Quotient of products

Average Error: 11.6 → 2.5

Time: 11.5s

Precision: binary64

\[[a1, a2]=\mathsf{sort}([a1, a2])\]

\[[b1, b2]=\mathsf{sort}([b1, b2])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.125263103070185 \cdot 10^{-282}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot \frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.355807106222864 \cdot 10^{+289}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{1}} \cdot \frac{a2}{b2}\\ \end{array}\]

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))
   (/ (/ a1 b1) (/ b2 a2))
   (if (<= (/ (* a1 a2) (* b1 b2)) -4.125263103070185e-282)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (/ (* (/ a1 b1) (/ a2 (* (cbrt b2) (cbrt b2)))) (cbrt b2))
       (if (<= (/ (* a1 a2) (* b1 b2)) 4.355807106222864e+289)
         (/ (* a1 a2) (* b1 b2))
         (* (/ (/ a1 b1) (cbrt 1.0)) (/ a2 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 = (a1 / b1) / (b2 / a2);
	} else if (((a1 * a2) / (b1 * b2)) <= -4.125263103070185e-282) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 0.0) {
		tmp = ((a1 / b1) * (a2 / (cbrt(b2) * cbrt(b2)))) / cbrt(b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 4.355807106222864e+289) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = ((a1 / b1) / cbrt(1.0)) * (a2 / b2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.6
Target	11.6
Herbie	2.5

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

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_3091 35.0

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

   4. Simplified 18.6

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

   6. Applied add-cube-cbrt_binary64_3182 19.4

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

   7. Applied associate-/r*_binary64_3091 19.4

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

   8. Simplified 14.2

      \[\leadsto \frac{\color{blue}{\frac{a1}{b1} \cdot \frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}}{\sqrt[3]{b2}}\]
   9. Using strategy rm

   10. Applied associate-/l*_binary64_3092 12.6

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

   11. Simplified 11.5

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.1252631030701847e-282 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 4.3558071062228637e289

   1. Initial program 0.7

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

   if -4.1252631030701847e-282 < (/.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 associate-/r*_binary64_3091 6.7

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

   4. Simplified 3.8

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

   6. Applied add-cube-cbrt_binary64_3182 4.0

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

   7. Applied associate-/r*_binary64_3091 4.0

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

   8. Simplified 3.2

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

   if 4.3558071062228637e289 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 58.9

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

   3. Applied associate-/r*_binary64_3091 47.0

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

   4. Simplified 14.8

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

   6. Applied add-cube-cbrt_binary64_3182 15.6

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

   7. Applied associate-/r*_binary64_3091 15.6

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

   8. Simplified 9.2

      \[\leadsto \frac{\color{blue}{\frac{a1}{b1} \cdot \frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}}{\sqrt[3]{b2}}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity_binary64_3147 9.2

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

   11. Applied cbrt-prod_binary64_3178 9.2

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

   12. Applied times-frac_binary64_3153 9.5

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

   13. Simplified 8.7

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.125263103070185 \cdot 10^{-282}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot \frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.355807106222864 \cdot 10^{+289}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{1}} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021091 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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