Quotient of products

Average Error: 11.2 → 2.3

Time: 4.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -3.8522383503833744 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;t_0 \leq -1.5469016511103522 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5.72839515998174 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b2}}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \end{array} \]

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 -3.8522383503833744e+283)
     (/ 1.0 (* (/ b1 a1) (/ b2 a2)))
     (if (<= t_0 -1.5469016511103522e-291)
       t_0
       (if (or (<= t_0 0.0) (not (<= t_0 5.72839515998174e+307)))
         (* (/ a1 (* (cbrt b2) (cbrt b2))) (/ (/ a2 (cbrt b2)) b1))
         (* (* a1 a2) (/ 1.0 (* b1 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 <= -3.8522383503833744e+283) {
		tmp = 1.0 / ((b1 / a1) * (b2 / a2));
	} else if (t_0 <= -1.5469016511103522e-291) {
		tmp = t_0;
	} else if ((t_0 <= 0.0) || !(t_0 <= 5.72839515998174e+307)) {
		tmp = (a1 / (cbrt(b2) * cbrt(b2))) * ((a2 / cbrt(b2)) / b1);
	} else {
		tmp = (a1 * a2) * (1.0 / (b1 * b2));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	11.3
Herbie	2.3

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -3.8522383503833744e283

   1. Initial program 53.7

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

   2. Applied clear-num_binary64 53.7

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

   3. Applied times-frac_binary64 12.2

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

   if -3.8522383503833744e283 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.54690165111035215e-291

   1. Initial program 0.8

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

   if -1.54690165111035215e-291 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0 or 5.72839515998174045e307 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 22.6

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

   2. Applied clear-num_binary64 22.8

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

   3. Applied associate-/l*_binary64 15.5

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

   4. Applied add-cube-cbrt_binary64 15.7

      \[\leadsto \frac{1}{\frac{b1}{\frac{a1 \cdot a2}{\color{blue}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}}}} \]

   5. Applied times-frac_binary64 6.6

      \[\leadsto \frac{1}{\frac{b1}{\color{blue}{\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}}}} \]

   6. Applied *-un-lft-identity_binary64 6.6

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot b1}}{\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}}} \]

   7. Applied times-frac_binary64 3.6

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{b1}{\frac{a2}{\sqrt[3]{b2}}}}} \]

   8. Applied add-cube-cbrt_binary64 3.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{b1}{\frac{a2}{\sqrt[3]{b2}}}} \]

   9. Applied times-frac_binary64 3.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{1}}{\frac{b1}{\frac{a2}{\sqrt[3]{b2}}}}} \]

   10. Simplified 3.3

       \[\leadsto \color{blue}{\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\sqrt[3]{1}}{\frac{b1}{\frac{a2}{\sqrt[3]{b2}}}} \]

   11. Simplified 3.2

       \[\leadsto \frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \color{blue}{\frac{\frac{a2}{\sqrt[3]{b2}}}{b1}} \]

   if 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.72839515998174045e307

   1. Initial program 0.7

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

   2. Applied div-inv_binary64 1.1

      \[\leadsto \color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -3.8522383503833744 \cdot 10^{+283}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1} \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.5469016511103522 \cdot 10^{-291}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0 \lor \neg \left(\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5.72839515998174 \cdot 10^{+307}\right):\\ \;\;\;\;\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b2}}}{b1}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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