Quotient of products

Average Error: 11.5 → 6.5

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -1.824827973319818 \cdot 10^{+202}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -3.8638100840414836 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq 0 \lor \neg \left(a1 \cdot a2 \leq 6.994370282012423 \cdot 10^{+231}\right):\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{a1 \cdot a2}{b1}} \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1}}\right) \cdot \frac{\sqrt[3]{\frac{a1 \cdot a2}{b1}}}{b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* a1 a2) -1.824827973319818e+202)
   (* a1 (/ a2 (* b1 b2)))
   (if (<= (* a1 a2) -3.8638100840414836e-76)
     (/ (/ (* a1 a2) b2) b1)
     (if (or (<= (* a1 a2) 0.0) (not (<= (* a1 a2) 6.994370282012423e+231)))
       (/ a1 (/ b1 (/ a2 b2)))
       (*
        (* (cbrt (/ (* a1 a2) b1)) (cbrt (/ (* a1 a2) b1)))
        (/ (cbrt (/ (* a1 a2) b1)) b2))))))

C

double code(double a1, double a2, double b1, double b2) {
	return (((double) (a1 * a2)) / ((double) (b1 * b2)));
}

↓

double code(double a1, double a2, double b1, double b2) {
	double tmp;
	if ((((double) (a1 * a2)) <= -1.824827973319818e+202)) {
		tmp = ((double) (a1 * (a2 / ((double) (b1 * b2)))));
	} else {
		double tmp_1;
		if ((((double) (a1 * a2)) <= -3.8638100840414836e-76)) {
			tmp_1 = ((((double) (a1 * a2)) / b2) / b1);
		} else {
			double tmp_2;
			if (((((double) (a1 * a2)) <= 0.0) || !(((double) (a1 * a2)) <= 6.994370282012423e+231))) {
				tmp_2 = (a1 / (b1 / (a2 / b2)));
			} else {
				tmp_2 = ((double) (((double) (((double) cbrt((((double) (a1 * a2)) / b1))) * ((double) cbrt((((double) (a1 * a2)) / b1))))) * (((double) cbrt((((double) (a1 * a2)) / b1))) / b2)));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.3
Herbie	6.5

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 a1 a2) < -1.8248279733198179e202

   1. Initial program 34.8

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

   3. Applied div-inv_binary64 34.8

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

   5. Applied associate-*l*_binary64 16.7

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

   6. Simplified 16.6

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

   if -1.8248279733198179e202 < (*.f64 a1 a2) < -3.86381008404148358e-76

   1. Initial program 3.9

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

   3. Applied div-inv_binary64 3.9

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

   5. Applied *-un-lft-identity_binary64 3.9

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

   6. Applied times-frac_binary64 3.8

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

   7. Applied associate-*r*_binary64 3.9

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

   8. Simplified 3.8

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

   10. Applied associate-*l/_binary64 3.7

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

   11. Simplified 3.6

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

   if -3.86381008404148358e-76 < (*.f64 a1 a2) < 0.0 or 6.99437028201242258e231 < (*.f64 a1 a2)

   1. Initial program 17.9

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

   3. Applied associate-/l*_binary64 10.1

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

   4. Simplified 6.6

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

   if 0.0 < (*.f64 a1 a2) < 6.99437028201242258e231

   1. Initial program 5.4

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

   3. Applied div-inv_binary64 5.8

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

   5. Applied *-un-lft-identity_binary64 5.8

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

   6. Applied times-frac_binary64 5.6

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

   7. Applied associate-*r*_binary64 5.7

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

   8. Simplified 5.7

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

   10. Applied add-cube-cbrt_binary64 6.4

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

   11. Applied associate-*l*_binary64 6.4

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

   12. Simplified 6.4

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \leq -1.824827973319818 \cdot 10^{+202}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;a1 \cdot a2 \leq -3.8638100840414836 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b2}}{b1}\\ \mathbf{elif}\;a1 \cdot a2 \leq 0 \lor \neg \left(a1 \cdot a2 \leq 6.994370282012423 \cdot 10^{+231}\right):\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{a1 \cdot a2}{b1}} \cdot \sqrt[3]{\frac{a1 \cdot a2}{b1}}\right) \cdot \frac{\sqrt[3]{\frac{a1 \cdot a2}{b1}}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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