Quotient of products

Average Error: 11.4 → 2.0

Time: 5.8s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -9.14 \cdot 10^{-322}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a1}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;t_0 \leq 1.5489696528649269 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))) (t_1 (* (/ a1 b1) (/ a2 b2))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -9.14e-322)
       t_0
       (if (<= t_0 0.0)
         (* (/ a2 (* (cbrt b2) (cbrt b2))) (/ (/ a1 b1) (cbrt b2)))
         (if (<= t_0 1.5489696528649269e+305) t_0 t_1))))))

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 t_1 = (a1 / b1) * (a2 / b2);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -9.14e-322) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a2 / (cbrt(b2) * cbrt(b2))) * ((a1 / b1) / cbrt(b2));
	} else if (t_0 <= 1.5489696528649269e+305) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.6
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)) < -inf.0 or 1.54896965286492688e305 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 63.3

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

   3. Applied times-frac_binary64 7.2

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.14021e-322 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.54896965286492688e305

   1. Initial program 0.8

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

   if -9.14021e-322 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.7

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

   3. Applied associate-/r*_binary64 6.4

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

   4. Simplified 3.9

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

   6. Applied add-cube-cbrt_binary64 4.1

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

   7. Applied times-frac_binary64 2.4

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -9.14 \cdot 10^{-322}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a1}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.5489696528649269 \cdot 10^{+305}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021209 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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