Quotient of products

Average Error: 11.6 → 2.8

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right)} \cdot \frac{\sqrt[3]{a2}}{\frac{\sqrt[3]{b2}}{\frac{a1}{\sqrt[3]{b1}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.19219955785124 \cdot 10^{-227} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 8.655475196185196 \cdot 10^{+273}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

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))
   (*
    (/
     (* (cbrt a2) (cbrt a2))
     (* (* (cbrt b2) (cbrt b2)) (* (cbrt b1) (cbrt b1))))
    (/ (cbrt a2) (/ (cbrt b2) (/ a1 (cbrt b1)))))
   (if (or (<= (/ (* a1 a2) (* b1 b2)) -3.19219955785124e-227)
           (and (not (<= (/ (* a1 a2) (* b1 b2)) 0.0))
                (<= (/ (* a1 a2) (* b1 b2)) 8.655475196185196e+273)))
     (/ (* a1 a2) (* b1 b2))
     (* (/ a1 b2) (/ a2 b1)))))

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 = ((cbrt(a2) * cbrt(a2)) / ((cbrt(b2) * cbrt(b2)) * (cbrt(b1) * cbrt(b1)))) * (cbrt(a2) / (cbrt(b2) / (a1 / cbrt(b1))));
	} else if ((((a1 * a2) / (b1 * b2)) <= -3.19219955785124e-227) || (!(((a1 * a2) / (b1 * b2)) <= 0.0) && (((a1 * a2) / (b1 * b2)) <= 8.655475196185196e+273))) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = (a1 / b2) * (a2 / b1);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.6
Target	11.3
Herbie	2.8

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

Derivation

1. Split input into 3 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 associate-/r*_binary64_2750 33.0

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

   4. Simplified 19.6

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

   6. Applied associate-/l*_binary64_2751 16.8

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

   8. Applied add-cube-cbrt_binary64_2841 17.5

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

   9. Applied *-un-lft-identity_binary64_2806 17.5

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

   10. Applied times-frac_binary64_2812 17.5

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

   11. Applied add-cube-cbrt_binary64_2841 17.8

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

   12. Applied times-frac_binary64_2812 17.7

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

   13. Applied add-cube-cbrt_binary64_2841 17.8

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

   14. Applied times-frac_binary64_2812 9.8

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

   15. Simplified 9.8

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.1921995578512398e-227 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 8.6554751961851957e273

   1. Initial program 0.8

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

   if -3.1921995578512398e-227 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0 or 8.6554751961851957e273 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 20.5

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

   3. Applied associate-/r*_binary64_2750 14.7

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

   4. Simplified 6.9

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

   6. Applied associate-/l*_binary64_2751 7.0

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

   8. Applied associate-/r/_binary64_2752 6.7

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

   9. Applied *-un-lft-identity_binary64_2806 6.7

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

   10. Applied times-frac_binary64_2812 4.7

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

   11. Simplified 4.5

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right)} \cdot \frac{\sqrt[3]{a2}}{\frac{\sqrt[3]{b2}}{\frac{a1}{\sqrt[3]{b1}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.19219955785124 \cdot 10^{-227} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 8.655475196185196 \cdot 10^{+273}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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