Quotient of products

Average Error: 10.9 → 2.3

Time: 11.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -2.705684065734731 \cdot 10^{+245}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq -7.6486 \cdot 10^{-320}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;t_0 \leq 4.4178988718779176 \cdot 10^{+307}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \]

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))))
   (if (<= t_0 -2.705684065734731e+245)
     (* a1 (/ (/ a2 b2) b1))
     (if (<= t_0 -7.6486e-320)
       t_0
       (if (<= t_0 0.0)
         (* (/ a1 b2) (/ a2 b1))
         (if (<= t_0 4.4178988718779176e+307) t_0 (/ (/ a2 b2) (/ b1 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 tmp;
	if (t_0 <= -2.705684065734731e+245) {
		tmp = a1 * ((a2 / b2) / b1);
	} else if (t_0 <= -7.6486e-320) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b2) * (a2 / b1);
	} else if (t_0 <= 4.4178988718779176e+307) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) / (b1 / a1);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.9
Target	11.2
Herbie	2.3

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.7056840657347309e245

   1. Initial program 40.3

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

   2. Applied egg-rr 18.0

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

   3. Applied egg-rr 17.5

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

   4. Applied egg-rr 19.1

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

   if -2.7056840657347309e245 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -7.64863e-320 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 4.41789887187791757e307

   1. Initial program 0.7

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

   if -7.64863e-320 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.5

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

   2. Applied egg-rr 2.0

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

   if 4.41789887187791757e307 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 64.0

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

   2. Applied egg-rr 4.6

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

   3. Applied egg-rr 4.4

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.705684065734731 \cdot 10^{+245}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -7.6486 \cdot 10^{-320}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 4.4178988718779176 \cdot 10^{+307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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