)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (/ a1 (/ b1 (/ a2 b2)))
       (if (<= (/ (* a1 a2) (* b1 b2)) 1.673913532993364e+303)
         (/ (* a1 a2) (* b1 b2))
         (/ 1.0 (/ (* b2 (/ b1 a1)) a2)))))))

C

double code(double a1, double a2, double b1, double b2) {
	return (((double) (a1 * a2)) / ((double) (b1 * b2)));
}

↓

double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -3.0886227990026322e+265)) {
		tmp = (a2 / ((double) (b1 * (b2 / a1))));
	} else {
		double tmp_1;
		if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= -4.9315844250235e-317)) {
			tmp_1 = (((double) (a1 * a2)) / ((double) (b1 * b2)));
		} else {
			double tmp_2;
			if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 0.0)) {
				tmp_2 = (a1 / (b1 / (a2 / b2)));
			} else {
				double tmp_3;
				if (((((double) (a1 * a2)) / ((double) (b1 * b2))) <= 1.673913532993364e+303)) {
					tmp_3 = (((double) (a1 * a2)) / ((double) (b1 * b2)));
				} else {
					tmp_3 = (1.0 / (((double) (b2 * (b1 / a1))) / a2));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.7
Herbie	3.3

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.08862279900263218e265

   1. Initial program 44.7

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

   3. Applied clear-num_binary64 44.7

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

   5. Applied associate-/r*_binary64 25.2

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

   6. Simplified 16.5

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

   7. Taylor expanded around 0 44.7

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

   8. Simplified 16.5

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

   10. Applied div-inv_binary64 16.6

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

   11. Applied associate-*l*_binary64 16.9

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

   12. Simplified 16.8

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

   if -3.08862279900263218e265 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.93158443e-317 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.673913532993364e303

   1. Initial program 0.8

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

   if -4.93158443e-317 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.2

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

   3. Applied associate-/l*_binary64 7.2

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

   4. Simplified 3.9

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

   if 1.673913532993364e303 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 62.8

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

   3. Applied clear-num_binary64 62.8

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

   5. Applied associate-/r*_binary64 45.6

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

   6. Simplified 11.7

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{b1}{a1} \cdot b2}}{a2}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.0886227990026322 \cdot 10^{+265}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.9315844250235 \cdot 10^{-317}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.673913532993364 \cdot 10^{+303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b2 \cdot \frac{b1}{a1}}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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