Quotient of products

Average Error: 11.2 → 4.0

Time: 5.1s

Precision: binary64

\[[a1, a2]=\mathsf{sort}([a1, a2])\]

\[[b1, b2]=\mathsf{sort}([b1, b2])\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -8.466466328893688 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{elif}\;t_0 \leq -1.913926 \cdot 10^{-318}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 1.9964924777458289 \cdot 10^{-292}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;t_0 \leq 3.336443235690429 \cdot 10^{+265}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -8.466466328893688e+180)
     (/ (/ a1 (* b1 b2)) (/ 1.0 a2))
     (if (<= t_0 -1.913926e-318)
       t_0
       (if (<= t_0 1.9964924777458289e-292)
         (* (/ a1 b1) (/ a2 b2))
         (if (<= t_0 3.336443235690429e+265)
           (/ 1.0 (/ (* b1 b2) (* a1 a2)))
           (/ 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 t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -8.466466328893688e+180) {
		tmp = (a1 / (b1 * b2)) / (1.0 / a2);
	} else if (t_0 <= -1.913926e-318) {
		tmp = t_0;
	} else if (t_0 <= 1.9964924777458289e-292) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= 3.336443235690429e+265) {
		tmp = 1.0 / ((b1 * b2) / (a1 * a2));
	} 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	11.2
Target	11.6
Herbie	4.0

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

Derivation

1. Split input into 5 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -8.46646632889368804e180

   1. Initial program 25.9

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

   2. Applied associate-/l*_binary64 19.7

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

   3. Applied div-inv_binary64 19.7

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

   4. Applied associate-/r*_binary64 19.4

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

   5. Simplified 19.4

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

   if -8.46646632889368804e180 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.9139263e-318

   1. Initial program 1.0

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

   2. Applied associate-/l*_binary64 6.5

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

   3. Taylor expanded in a1 around 0 1.0

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

   if -1.9139263e-318 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.99649247774582888e-292

   1. Initial program 13.3

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

   2. Applied times-frac_binary64 2.5

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

   if 1.99649247774582888e-292 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 3.3364432356904289e265

   1. Initial program 0.7

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

   2. Applied clear-num_binary64 0.8

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

   if 3.3364432356904289e265 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 53.9

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

   2. Applied associate-/l*_binary64 42.9

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

   3. Applied div-inv_binary64 42.9

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

   4. Applied add-cube-cbrt_binary64 43.2

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

   5. Applied times-frac_binary64 44.8

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

   6. Applied frac-times_binary64 43.2

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

   7. Simplified 42.9

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

   8. Simplified 42.9

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

   9. Applied associate-/l*_binary64 14.4

      \[\leadsto \frac{a1}{\color{blue}{\frac{b2}{\frac{a2}{b1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -8.466466328893688 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.913926 \cdot 10^{-318}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.9964924777458289 \cdot 10^{-292}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.336443235690429 \cdot 10^{+265}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022055 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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