Quotient of products

Average Error: 11.2 → 4.2

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.20887730078496 \cdot 10^{+277}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.7900592547123306 \cdot 10^{-197}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.6872465957563077 \cdot 10^{+266}\right):\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) -1.20887730078496e+277)
   (* a1 (/ (/ a2 b1) b2))
   (if (<= (/ (* a1 a2) (* b1 b2)) -2.7900592547123306e-197)
     (* (* a1 a2) (/ 1.0 (* b1 b2)))
     (if (or (<= (/ (* a1 a2) (* b1 b2)) 0.0)
             (not (<= (/ (* a1 a2) (* b1 b2)) 1.6872465957563077e+266)))
       (* a1 (/ (/ a2 b1) b2))
       (/ (* 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 tmp;
	if (((a1 * a2) / (b1 * b2)) <= -1.20887730078496e+277) {
		tmp = a1 * ((a2 / b1) / b2);
	} else if (((a1 * a2) / (b1 * b2)) <= -2.7900592547123306e-197) {
		tmp = (a1 * a2) * (1.0 / (b1 * b2));
	} else if ((((a1 * a2) / (b1 * b2)) <= 0.0) || !(((a1 * a2) / (b1 * b2)) <= 1.6872465957563077e+266)) {
		tmp = a1 * ((a2 / b1) / b2);
	} 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	11.2
Target	11.3
Herbie	4.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.20887730078496e277 or -2.79005925471233059e-197 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0 or 1.68724659575630769e266 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 22.7

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

   3. Applied div-inv_binary64_1796 22.9

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}}\]
   4. Using strategy rm

   5. Applied associate-*l*_binary64_1740 16.0

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

   6. Simplified 15.8

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

   8. Applied associate-/r*_binary64_1743 7.9

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

   if -1.20887730078496e277 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.79005925471233059e-197

   1. Initial program 0.8

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

   3. Applied div-inv_binary64_1796 1.1

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.68724659575630769e266

   1. Initial program 0.8

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

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.20887730078496 \cdot 10^{+277}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.7900592547123306 \cdot 10^{-197}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.6872465957563077 \cdot 10^{+266}\right):\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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