Quotient of products

Average Error: 11.2 → 3.5

Time: 4.5s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;t_0 \leq -5.0856438256446027 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -5.6645791 \cdot 10^{-316}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 4.373228051816653 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \end{array} \]

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 -5.0856438256446027e+222)
     t_1
     (if (<= t_0 -5.6645791e-316)
       t_0
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 4.373228051816653e+239) t_0 (/ a2 (* b1 (/ b2 a1)))))))))

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 <= -5.0856438256446027e+222) {
		tmp = t_1;
	} else if (t_0 <= -5.6645791e-316) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 4.373228051816653e+239) {
		tmp = t_0;
	} else {
		tmp = a2 / (b1 * (b2 / a1));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	11.3
Herbie	3.5

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.0856438256446027e222 or -5.664579081e-316 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 16.9

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

   2. Applied times-frac_binary64 4.6

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

   if -5.0856438256446027e222 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.664579081e-316 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 4.37322805181665277e239

   1. Initial program 0.8

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

   if 4.37322805181665277e239 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 48.7

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

   2. Applied associate-/r*_binary64 40.3

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

   3. Simplified 15.0

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

   4. Applied associate-/l*_binary64 15.0

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

   5. Applied associate-/r/_binary64 15.2

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

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.0856438256446027 \cdot 10^{+222}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.6645791 \cdot 10^{-316}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.373228051816653 \cdot 10^{+239}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1 \cdot \frac{b2}{a1}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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