Quotient of products

Average Error: 11.0 → 2.0

Time: 7.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{elif}\;t_0 \leq -6.165001 \cdot 10^{-317} \lor \neg \left(t_0 \leq 0\right) \land t_0 \leq 8.745338238018392 \cdot 10^{+303}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

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 (/ 1.0 (/ b2 (/ a2 b1))))
     (if (or (<= t_0 -6.165001e-317)
             (and (not (<= t_0 0.0)) (<= t_0 8.745338238018392e+303)))
       t_0
       (* (/ a1 b1) (/ a2 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 t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = a1 * (1.0 / (b2 / (a2 / b1)));
	} else if ((t_0 <= -6.165001e-317) || (!(t_0 <= 0.0) && (t_0 <= 8.745338238018392e+303))) {
		tmp = t_0;
	} else {
		tmp = (a1 / b1) * (a2 / b2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.0
Target	11.6
Herbie	2.0

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

Derivation

1. Split input into 3 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 div-inv_binary64 64.0

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

   4. Simplified 64.0

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

   6. Applied associate-*l*_binary64 29.8

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

   7. Simplified 29.7

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

   9. Applied clear-num_binary64 29.8

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

   10. Simplified 15.9

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -6.16500103e-317 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 8.7453382380183924e303

   1. Initial program 0.7

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

   if -6.16500103e-317 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0 or 8.7453382380183924e303 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 22.3

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

   3. Applied times-frac_binary64 2.8

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -6.165001 \cdot 10^{-317} \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0\right) \land \frac{a1 \cdot a2}{b1 \cdot b2} \leq 8.745338238018392 \cdot 10^{+303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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