Quotient of products

Average Error: 11.2 → 1.8

Time: 5.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{\frac{a2}{b2}}\\ \frac{t_1 \cdot t_1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{t_1}} \end{array}\\ \mathbf{elif}\;t_0 \leq 5.928194317570427 \cdot 10^{+301}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

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))
     (*
      (/ (* (cbrt a2) (cbrt a2)) (* (cbrt b2) (cbrt b2)))
      (/ a1 (/ b1 (/ (cbrt a2) (cbrt b2)))))
     (if (<= t_0 -5e-324)
       t_0
       (if (<= t_0 0.0)
         (let* ((t_1 (cbrt (/ a2 b2))))
           (*
            (/ (* t_1 t_1) (* (cbrt b1) (cbrt b1)))
            (/ a1 (/ (cbrt b1) t_1))))
         (if (<= t_0 5.928194317570427e+301) 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)) {
		tmp = ((cbrt(a2) * cbrt(a2)) / (cbrt(b2) * cbrt(b2))) * (a1 / (b1 / (cbrt(a2) / cbrt(b2))));
	} else if (t_0 <= -5e-324) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		double t_1 = cbrt(a2 / b2);
		tmp = ((t_1 * t_1) / (cbrt(b1) * cbrt(b1))) * (a1 / (cbrt(b1) / t_1));
	} else if (t_0 <= 5.928194317570427e+301) {
		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	11.2
Target	11.3
Herbie	1.8

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

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 28.0

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

   3. Simplified 16.9

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

   4. Applied add-cube-cbrt_binary64 17.6

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

   5. Applied add-cube-cbrt_binary64 17.8

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

   6. Applied times-frac_binary64 17.8

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

   7. Applied *-un-lft-identity_binary64 17.8

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

   8. Applied times-frac_binary64 13.9

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

   9. Applied *-un-lft-identity_binary64 13.9

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

   10. Applied times-frac_binary64 10.7

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

   11. Simplified 10.7

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -4.94066e-324 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 5.92819431757042689e301

   1. Initial program 0.7

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

   2. Applied associate-/l*_binary64 7.8

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

   3. Simplified 14.4

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

   4. Taylor expanded in a1 around 0 0.7

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

   if -4.94066e-324 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.5

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

   2. Applied associate-/l*_binary64 6.9

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

   3. Simplified 3.7

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

   4. Applied add-cube-cbrt_binary64 3.9

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

   5. Applied add-cube-cbrt_binary64 3.9

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

   6. Applied times-frac_binary64 3.9

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

   7. Applied *-un-lft-identity_binary64 3.9

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

   8. Applied times-frac_binary64 1.9

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

   9. Simplified 1.8

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

   if 5.92819431757042689e301 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 62.0

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

   2. Applied times-frac_binary64 5.9

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{a1}{\frac{b1}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a2}{b2}} \cdot \sqrt[3]{\frac{a2}{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\sqrt[3]{\frac{a2}{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 5.928194317570427 \cdot 10^{+301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022094 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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