Quotient of products

Average Error: 11.5 → 3.1

Time: 5.0s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a2}{\sqrt[3]{b1}}\\ t_1 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_2 := \frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}\\ t_3 := \frac{t_2 \cdot t_0}{b2}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -7.610924163598907 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 1.947699555580729 \cdot 10^{+288}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_0}{b2}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ a2 (cbrt b1)))
        (t_1 (/ (* a1 a2) (* b1 b2)))
        (t_2 (/ a1 (* (cbrt b1) (cbrt b1))))
        (t_3 (/ (* t_2 t_0) b2)))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -7.610924163598907e-298)
       t_1
       (if (<= t_1 0.0)
         t_3
         (if (<= t_1 1.947699555580729e+288)
           (* (* a1 a2) (/ (/ 1.0 b1) b2))
           (* t_2 (/ t_0 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 = a2 / cbrt(b1);
	double t_1 = (a1 * a2) / (b1 * b2);
	double t_2 = a1 / (cbrt(b1) * cbrt(b1));
	double t_3 = (t_2 * t_0) / b2;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -7.610924163598907e-298) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = t_3;
	} else if (t_1 <= 1.947699555580729e+288) {
		tmp = (a1 * a2) * ((1.0 / b1) / b2);
	} else {
		tmp = t_2 * (t_0 / b2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.7
Herbie	3.1

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or -7.61092416359890659e-298 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 18.8

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

   2. Applied associate-/r*_binary64 9.9

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

   3. Applied add-cube-cbrt_binary64 10.1

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

   4. Applied times-frac_binary64 5.2

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -7.61092416359890659e-298

   1. Initial program 0.9

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.94769955558072912e288

   1. Initial program 1.0

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

   2. Applied associate-/r*_binary64 7.6

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

   3. Applied *-un-lft-identity_binary64 7.6

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

   4. Applied div-inv_binary64 7.6

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

   5. Applied times-frac_binary64 1.5

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

   if 1.94769955558072912e288 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 58.6

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

   2. Applied associate-/r*_binary64 47.6

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

   3. Applied *-un-lft-identity_binary64 47.6

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

   4. Applied add-cube-cbrt_binary64 47.8

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

   5. Applied times-frac_binary64 13.6

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

   6. Applied times-frac_binary64 6.8

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

   7. Simplified 6.8

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a2}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -7.610924163598907 \cdot 10^{-298}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a2}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.947699555580729 \cdot 10^{+288}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{b2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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