Quotient of products

Average Error: 11.2 → 3.7

Time: 7.0s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -6.916235610315 \cdot 10^{-312}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5.041403853359531 \cdot 10^{-289}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq 7.488614284566524 \cdot 10^{+190}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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) (* (cbrt b2) (cbrt b2))) (/ a2 (cbrt b2)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -6.916235610315e-312)
       t_0
       (if (<= t_0 5.041403853359531e-289)
         (/ (* a1 (/ a2 b2)) b1)
         (if (<= t_0 7.488614284566524e+190)
           (* (* a1 a2) (/ (/ 1.0 b1) b2))
           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) / (cbrt(b2) * cbrt(b2))) * (a2 / cbrt(b2));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -6.916235610315e-312) {
		tmp = t_0;
	} else if (t_0 <= 5.041403853359531e-289) {
		tmp = (a1 * (a2 / b2)) / b1;
	} else if (t_0 <= 7.488614284566524e+190) {
		tmp = (a1 * a2) * ((1.0 / b1) / b2);
	} 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	11.2
Target	11.4
Herbie	3.7

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or 7.48861428456652416e190 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 46.1

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

   2. Applied times-frac_binary64 11.7

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

   3. Applied add-cube-cbrt_binary64 12.5

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

   4. Applied *-un-lft-identity_binary64 12.5

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

   5. Applied times-frac_binary64 12.5

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

   6. Applied associate-*r*_binary64 14.2

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

   7. Simplified 14.2

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -6.9162356103149e-312

   1. Initial program 0.8

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

   if -6.9162356103149e-312 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.0414038533595308e-289

   1. Initial program 12.5

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

   2. Applied times-frac_binary64 2.3

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

   3. Applied associate-*l/_binary64 3.7

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

   if 5.0414038533595308e-289 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 7.48861428456652416e190

   1. Initial program 0.8

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

   2. Applied associate-/r*_binary64 6.6

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

   3. Applied *-un-lft-identity_binary64 6.6

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

   4. Applied div-inv_binary64 6.7

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

   5. Applied times-frac_binary64 1.1

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -6.916235610315 \cdot 10^{-312}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5.041403853359531 \cdot 10^{-289}:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 7.488614284566524 \cdot 10^{+190}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a2}{\sqrt[3]{b2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022019 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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