Quotient of products

Average Error: 11.4 → 2.5

Time: 6.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -8.400079601776277 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \left(\left(\frac{\sqrt[3]{a1}}{b1} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)\right) \cdot \frac{\sqrt[3]{a2}}{b2}\right)\\ \mathbf{elif}\;t_0 \leq 3.589237135653206 \cdot 10^{+299}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))) (t_1 (* (/ a1 b1) (/ a2 b2))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -8.400079601776277e-290)
       t_0
       (if (<= t_0 0.0)
         (*
          (* (cbrt a1) (cbrt a1))
          (* (* (/ (cbrt a1) b1) (* (cbrt a2) (cbrt a2))) (/ (cbrt a2) b2)))
         (if (<= t_0 3.589237135653206e+299) t_0 t_1))))))

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 t_1 = (a1 / b1) * (a2 / b2);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -8.400079601776277e-290) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (cbrt(a1) * cbrt(a1)) * (((cbrt(a1) / b1) * (cbrt(a2) * cbrt(a2))) * (cbrt(a2) / b2));
	} else if (t_0 <= 3.589237135653206e+299) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.6
Herbie	2.5

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or 3.58923713565320618e299 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 62.5

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

   2. Applied times-frac_binary64 8.3

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -8.4000796017762771e-290 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 3.58923713565320618e299

   1. Initial program 0.9

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

   if -8.4000796017762771e-290 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.1

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

   2. Applied times-frac_binary64 2.7

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

   3. Applied *-un-lft-identity_binary64 2.7

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

   4. Applied add-cube-cbrt_binary64 2.9

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

   5. Applied times-frac_binary64 2.9

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

   6. Applied associate-*l*_binary64 3.8

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

   7. Applied *-un-lft-identity_binary64 3.8

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

   8. Applied add-cube-cbrt_binary64 3.8

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

   9. Applied times-frac_binary64 3.8

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

   10. Applied associate-*r*_binary64 3.4

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

   11. Simplified 3.4

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -8.400079601776277 \cdot 10^{-290}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \left(\left(\frac{\sqrt[3]{a1}}{b1} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)\right) \cdot \frac{\sqrt[3]{a2}}{b2}\right)\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 3.589237135653206 \cdot 10^{+299}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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