Quotient of products

Average Error: 10.9 → 2.9

Time: 2.5min

Precision: binary64

Cost: 3140

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -4.2694800775142445 \cdot 10^{+241}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1.1223206066020718 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}}{\sqrt[3]{\sqrt{-b1 \cdot b2}}} \cdot \left(\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)\right)}{\sqrt[3]{\sqrt{-b1 \cdot b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.330746503442893 \cdot 10^{-288}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 7.94098457665328 \cdot 10^{+210}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}}{\frac{\sqrt[3]{-b1 \cdot b2}}{\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* b1 b2) -4.2694800775142445e+241)
   (/ (* a2 (/ a1 b1)) b2)
   (if (<= (* b1 b2) -1.1223206066020718e-235)
     (/
      (*
       (/
        (* (cbrt a1) (/ (cbrt a1) (cbrt (- (* b1 b2)))))
        (cbrt (sqrt (- (* b1 b2)))))
       (* (/ (cbrt a1) (cbrt (- (* b1 b2)))) (- a2)))
      (cbrt (sqrt (- (* b1 b2)))))
     (if (<= (* b1 b2) 5.330746503442893e-288)
       (/ (* a2 (/ a1 b1)) b2)
       (if (<= (* b1 b2) 7.94098457665328e+210)
         (/
          (/ (* (cbrt a1) (cbrt a1)) (cbrt (- (* b1 b2))))
          (/
           (cbrt (- (* b1 b2)))
           (* (/ (cbrt a1) (cbrt (- (* b1 b2)))) (- a2))))
         (* (/ a1 b1) (/ 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 ((b1 * b2) <= -4.2694800775142445e+241) {
		tmp = (a2 * (a1 / b1)) / b2;
	} else if ((b1 * b2) <= -1.1223206066020718e-235) {
		tmp = (((cbrt(a1) * (cbrt(a1) / cbrt(-(b1 * b2)))) / cbrt(sqrt(-(b1 * b2)))) * ((cbrt(a1) / cbrt(-(b1 * b2))) * -a2)) / cbrt(sqrt(-(b1 * b2)));
	} else if ((b1 * b2) <= 5.330746503442893e-288) {
		tmp = (a2 * (a1 / b1)) / b2;
	} else if ((b1 * b2) <= 7.94098457665328e+210) {
		tmp = ((cbrt(a1) * cbrt(a1)) / cbrt(-(b1 * b2))) / (cbrt(-(b1 * b2)) / ((cbrt(a1) / cbrt(-(b1 * b2))) * -a2));
	} else {
		tmp = (a1 / b1) * (a2 / b2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.9
Target	11.2
Herbie	2.9

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

Alternative 1
Accuracy	2.9
Cost	3140

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -4.2694800775142445 \cdot 10^{+241}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1.1223206066020718 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}}{\sqrt[3]{\sqrt{-b1 \cdot b2}}} \cdot \left(\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)\right)}{\sqrt[3]{\sqrt{-b1 \cdot b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.330746503442893 \cdot 10^{-288}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 7.94098457665328 \cdot 10^{+210}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}\right) \cdot \frac{\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)}{\sqrt[3]{-b1 \cdot b2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Alternative 2
Accuracy	2.9
Cost	3078

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -4.2694800775142445 \cdot 10^{+241}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1.1223206066020718 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}}{\sqrt[3]{\sqrt{-b1 \cdot b2}}} \cdot \left(\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)\right)}{\sqrt[3]{\sqrt{-b1 \cdot b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.330746503442893 \cdot 10^{-288} \lor \neg \left(b1 \cdot b2 \leq 7.94098457665328 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}\right) \cdot \frac{\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)}{\sqrt[3]{-b1 \cdot b2}}\\ \end{array}\]

