Quotient of products

Average Error: 11.4 → 2.6

Time: 12.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;t_0 \leq -1.3142776653186078 \cdot 10^{-294}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{a1} \cdot \sqrt[3]{a1}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\left(a2 \cdot \frac{t_1}{b1}\right) \cdot \frac{\sqrt[3]{a1}}{b2}\\ \mathbf{elif}\;t_0 \leq 5.037792537121833 \cdot 10^{+183}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \end{array}\\ \end{array} \]

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))
     (* a1 (/ 1.0 (/ b1 (/ a2 b2))))
     (if (<= t_0 -1.3142776653186078e-294)
       (* (* a1 a2) (/ 1.0 (* b1 b2)))
       (let* ((t_1 (* (cbrt a1) (cbrt a1))))
         (if (<= t_0 0.0)
           (* (* a2 (/ t_1 b1)) (/ (cbrt a1) b2))
           (if (<= t_0 5.037792537121833e+183)
             t_0
             (*
              (/
               t_1
               (/
                (* (cbrt b1) (cbrt b1))
                (/ (* (cbrt a2) (cbrt a2)) (* (cbrt b2) (cbrt b2)))))
              (/ (cbrt a1) (/ (cbrt b1) (/ (cbrt a2) (cbrt 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 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = a1 * (1.0 / (b1 / (a2 / b2)));
	} else if (t_0 <= -1.3142776653186078e-294) {
		tmp = (a1 * a2) * (1.0 / (b1 * b2));
	} else {
		double t_1 = cbrt(a1) * cbrt(a1);
		double tmp_1;
		if (t_0 <= 0.0) {
			tmp_1 = (a2 * (t_1 / b1)) * (cbrt(a1) / b2);
		} else if (t_0 <= 5.037792537121833e+183) {
			tmp_1 = t_0;
		} else {
			tmp_1 = (t_1 / ((cbrt(b1) * cbrt(b1)) / ((cbrt(a2) * cbrt(a2)) / (cbrt(b2) * cbrt(b2))))) * (cbrt(a1) / (cbrt(b1) / (cbrt(a2) / cbrt(b2))));
		}
		tmp = tmp_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	2.6

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

Derivation

1. Split input into 5 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. Using strategy rm

   3. Applied associate-/l*_binary64 29.6

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

   4. Simplified 16.6

      \[\leadsto \frac{a1}{\color{blue}{\frac{b1}{\frac{a2}{b2}}}} \]
   5. Using strategy rm

   6. Applied div-inv_binary64 17.1

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.3142776653186078e-294

   1. Initial program 0.7

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Using strategy rm

   3. Applied div-inv_binary64 1.1

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

   if -1.3142776653186078e-294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.4

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Using strategy rm

   3. Applied associate-/l*_binary64 7.7

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

   4. Simplified 3.8

      \[\leadsto \frac{a1}{\color{blue}{\frac{b1}{\frac{a2}{b2}}}} \]
   5. Using strategy rm

   6. Applied associate-/r/_binary64 4.1

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

   7. Applied add-cube-cbrt_binary64 4.2

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

   8. Applied times-frac_binary64 3.4

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

   9. Simplified 3.2

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

   if -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.0377925371218328e183

   1. Initial program 0.6

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

   if 5.0377925371218328e183 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 40.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Using strategy rm

   3. Applied associate-/l*_binary64 34.6

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

   4. Simplified 15.2

      \[\leadsto \frac{a1}{\color{blue}{\frac{b1}{\frac{a2}{b2}}}} \]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64 16.0

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

   7. Applied add-cube-cbrt_binary64 16.2

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

   8. Applied times-frac_binary64 16.2

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

   9. Applied add-cube-cbrt_binary64 16.3

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

   10. Applied times-frac_binary64 13.9

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

   11. Applied add-cube-cbrt_binary64 14.1

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

   12. Applied times-frac_binary64 4.8

       \[\leadsto \color{blue}{\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;a1 \cdot \frac{1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.3142776653186078 \cdot 10^{-294}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{1}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\left(a2 \cdot \frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b1}\right) \cdot \frac{\sqrt[3]{a1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5.037792537121833 \cdot 10^{+183}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021190 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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