Quotient of products

Average Error: 11.5 → 2.5

Time: 4.8s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.886776762190504 \cdot 10^{-308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.3437713114153212 \cdot 10^{+299}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \end{array}\]

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) (/ a2 b2))
   (if (<= (/ (* a1 a2) (* b1 b2)) -5.886776762190504e-308)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (/ (* a1 (/ a2 b1)) b2)
       (if (<= (/ (* a1 a2) (* b1 b2)) 2.3437713114153212e+299)
         (/ (* a1 a2) (* b1 b2))
         (* (/ a1 (* (cbrt b2) (cbrt b2))) (/ (/ a2 b1) (cbrt 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) * (a2 / b2);
	} else if (((a1 * a2) / (b1 * b2)) <= -5.886776762190504e-308) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 0.0) {
		tmp = (a1 * (a2 / b1)) / b2;
	} else if (((a1 * a2) / (b1 * b2)) <= 2.3437713114153212e+299) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = (a1 / (cbrt(b2) * cbrt(b2))) * ((a2 / b1) / cbrt(b2));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.4
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 times-frac_binary64 12.8

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.88677676219050384e-308 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.343771311415321e299

   1. Initial program 0.8

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

   if -5.88677676219050384e-308 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.0

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

   3. Applied associate-/r*_binary64 6.4

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

   4. Simplified 3.1

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

   6. Applied div-inv_binary64 3.1

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

   7. Applied associate-*r*_binary64 6.4

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

   8. Simplified 6.4

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

   10. Applied associate-*l*_binary64 3.3

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

   11. Simplified 3.3

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

   13. Applied *-un-lft-identity_binary64 3.3

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

   if 2.343771311415321e299 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 62.3

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

   3. Applied associate-/r*_binary64 47.5

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

   4. Simplified 13.0

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

   6. Applied div-inv_binary64 13.0

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

   7. Applied associate-*r*_binary64 47.6

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

   8. Simplified 47.6

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

   10. Applied associate-*l*_binary64 12.6

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

   11. Simplified 12.5

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

   13. Applied add-cube-cbrt_binary64 13.3

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

   14. Applied times-frac_binary64 6.5

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.886776762190504 \cdot 10^{-308}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.3437713114153212 \cdot 10^{+299}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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