Alternative 3
Accuracy	2.8
Cost	3078

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -4.2694800775142445 \cdot 10^{+241}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1.1223206066020718 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}}{\sqrt[3]{\sqrt{-b1 \cdot b2}}} \cdot \frac{\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)}{\sqrt[3]{\sqrt{-b1 \cdot b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.330746503442893 \cdot 10^{-288} \lor \neg \left(b1 \cdot b2 \leq 7.94098457665328 \cdot 10^{+210}\right):\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}\right) \cdot \frac{\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)}{\sqrt[3]{-b1 \cdot b2}}\\ \end{array}\]

Alternative 4
Accuracy	8.0
Cost	2752

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

Alternative 5
Accuracy	8.1
Cost	2816

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

Alternative 6
Accuracy	8.4
Cost	1856

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

Alternative 7
Accuracy	8.5
Cost	1856

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

Alternative 8
Accuracy	10.7
Cost	1600

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 b1 b2) < -4.2694800775142445e241 or -1.1223206066020718e-235 < (*.f64 b1 b2) < 5.33074650344289276e-288

   1. Initial program 28.7

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

   3. Applied associate-/r*_binary64_3091 11.4

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

   4. Simplified 4.9

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

   if -4.2694800775142445e241 < (*.f64 b1 b2) < -1.1223206066020718e-235

   1. Initial program 4.6

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

   3. Applied frac-2neg_binary64_3158 4.6

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

   5. Applied add-cube-cbrt_binary64_3182 5.4

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

   6. Applied associate-/r*_binary64_3091 5.4

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

   7. Simplified 5.1

      \[\leadsto \frac{\color{blue}{\frac{a1}{\sqrt[3]{-b1 \cdot b2} \cdot \sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)}}{\sqrt[3]{-b1 \cdot b2}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt_binary64_3182 5.3

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

   10. Applied times-frac_binary64_3153 5.3

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

   11. Applied associate-*l*_binary64_3088 2.6

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

   12. Simplified 2.6

       \[\leadsto \frac{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \color{blue}{\left(\left(-a2\right) \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}\right)}}{\sqrt[3]{-b1 \cdot b2}}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt_binary64_3169 2.6

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

   15. Applied cbrt-prod_binary64_3178 2.6

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

   16. Applied associate-/r*_binary64_3091 2.6

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

   17. Simplified 1.9

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

   if 5.33074650344289276e-288 < (*.f64 b1 b2) < 7.94098457665328e210

   1. Initial program 4.5

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

   3. Applied frac-2neg_binary64_3158 4.5

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

   5. Applied add-cube-cbrt_binary64_3182 5.4

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

   6. Applied associate-/r*_binary64_3091 5.4

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

   7. Simplified 4.7

      \[\leadsto \frac{\color{blue}{\frac{a1}{\sqrt[3]{-b1 \cdot b2} \cdot \sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)}}{\sqrt[3]{-b1 \cdot b2}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt_binary64_3182 4.9

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

   10. Applied times-frac_binary64_3153 4.9

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

   11. Applied associate-*l*_binary64_3088 2.2

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

   12. Simplified 2.2

       \[\leadsto \frac{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \color{blue}{\left(\left(-a2\right) \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}\right)}}{\sqrt[3]{-b1 \cdot b2}}\]
   13. Using strategy rm

   14. Applied associate-/l*_binary64_3092 2.1

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

   if 7.94098457665328e210 < (*.f64 b1 b2)

   1. Initial program 15.8

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

   3. Applied times-frac_binary64_3153 4.7

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -4.2694800775142445 \cdot 10^{+241}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -1.1223206066020718 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}}{\sqrt[3]{\sqrt{-b1 \cdot b2}}} \cdot \left(\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)\right)}{\sqrt[3]{\sqrt{-b1 \cdot b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.330746503442893 \cdot 10^{-288}:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b1}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 7.94098457665328 \cdot 10^{+210}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}}}{\frac{\sqrt[3]{-b1 \cdot b2}}{\frac{\sqrt[3]{a1}}{\sqrt[3]{-b1 \cdot b2}} \cdot \left(-a2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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