Quotient of products

Average Error: 10.7 → 2.7

Time: 4.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -1.078471501403316 \cdot 10^{+259}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{elif}\;t_0 \leq -1.9812476037083025 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;t_0 \leq 2.8284868700362058 \cdot 10^{+298}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array} \]

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 -1.078471501403316e+259)
     (/ (/ a1 (/ b1 a2)) b2)
     (if (<= t_0 -1.9812476037083025e-288)
       t_0
       (if (<= t_0 0.0)
         (/ (/ a1 b2) (/ b1 a2))
         (if (<= t_0 2.8284868700362058e+298) t_0 (/ (/ a1 b1) (/ b2 a2))))))))

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 <= -1.078471501403316e+259) {
		tmp = (a1 / (b1 / a2)) / b2;
	} else if (t_0 <= -1.9812476037083025e-288) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a1 / b2) / (b1 / a2);
	} else if (t_0 <= 2.8284868700362058e+298) {
		tmp = t_0;
	} else {
		tmp = (a1 / b1) / (b2 / a2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.7
Target	11.3
Herbie	2.7

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.07847150140331607e259

   1. Initial program 41.8

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

   2. Applied associate-/r*_binary64 25.9

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

   3. Applied associate-/l*_binary64 18.0

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

   if -1.07847150140331607e259 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.98124760370830246e-288 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.8284868700362058e298

   1. Initial program 0.8

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

   if -1.98124760370830246e-288 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 12.6

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

   2. Applied associate-/r*_binary64 6.4

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

   3. Applied associate-/l*_binary64 3.8

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

   4. Applied associate-/l/_binary64 4.2

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

   5. Applied associate-/r*_binary64 2.7

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

   if 2.8284868700362058e298 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 60.9

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

   2. Applied associate-/r*_binary64 47.4

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

   3. Applied associate-/l*_binary64 14.5

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

   4. Applied associate-/r/_binary64 14.3

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

   5. Applied associate-/l*_binary64 7.8

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.078471501403316 \cdot 10^{+259}:\\ \;\;\;\;\frac{\frac{a1}{\frac{b1}{a2}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.9812476037083025 \cdot 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.8284868700362058 \cdot 10^{+298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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