Quotient of products

Average Error: 11.4 → 2.4

Time: 5.6s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{b1} \cdot \sqrt[3]{b1}\\ t_1 := \sqrt[3]{b2} \cdot \sqrt[3]{b2}\\ t_2 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{t_0 \cdot t_1} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{a2}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;t_2 \leq -2.653892108765896 \cdot 10^{-209}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{t_1}}{t_0} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (* (cbrt b1) (cbrt b1)))
        (t_1 (* (cbrt b2) (cbrt b2)))
        (t_2 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_2 (- INFINITY))
     (*
      (/ (* (cbrt a1) (cbrt a1)) (* t_0 t_1))
      (/ (cbrt a1) (/ (cbrt b1) (/ a2 (cbrt b2)))))
     (if (<= t_2 -2.653892108765896e-209)
       t_2
       (*
        (/ (/ (* (cbrt a2) (cbrt a2)) t_1) t_0)
        (/ 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 = cbrt(b1) * cbrt(b1);
	double t_1 = cbrt(b2) * cbrt(b2);
	double t_2 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((cbrt(a1) * cbrt(a1)) / (t_0 * t_1)) * (cbrt(a1) / (cbrt(b1) / (a2 / cbrt(b2))));
	} else if (t_2 <= -2.653892108765896e-209) {
		tmp = t_2;
	} else {
		tmp = (((cbrt(a2) * cbrt(a2)) / t_1) / t_0) * (a1 / (cbrt(b1) / (cbrt(a2) / cbrt(b2))));
	}
	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	2.4

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

Derivation

1. Split input into 3 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 29.1

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

   3. Simplified 16.2

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

   4. Applied add-cube-cbrt_binary64 17.0

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

   5. Applied *-un-lft-identity_binary64 17.0

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

   6. Applied times-frac_binary64 17.0

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

   7. Applied add-cube-cbrt_binary64 17.2

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

   8. Applied times-frac_binary64 15.4

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

   9. Applied add-cube-cbrt_binary64 15.5

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

   10. Applied times-frac_binary64 6.0

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

   11. Simplified 6.0

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -2.6538921087658958e-209

   1. Initial program 0.8

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

   if -2.6538921087658958e-209 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 12.8

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

   2. Applied associate-/l*_binary64 11.9

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

   3. Simplified 9.7

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

   4. Applied add-cube-cbrt_binary64 10.2

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

   5. Applied add-cube-cbrt_binary64 10.3

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

   6. Applied times-frac_binary64 10.3

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

   7. Applied add-cube-cbrt_binary64 10.4

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

   8. Applied times-frac_binary64 7.5

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

   9. Applied *-un-lft-identity_binary64 7.5

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

   10. Applied times-frac_binary64 2.8

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

   11. Simplified 2.8

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right) \cdot \left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right)} \cdot \frac{\sqrt[3]{a1}}{\frac{\sqrt[3]{b1}}{\frac{a2}{\sqrt[3]{b2}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -2.653892108765896 \cdot 10^{-209}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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