Quotient of products

Average Error: 11.3 → 2.0

Time: 10.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.417901669555667 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.542552855874417 \cdot 10^{-297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b2}}{a2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.7044719015567174 \cdot 10^{+305}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) -3.417901669555667e+301)
   (/ (/ a1 b2) (/ b1 a2))
   (if (<= (/ (* a1 a2) (* b1 b2)) -2.542552855874417e-297)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 0.0)
       (/
        1.0
        (*
         (/
          (* (cbrt b1) (cbrt b1))
          (/ (* (cbrt a1) (cbrt a1)) (* (cbrt b2) (cbrt b2))))
         (/ (cbrt b1) (/ (cbrt a1) (/ (cbrt b2) a2)))))
       (if (<= (/ (* a1 a2) (* b1 b2)) 1.7044719015567174e+305)
         (/ (* a1 a2) (* b1 b2))
         (/ (/ a1 b2) (/ b1 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)) <= -3.417901669555667e+301) {
		tmp = (a1 / b2) / (b1 / a2);
	} else if (((a1 * a2) / (b1 * b2)) <= -2.542552855874417e-297) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 0.0) {
		tmp = 1.0 / (((cbrt(b1) * cbrt(b1)) / ((cbrt(a1) * cbrt(a1)) / (cbrt(b2) * cbrt(b2)))) * (cbrt(b1) / (cbrt(a1) / (cbrt(b2) / a2))));
	} else if (((a1 * a2) / (b1 * b2)) <= 1.7044719015567174e+305) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = (a1 / b2) / (b1 / a2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.5
Herbie	2.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.41790166955566683e301 or 1.7044719015567174e305 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 63.0

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

   3. Applied clear-num_binary64_2805 63.0

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

   5. Applied associate-/l*_binary64_2751 44.0

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

   6. Simplified 13.9

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

   8. Applied associate-/r/_binary64_2752 14.1

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

   9. Applied *-un-lft-identity_binary64_2806 14.1

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

   10. Applied times-frac_binary64_2812 7.2

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

   11. Applied associate-/r*_binary64_2750 7.2

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

   12. Simplified 7.1

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

   if -3.41790166955566683e301 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.5425528558744169e-297 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.7044719015567174e305

   1. Initial program 0.8

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

   if -2.5425528558744169e-297 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.3

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

   3. Applied clear-num_binary64_2805 13.7

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

   5. Applied associate-/l*_binary64_2751 7.1

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

   6. Simplified 4.5

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

   8. Applied *-un-lft-identity_binary64_2806 4.5

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

   9. Applied add-cube-cbrt_binary64_2841 4.7

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

   10. Applied times-frac_binary64_2812 4.7

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

   11. Applied add-cube-cbrt_binary64_2841 4.7

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

   12. Applied times-frac_binary64_2812 4.2

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

   13. Applied add-cube-cbrt_binary64_2841 4.2

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

   14. Applied times-frac_binary64_2812 2.7

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

   15. Simplified 2.7

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.417901669555667 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.542552855874417 \cdot 10^{-297}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b2}}{a2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.7044719015567174 \cdot 10^{+305}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020298 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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