Quotient of products

Average Error: 11.5 → 5.0

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.8142638068678418 \cdot 10^{-179}:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)} \cdot \frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.479436261727929 \cdot 10^{+303}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)} \cdot \frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}}}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) -2.8142638068678418e-179)
   (/
    1.0
    (*
     (/
      (* (cbrt b1) (cbrt b1))
      (* (* (cbrt a1) (cbrt a1)) (* (cbrt a2) (cbrt a2))))
     (/ (cbrt b1) (/ (cbrt a1) (/ b2 (cbrt a2))))))
   (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
     (/ (/ a1 b1) (/ b2 a2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 3.479436261727929e+303)
       (/ 1.0 (/ (* b1 b2) (* a1 a2)))
       (/
        1.0
        (*
         (/
          (* (cbrt b1) (cbrt b1))
          (* (* (cbrt a1) (cbrt a1)) (* (cbrt a2) (cbrt a2))))
         (/ (cbrt b1) (/ (cbrt a1) (/ b2 (cbrt 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)) <= -2.8142638068678418e-179) {
		tmp = 1.0 / (((cbrt(b1) * cbrt(b1)) / ((cbrt(a1) * cbrt(a1)) * (cbrt(a2) * cbrt(a2)))) * (cbrt(b1) / (cbrt(a1) / (b2 / cbrt(a2)))));
	} else if (((a1 * a2) / (b1 * b2)) <= 0.0) {
		tmp = (a1 / b1) / (b2 / a2);
	} else if (((a1 * a2) / (b1 * b2)) <= 3.479436261727929e+303) {
		tmp = 1.0 / ((b1 * b2) / (a1 * a2));
	} else {
		tmp = 1.0 / (((cbrt(b1) * cbrt(b1)) / ((cbrt(a1) * cbrt(a1)) * (cbrt(a2) * cbrt(a2)))) * (cbrt(b1) / (cbrt(a1) / (b2 / cbrt(a2)))));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.7
Herbie	5.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.8142638068678418e-179 or 3.47943626172792874e303 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 20.0

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

   3. Applied clear-num_binary64 20.1

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

   5. Applied associate-/l*_binary64 18.2

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

   6. Simplified 14.2

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

   8. Applied add-cube-cbrt_binary64 15.0

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

   9. Applied *-un-lft-identity_binary64 15.0

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

   10. Applied times-frac_binary64 15.0

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

   11. Applied add-cube-cbrt_binary64 15.2

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

   12. Applied times-frac_binary64 12.2

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

   13. Applied add-cube-cbrt_binary64 12.4

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

   14. Applied times-frac_binary64 7.5

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

   15. Simplified 7.5

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

   if -2.8142638068678418e-179 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 11.7

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

   3. Applied clear-num_binary64 12.0

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

   5. Applied times-frac_binary64 6.3

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

   6. Applied associate-/r*_binary64 5.8

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

   7. Simplified 5.5

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 3.47943626172792874e303

   1. Initial program 0.9

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

   3. Applied clear-num_binary64 1.4

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

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.8142638068678418 \cdot 10^{-179}:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)} \cdot \frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.479436261727929 \cdot 10^{+303}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)} \cdot \frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a1}}{\frac{b2}{\sqrt[3]{a2}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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