Quotient of products

Average Error: 10.8 → 2.9

Time: 10.3s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{1}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a1}}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.5261378074277998 \cdot 10^{+292}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{b2}}}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
   (*
    (/
     (* (cbrt a1) (cbrt a1))
     (/ 1.0 (/ (* (cbrt a2) (cbrt a2)) (* (cbrt b2) (cbrt b2)))))
    (/ (cbrt a1) (/ b1 (/ (cbrt a2) (cbrt b2)))))
   (if (<= (/ (* a1 a2) (* b1 b2)) 3.5261378074277998e+292)
     (/ (* a1 a2) (* b1 b2))
     (*
      (/ (* (cbrt a2) (cbrt a2)) (* (cbrt b1) (cbrt b1)))
      (/ a1 (/ (cbrt b1) (/ (cbrt 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)) <= 0.0) {
		tmp = ((cbrt(a1) * cbrt(a1)) / (1.0 / ((cbrt(a2) * cbrt(a2)) / (cbrt(b2) * cbrt(b2))))) * (cbrt(a1) / (b1 / (cbrt(a2) / cbrt(b2))));
	} else if (((a1 * a2) / (b1 * b2)) <= 3.5261378074277998e+292) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = ((cbrt(a2) * cbrt(a2)) / (cbrt(b1) * cbrt(b1))) * (a1 / (cbrt(b1) / (cbrt(a2) / b2)));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.8
Target	11.4
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 9.9

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

   3. Applied associate-/l*_binary64 9.0

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

   4. Simplified 9.7

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

   6. Applied add-cube-cbrt_binary64 10.1

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

   7. Applied add-cube-cbrt_binary64 10.3

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

   8. Applied times-frac_binary64 10.3

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

   9. Applied *-un-lft-identity_binary64 10.3

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

   10. Applied times-frac_binary64 7.4

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

   11. Applied add-cube-cbrt_binary64 7.5

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

   12. Applied times-frac_binary64 3.4

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

   if 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 3.52613780742779975e292

   1. Initial program 0.7

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

   if 3.52613780742779975e292 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 60.5

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

   3. Applied associate-/l*_binary64 44.8

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

   4. Simplified 13.5

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

   6. Applied *-un-lft-identity_binary64 13.5

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

   7. Applied add-cube-cbrt_binary64 14.3

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

   8. Applied times-frac_binary64 14.3

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

   9. Applied add-cube-cbrt_binary64 14.5

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

   10. Applied times-frac_binary64 12.8

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

   11. Applied *-un-lft-identity_binary64 12.8

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

   12. Applied times-frac_binary64 6.9

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

   13. Simplified 6.9

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{1}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a1}}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.5261378074277998 \cdot 10^{+292}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{b2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021126 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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