Quotient of products

Average Error: 11.3 → 2.7

Time: 8.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.670346478966627 \cdot 10^{-291}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{1}{\left(b2 \cdot \frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{a1}\right) \cdot \frac{\sqrt[3]{b1}}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.4779041765760598 \cdot 10^{+294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) (- INFINITY))
   (/ (/ a1 b2) (/ b1 a2))
   (if (<= (/ (* a1 a2) (* b1 b2)) -2.670346478966627e-291)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (/ 1.0 (* (* b2 (/ (* (cbrt b1) (cbrt b1)) a1)) (/ (cbrt b1) a2)))
       (if (<= (/ (* a1 a2) (* b1 b2)) 2.4779041765760598e+294)
         (/ (* a1 a2) (* b1 b2))
         (/ (/ a1 b2) (/ b1 a2)))))))

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 / b2) / (b1 / a2);
	} else if (((a1 * a2) / (b1 * b2)) <= -2.670346478966627e-291) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 0.0) {
		tmp = 1.0 / ((b2 * ((cbrt(b1) * cbrt(b1)) / a1)) * (cbrt(b1) / a2));
	} else if (((a1 * a2) / (b1 * b2)) <= 2.4779041765760598e+294) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = (a1 / b2) / (b1 / a2);
	}
	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	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 or 2.47790417657605985e294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 61.3

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

   3. Applied clear-num_binary64_5874 61.3

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

   5. Applied associate-/l*_binary64_5820 44.2

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

   6. Simplified 15.1

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

   8. Applied associate-/r/_binary64_5821 14.8

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

   9. Applied *-un-lft-identity_binary64_5875 14.8

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

   10. Applied times-frac_binary64_5881 7.7

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

   11. Applied associate-/r*_binary64_5819 7.6

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

   12. Simplified 7.5

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.67034647896662683e-291 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.47790417657605985e294

   1. Initial program 0.9

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

   if -2.67034647896662683e-291 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.6

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

   3. Applied clear-num_binary64_5874 14.0

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

   5. Applied associate-/l*_binary64_5820 7.2

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

   6. Simplified 4.7

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

   8. Applied associate-/r/_binary64_5821 4.6

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

   9. Applied add-cube-cbrt_binary64_5910 4.8

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right) \cdot \sqrt[3]{b1}}}{\frac{a1}{b2} \cdot a2}}\]

   10. Applied times-frac_binary64_5881 4.4

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{a1}{b2}} \cdot \frac{\sqrt[3]{b1}}{a2}}}\]

   11. Simplified 4.4

       \[\leadsto \frac{1}{\color{blue}{\left(\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{a1} \cdot b2\right)} \cdot \frac{\sqrt[3]{b1}}{a2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.670346478966627 \cdot 10^{-291}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{1}{\left(b2 \cdot \frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{a1}\right) \cdot \frac{\sqrt[3]{b1}}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.4779041765760598 \cdot 10^{+294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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