Quotient of products

Average Error: 11.3 → 4.1

Time: 4.9s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -1.1144896539576276 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq 5.658058179724235 \cdot 10^{+290}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array} \]

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 -1.1144896539576276e-302)
     t_0
     (if (<= t_0 0.0)
       (/ (* a2 (/ a1 b2)) b1)
       (if (<= t_0 5.658058179724235e+290) t_0 (* (/ a1 b2) (/ a2 b1)))))))

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 <= -1.1144896539576276e-302) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (a2 * (a1 / b2)) / b1;
	} else if (t_0 <= 5.658058179724235e+290) {
		tmp = t_0;
	} else {
		tmp = (a1 / b2) * (a2 / b1);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.6
Herbie	4.1

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.114489653957628e-302 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.65805817972423479e290

   1. Initial program 4.1

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

   if -1.114489653957628e-302 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.7

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

   3. Applied div-inv_binary64 13.7

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

   4. Simplified 13.7

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

   6. Applied un-div-inv_binary64 13.7

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

   7. Applied associate-/r*_binary64 6.6

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

   8. Simplified 3.3

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

   if 5.65805817972423479e290 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 59.2

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

   3. Applied div-inv_binary64 59.3

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

   4. Simplified 59.3

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

   6. Applied un-div-inv_binary64 59.2

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

   7. Applied times-frac_binary64 7.6

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.1144896539576276 \cdot 10^{-302}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a2 \cdot \frac{a1}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5.658058179724235 \cdot 10^{+290}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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