Quotient of products

Average Error: 11.8 → 2.1

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b2}\right) \cdot \frac{1}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.3109376793505 \cdot 10^{-312}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 8.994764357597508 \cdot 10^{+307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \left(\frac{a2}{b2} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{b1}}\right)\\ \end{array}\]

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))
   (* (* a1 (/ a2 b2)) (/ 1.0 b1))
   (if (<= (/ (* a1 a2) (* b1 b2)) -1.3109376793505e-312)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (* (/ a2 b2) (/ a1 b1))
       (if (<= (/ (* a1 a2) (* b1 b2)) 8.994764357597508e+307)
         (/ (* a1 a2) (* b1 b2))
         (*
          (/ (* (cbrt a1) (cbrt a1)) (* (cbrt b1) (cbrt b1)))
          (* (/ a2 b2) (/ (cbrt a1) (cbrt b1)))))))))

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 = (a1 * (a2 / b2)) * (1.0 / b1);
	} else if (((a1 * a2) / (b1 * b2)) <= -1.3109376793505e-312) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 0.0) {
		tmp = (a2 / b2) * (a1 / b1);
	} else if (((a1 * a2) / (b1 * b2)) <= 8.994764357597508e+307) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = ((cbrt(a1) * cbrt(a1)) / (cbrt(b1) * cbrt(b1))) * ((a2 / b2) * (cbrt(a1) / cbrt(b1)));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.8
Target	11.3
Herbie	2.1

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

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 times-frac_binary64 10.7

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

   5. Applied associate-*l/_binary64 19.5

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

   7. Applied div-inv_binary64 19.5

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.3109376793505e-312 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 8.9947643575975084e307

   1. Initial program 0.7

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

   if -1.3109376793505e-312 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 12.9

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

   3. Applied times-frac_binary64 2.2

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

   if 8.9947643575975084e307 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 63.9

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

   3. Applied times-frac_binary64 5.8

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

   5. Applied add-cube-cbrt_binary64 6.8

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

   6. Applied add-cube-cbrt_binary64 6.9

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

   7. Applied times-frac_binary64 7.0

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

   8. Applied associate-*l*_binary64 4.2

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

   9. Simplified 4.2

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b2}\right) \cdot \frac{1}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.3109376793505 \cdot 10^{-312}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 8.994764357597508 \cdot 10^{+307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \left(\frac{a2}{b2} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{b1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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