Quotient of products

Average Error: 11.2 → 3.2

Time: 5.6s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;t_0 \leq -1.1429146543765217 \cdot 10^{-267}:\\ \;\;\;\;\frac{a1 \cdot a2}{\frac{b1}{\frac{1}{b2}}}\\ \mathbf{elif}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 8.720832817076526 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (- INFINITY))
     (* (/ a1 b1) (/ a2 b2))
     (if (<= t_0 -1.1429146543765217e-267)
       (/ (* a1 a2) (/ b1 (/ 1.0 b2)))
       (if (or (<= t_0 0.0) (not (<= t_0 8.720832817076526e+201)))
         (* (/ a2 b1) (/ a1 b2))
         t_0)))))

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 <= -((double) INFINITY)) {
		tmp = (a1 / b1) * (a2 / b2);
	} else if (t_0 <= -1.1429146543765217e-267) {
		tmp = (a1 * a2) / (b1 / (1.0 / b2));
	} else if ((t_0 <= 0.0) || !(t_0 <= 8.720832817076526e+201)) {
		tmp = (a2 / b1) * (a1 / b2);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	11.4
Herbie	3.2

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied times-frac_binary64 12.8

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.1429146543765217e-267

   1. Initial program 0.8

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

   3. Applied associate-/l*_binary64 8.1

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

   4. Simplified 14.2

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

   6. Applied div-inv_binary64 14.3

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

   7. Applied *-un-lft-identity_binary64 14.3

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

   8. Applied times-frac_binary64 8.3

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

   9. Applied associate-/r*_binary64 0.9

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

   10. Simplified 0.9

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

   if -1.1429146543765217e-267 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0 or 8.72083281707652592e201 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 19.6

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

   3. Applied associate-/l*_binary64 14.3

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

   4. Simplified 7.3

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

   6. Applied associate-/r/_binary64 7.3

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

   7. Applied *-un-lft-identity_binary64 7.3

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

   8. Applied times-frac_binary64 5.5

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

   9. Simplified 5.4

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 8.72083281707652592e201

   1. Initial program 0.7

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.1429146543765217 \cdot 10^{-267}:\\ \;\;\;\;\frac{a1 \cdot a2}{\frac{b1}{\frac{1}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 8.720832817076526 \cdot 10^{+201}\right):\\ \;\;\;\;\frac{a2}{b1} \cdot \frac{a1}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021196 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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