Quotient of products

Average Error: 11.0 → 2.6

Time: 6.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{\frac{a2}{b2}}\\ \left(\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \left(t_1 \cdot t_1\right)\right) \cdot \frac{\sqrt[3]{a1}}{\frac{b1}{t_1}} \end{array}\\ \mathbf{elif}\;t_0 \leq -1.5248764427209305 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;t_0 \leq 2.4128988234850523 \cdot 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 (- INFINITY))
     (let* ((t_1 (cbrt (/ a2 b2))))
       (* (* (* (cbrt a1) (cbrt a1)) (* t_1 t_1)) (/ (cbrt a1) (/ b1 t_1))))
     (if (<= t_0 -1.5248764427209305e-302)
       t_0
       (if (<= t_0 0.0)
         (* (/ 1.0 (/ b1 a2)) (/ a1 b2))
         (if (<= t_0 2.4128988234850523e+246) t_0 (* (/ a2 b2) (/ a1 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 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		double t_1_1 = cbrt(a2 / b2);
		tmp = ((cbrt(a1) * cbrt(a1)) * (t_1_1 * t_1_1)) * (cbrt(a1) / (b1 / t_1_1));
	} else if (t_0 <= -1.5248764427209305e-302) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / (b1 / a2)) * (a1 / b2);
	} else if (t_0 <= 2.4128988234850523e+246) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) * (a1 / 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	2.6

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

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-/l*_binary64 31.4

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

   3. Simplified 17.0

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

   4. Applied add-cube-cbrt_binary64 17.7

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

   5. Applied *-un-lft-identity_binary64 17.7

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

   6. Applied times-frac_binary64 17.7

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

   7. Applied add-cube-cbrt_binary64 17.9

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

   8. Applied times-frac_binary64 10.6

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

   9. Simplified 10.7

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.5248764427209305e-302 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.4128988234850523e246

   1. Initial program 0.8

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

   2. Applied times-frac_binary64 15.9

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

   3. Taylor expanded in a1 around 0 0.8

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

   if -1.5248764427209305e-302 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.0

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

   2. Applied associate-/l*_binary64 7.0

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

   3. Simplified 3.9

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

   4. Applied associate-/r/_binary64 4.0

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

   5. Applied *-un-lft-identity_binary64 4.0

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

   6. Applied times-frac_binary64 2.6

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

   if 2.4128988234850523e246 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 48.4

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

   2. Applied times-frac_binary64 11.3

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\left(\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \left(\sqrt[3]{\frac{a2}{b2}} \cdot \sqrt[3]{\frac{a2}{b2}}\right)\right) \cdot \frac{\sqrt[3]{a1}}{\frac{b1}{\sqrt[3]{\frac{a2}{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.5248764427209305 \cdot 10^{-302}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{1}{\frac{b1}{a2}} \cdot \frac{a1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.4128988234850523 \cdot 10^{+246}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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