Quotient of products

Average Error: 11.6 → 4.1

Time: 6.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -7.598920629217912 \cdot 10^{+198}:\\ \;\;\;\;\frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -9.536629910005653 \cdot 10^{-213}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.193948502327851 \cdot 10^{-106} \lor \neg \left(b1 \cdot b2 \leq 8.619819618703962 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{b2}}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* b1 b2) -7.598920629217912e+198)
   (* (/ 1.0 (/ b1 a2)) (/ a1 b2))
   (if (<= (* b1 b2) -9.536629910005653e-213)
     (* a1 (/ a2 (* b1 b2)))
     (if (or (<= (* b1 b2) 5.193948502327851e-106)
             (not (<= (* b1 b2) 8.619819618703962e+164)))
       (*
        (/
         (* (cbrt a1) (cbrt a1))
         (/ (* (cbrt b1) (cbrt b1)) (* (cbrt a2) (cbrt a2))))
        (/ (cbrt a1) (/ (cbrt b1) (/ (cbrt a2) b2))))
       (* a2 (/ a1 (* b1 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) <= -7.598920629217912e+198) {
		tmp = (1.0 / (b1 / a2)) * (a1 / b2);
	} else if ((b1 * b2) <= -9.536629910005653e-213) {
		tmp = a1 * (a2 / (b1 * b2));
	} else if (((b1 * b2) <= 5.193948502327851e-106) || !((b1 * b2) <= 8.619819618703962e+164)) {
		tmp = ((cbrt(a1) * cbrt(a1)) / ((cbrt(b1) * cbrt(b1)) / (cbrt(a2) * cbrt(a2)))) * (cbrt(a1) / (cbrt(b1) / (cbrt(a2) / b2)));
	} else {
		tmp = a2 * (a1 / (b1 * b2));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.6
Target	11.0
Herbie	4.1

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 b1 b2) < -7.5989206292179123e198

   1. Initial program 15.8

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

   3. Applied associate-/l*_binary64_3774 16.2

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

   4. Simplified 8.5

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

   6. Applied associate-/r/_binary64_3775 8.6

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

   7. Applied *-un-lft-identity_binary64_3829 8.6

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

   8. Applied times-frac_binary64_3835 4.7

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

   if -7.5989206292179123e198 < (*.f64 b1 b2) < -9.53662991000565299e-213

   1. Initial program 5.1

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

   3. Applied associate-/l*_binary64_3774 3.9

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

   4. Simplified 10.4

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

   6. Applied div-inv_binary64_3826 10.7

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

   7. Simplified 4.0

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

   if -9.53662991000565299e-213 < (*.f64 b1 b2) < 5.1939485023278511e-106 or 8.6198196187039622e164 < (*.f64 b1 b2)

   1. Initial program 19.8

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

   3. Applied associate-/l*_binary64_3774 19.4

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

   4. Simplified 11.7

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

   6. Applied *-un-lft-identity_binary64_3829 11.7

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

   7. Applied add-cube-cbrt_binary64_3864 12.3

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

   8. Applied times-frac_binary64_3835 12.3

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

   9. Applied add-cube-cbrt_binary64_3864 12.4

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

   10. Applied times-frac_binary64_3835 11.0

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

   11. Applied add-cube-cbrt_binary64_3864 11.1

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

   12. Applied times-frac_binary64_3835 4.3

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

   if 5.1939485023278511e-106 < (*.f64 b1 b2) < 8.6198196187039622e164

   1. Initial program 3.8

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

   3. Applied associate-/l*_binary64_3774 3.2

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

   4. Simplified 10.8

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

   6. Applied div-inv_binary64_3826 10.9

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

   7. Applied *-un-lft-identity_binary64_3829 10.9

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

   8. Applied times-frac_binary64_3835 3.3

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

   9. Applied *-un-lft-identity_binary64_3829 3.3

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

   10. Applied times-frac_binary64_3835 3.6

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

   11. Simplified 3.5

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

   12. Simplified 3.5

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -7.598920629217912 \cdot 10^{+198}:\\ \;\;\;\;\frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -9.536629910005653 \cdot 10^{-213}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 5.193948502327851 \cdot 10^{-106} \lor \neg \left(b1 \cdot b2 \leq 8.619819618703962 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{b2}}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021026 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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