Quotient of products

Average Error: 11.0 → 2.5

Time: 5.2s

Precision: binary64

\[[a1, a2] = \mathsf{sort}([a1, a2]) \[b1, b2] = \mathsf{sort}([b1, b2]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a2}{\sqrt[3]{b1}}\\ t_1 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_2 := \frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2 \cdot \frac{t_0}{b2}\\ \mathbf{elif}\;t_1 \leq -2.0541893291628116 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{t_2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{t_0}{\sqrt[3]{b2}}\\ \mathbf{elif}\;t_1 \leq 1.2274224996005442 \cdot 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b2}{a1 \cdot \frac{a2}{b1}}}\\ \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ a2 (cbrt b1)))
        (t_1 (/ (* a1 a2) (* b1 b2)))
        (t_2 (/ a1 (* (cbrt b1) (cbrt b1)))))
   (if (<= t_1 (- INFINITY))
     (* t_2 (/ t_0 b2))
     (if (<= t_1 -2.0541893291628116e-288)
       t_1
       (if (<= t_1 0.0)
         (* (/ t_2 (* (cbrt b2) (cbrt b2))) (/ t_0 (cbrt b2)))
         (if (<= t_1 1.2274224996005442e+303)
           t_1
           (/ 1.0 (/ b2 (* a1 (/ 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 = a2 / cbrt(b1);
	double t_1 = (a1 * a2) / (b1 * b2);
	double t_2 = a1 / (cbrt(b1) * cbrt(b1));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * (t_0 / b2);
	} else if (t_1 <= -2.0541893291628116e-288) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (t_2 / (cbrt(b2) * cbrt(b2))) * (t_0 / cbrt(b2));
	} else if (t_1 <= 1.2274224996005442e+303) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (b2 / (a1 * (a2 / b1)));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.0
Target	11.1
Herbie	2.5

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

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. Applied associate-/r*_binary64 36.7

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

   3. Applied *-un-lft-identity_binary64 36.7

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

   4. Applied add-cube-cbrt_binary64 37.2

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

   5. Applied times-frac_binary64 17.9

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

   6. Applied times-frac_binary64 10.4

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

   7. Simplified 10.4

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.05418932916281163e-288 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.2274224996005442e303

   1. Initial program 0.8

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

   2. Applied associate-/r*_binary64 7.9

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

   3. Taylor expanded in a1 around 0 0.8

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

   if -2.05418932916281163e-288 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.0

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

   2. Applied associate-/r*_binary64 6.7

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

   3. Applied add-cube-cbrt_binary64 6.8

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

   4. Applied add-cube-cbrt_binary64 6.9

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

   5. Applied times-frac_binary64 3.8

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

   6. Applied times-frac_binary64 2.4

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

   if 1.2274224996005442e303 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 63.3

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

   2. Applied associate-/r*_binary64 46.4

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

   3. Applied *-un-lft-identity_binary64 46.4

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

   4. Applied times-frac_binary64 14.2

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

   5. Simplified 14.2

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

   6. Applied clear-num_binary64 14.2

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.0541893291628116 \cdot 10^{-288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.2274224996005442 \cdot 10^{+303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b2}{a1 \cdot \frac{a2}{b1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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