Quotient of products

Average Error: 11.3 → 3.2

Time: 8.3s

Precision: binary64

\[[a1, a2] = \mathsf{sort}([a1, a2]) \[b1, b2] = \mathsf{sort}([b1, b2]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1.0778478930048488 \cdot 10^{-249}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 1.154926780129537 \cdot 10^{+270}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \end{array} \]

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 -1.0778478930048488e-249)
       t_0
       (if (<= t_0 0.0)
         t_1
         (if (<= t_0 1.154926780129537e+270) t_0 (/ (* a1 (/ a2 b1)) b2)))))))

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 <= -1.0778478930048488e-249) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 1.154926780129537e+270) {
		tmp = t_0;
	} else {
		tmp = (a1 * (a2 / b1)) / b2;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.5
Herbie	3.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or -1.0778478930048488e-249 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 16.8

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

   2. Applied times-frac_binary64 4.3

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.0778478930048488e-249 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.15492678012953692e270

   1. Initial program 0.8

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

   if 1.15492678012953692e270 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 55.6

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

   2. Applied associate-/r*_binary64 45.1

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

   3. Applied *-un-lft-identity_binary64 45.1

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

   4. Applied times-frac_binary64 14.5

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

   5. Simplified 14.5

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.0778478930048488 \cdot 10^{-249}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.154926780129537 \cdot 10^{+270}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022066 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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