Quotient of products

Average Error: 11.9 → 1.8

Time: 7.1s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.217247420283257 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.300420913112 \cdot 10^{-310}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 6.2999843586282166 \cdot 10^{+302}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{\frac{b1}{\frac{1}{b2}}}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) -4.217247420283257e+302)
   (*
    (/
     1.0
     (/
      (* (cbrt b1) (cbrt b1))
      (/ (* (cbrt a2) (cbrt a2)) (* (cbrt b2) (cbrt b2)))))
    (/ a1 (/ (cbrt b1) (/ (cbrt a2) (cbrt b2)))))
   (if (<= (/ (* a1 a2) (* b1 b2)) -2.300420913112e-310)
     (/ (* a1 a2) (* b1 b2))
     (if (or (<= (/ (* a1 a2) (* b1 b2)) 0.0)
             (not (<= (/ (* a1 a2) (* b1 b2)) 6.2999843586282166e+302)))
       (* (/ a1 b1) (/ a2 b2))
       (/ (* a1 a2) (/ b1 (/ 1.0 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)) <= -4.217247420283257e+302) {
		tmp = (1.0 / ((cbrt(b1) * cbrt(b1)) / ((cbrt(a2) * cbrt(a2)) / (cbrt(b2) * cbrt(b2))))) * (a1 / (cbrt(b1) / (cbrt(a2) / cbrt(b2))));
	} else if (((a1 * a2) / (b1 * b2)) <= -2.300420913112e-310) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if ((((a1 * a2) / (b1 * b2)) <= 0.0) || !(((a1 * a2) / (b1 * b2)) <= 6.2999843586282166e+302)) {
		tmp = (a1 / b1) * (a2 / b2);
	} else {
		tmp = (a1 * a2) / (b1 / (1.0 / b2));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.9
Target	11.1
Herbie	1.8

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.217247420283257e302

   1. Initial program 60.6

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

   3. Applied associate-/l*_binary64 27.4

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

   4. Simplified 17.7

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

   6. Applied add-cube-cbrt_binary64 18.5

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

   7. Applied add-cube-cbrt_binary64 18.6

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

   8. Applied times-frac_binary64 18.6

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

   9. Applied add-cube-cbrt_binary64 18.7

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

   10. Applied times-frac_binary64 14.5

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

   11. Applied *-un-lft-identity_binary64 14.5

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

   12. Applied times-frac_binary64 6.0

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

   if -4.217247420283257e302 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.300420913112001e-310

   1. Initial program 0.7

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

   if -2.300420913112001e-310 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0 or 6.2999843586282166e302 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 24.0

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

   3. Applied times-frac_binary64 3.0

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

   if 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 6.2999843586282166e302

   1. Initial program 0.8

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

   3. Applied associate-/l*_binary64 8.0

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

   4. Simplified 14.2

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

   6. Applied div-inv_binary64 14.3

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

   7. Applied *-un-lft-identity_binary64 14.3

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

   8. Applied times-frac_binary64 8.1

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

   9. Applied associate-/r*_binary64 0.9

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

   10. Simplified 0.8

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -4.217247420283257 \cdot 10^{+302}:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.300420913112 \cdot 10^{-310}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 6.2999843586282166 \cdot 10^{+302}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{\frac{b1}{\frac{1}{b2}}}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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