Quotient of products

Average Error: 11.7 → 5.1

Time: 6.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -\infty:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \leq -4.072413297069987 \cdot 10^{-246}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2.4232841084 \cdot 10^{-313}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2.837907807204954 \cdot 10^{+169}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (* b1 b2) (- INFINITY))
   (* (/ 1.0 b1) (/ a1 (/ 1.0 (/ a2 b2))))
   (if (<= (* b1 b2) -4.072413297069987e-246)
     (* a1 (/ a2 (* b1 b2)))
     (if (<= (* b1 b2) 2.4232841084e-313)
       (* (* a1 (/ a2 b1)) (/ 1.0 b2))
       (if (<= (* b1 b2) 2.837907807204954e+169)
         (/ (* a1 a2) (* b1 b2))
         (* (/ a2 b1) (/ a1 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 ((b1 * b2) <= -((double) INFINITY)) {
		tmp = (1.0 / b1) * (a1 / (1.0 / (a2 / b2)));
	} else if ((b1 * b2) <= -4.072413297069987e-246) {
		tmp = a1 * (a2 / (b1 * b2));
	} else if ((b1 * b2) <= 2.4232841084e-313) {
		tmp = (a1 * (a2 / b1)) * (1.0 / b2);
	} else if ((b1 * b2) <= 2.837907807204954e+169) {
		tmp = (a1 * a2) / (b1 * b2);
	} else {
		tmp = (a2 / b1) * (a1 / b2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.7
Target	11.3
Herbie	5.1

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

Derivation

1. Split input into 5 regimes

2. if (*.f64 b1 b2) < -inf.0

   1. Initial program 23.4

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

   2. Applied associate-/l*_binary64 23.2

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

   3. Simplified 8.7

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

   4. Applied div-inv_binary64 8.8

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

   5. Applied *-un-lft-identity_binary64 8.8

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

   6. Applied times-frac_binary64 2.1

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

   if -inf.0 < (*.f64 b1 b2) < -4.07241329706998716e-246

   1. Initial program 5.1

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

   2. Applied associate-/l*_binary64 4.8

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

   3. Simplified 10.7

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

   4. Applied div-inv_binary64 10.9

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

   5. Simplified 4.9

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

   if -4.07241329706998716e-246 < (*.f64 b1 b2) < 2.42328410838e-313

   1. Initial program 50.9

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

   2. Applied associate-/l*_binary64 50.1

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

   3. Simplified 17.8

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

   4. Applied associate-/r/_binary64 17.6

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

   5. Applied *-un-lft-identity_binary64 17.6

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

   6. Applied times-frac_binary64 8.8

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

   7. Applied div-inv_binary64 8.8

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

   8. Applied associate-*r*_binary64 8.6

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

   9. Simplified 8.5

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

   if 2.42328410838e-313 < (*.f64 b1 b2) < 2.83790780720495407e169

   1. Initial program 5.5

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

   if 2.83790780720495407e169 < (*.f64 b1 b2)

   1. Initial program 15.0

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

   2. Applied associate-/l*_binary64 14.1

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

   3. Simplified 8.2

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

   4. Applied associate-/r/_binary64 8.1

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

   5. Applied *-un-lft-identity_binary64 8.1

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

   6. Applied times-frac_binary64 4.8

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

   7. Simplified 4.6

      \[\leadsto \color{blue}{\frac{a2}{b1}} \cdot \frac{a1}{b2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -\infty:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \leq -4.072413297069987 \cdot 10^{-246}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2.4232841084 \cdot 10^{-313}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 2.837907807204954 \cdot 10^{+169}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022103 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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