Quotient of products

Average Error: 12.0 → 3.2

Time: 5.8s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{elif}\;t_0 \leq -1.6989813180162923 \cdot 10^{-283}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 1.3349460866107138 \cdot 10^{+220}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{\frac{b2}{\frac{1}{b1}}}\\ \end{array} \]

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 (- INFINITY))
     (/ (/ a1 (/ b1 a2)) b2)
     (if (<= t_0 -1.6989813180162923e-283)
       t_0
       (if (or (<= t_0 0.0) (not (<= t_0 1.3349460866107138e+220)))
         (* (/ a1 b1) (/ a2 b2))
         (/ (* a1 a2) (/ b2 (/ 1.0 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 <= -((double) INFINITY)) {
		tmp = (a1 / (b1 / a2)) / b2;
	} else if (t_0 <= -1.6989813180162923e-283) {
		tmp = t_0;
	} else if ((t_0 <= 0.0) || !(t_0 <= 1.3349460866107138e+220)) {
		tmp = (a1 / b1) * (a2 / b2);
	} else {
		tmp = (a1 * a2) / (b2 / (1.0 / b1));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	12.0
Target	11.6
Herbie	3.2

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0

   1. Initial program 64.0

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

   2. Applied associate-/r*_binary64 33.2

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

   3. Applied associate-/l*_binary64 15.2

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.6989813180162923e-283

   1. Initial program 0.7

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

   2. Applied associate-/r*_binary64 8.5

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

   3. Applied div-inv_binary64 8.6

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

   4. Applied frac-times_binary64 0.7

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

   if -1.6989813180162923e-283 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0 or 1.33494608661071378e220 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 21.7

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

   2. Applied times-frac_binary64 5.1

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.33494608661071378e220

   1. Initial program 1.0

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

   2. Applied associate-/r*_binary64 7.8

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

   3. Applied div-inv_binary64 7.9

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

   4. Applied associate-/l*_binary64 1.0

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{\frac{b2}{\frac{1}{b1}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.6989813180162923 \cdot 10^{-283}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.3349460866107138 \cdot 10^{+220}\right):\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{\frac{b2}{\frac{1}{b1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021357 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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