Henrywood and Agarwal, Equation (3)

Average Error: 19.5 → 12.2

Time: 4.8s

Precision: binary64

Math

\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -6.6560613513075085 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\frac{V}{\frac{\sqrt[3]{A}}{\ell}}}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 2.23 \cdot 10^{-321}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 1.0135061622844712 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

↓

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -6.6560613513075085e-183)
   (* (sqrt (/ (* (cbrt A) (cbrt A)) (/ V (/ (cbrt A) l)))) c0)
   (if (<= (* V l) 2.23e-321)
     (* c0 (sqrt (/ 1.0 (/ V (/ A l)))))
     (if (<= (* V l) 1.0135061622844712e+304)
       (* c0 (/ (sqrt A) (sqrt (* V l))))
       (* c0 (sqrt (* (/ A l) (/ 1.0 V))))))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

↓

double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -6.6560613513075085e-183) {
		tmp = sqrt(((cbrt(A) * cbrt(A)) / (V / (cbrt(A) / l)))) * c0;
	} else if ((V * l) <= 2.23e-321) {
		tmp = c0 * sqrt((1.0 / (V / (A / l))));
	} else if ((V * l) <= 1.0135061622844712e+304) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = c0 * sqrt(((A / l) * (1.0 / V)));
	}
	return tmp;
}

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

↓

public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -6.6560613513075085e-183) {
		tmp = Math.sqrt(((Math.cbrt(A) * Math.cbrt(A)) / (V / (Math.cbrt(A) / l)))) * c0;
	} else if ((V * l) <= 2.23e-321) {
		tmp = c0 * Math.sqrt((1.0 / (V / (A / l))));
	} else if ((V * l) <= 1.0135061622844712e+304) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = c0 * Math.sqrt(((A / l) * (1.0 / V)));
	}
	return tmp;
}

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

↓

function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -6.6560613513075085e-183)
		tmp = Float64(sqrt(Float64(Float64(cbrt(A) * cbrt(A)) / Float64(V / Float64(cbrt(A) / l)))) * c0);
	elseif (Float64(V * l) <= 2.23e-321)
		tmp = Float64(c0 * sqrt(Float64(1.0 / Float64(V / Float64(A / l)))));
	elseif (Float64(V * l) <= 1.0135061622844712e+304)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) * Float64(1.0 / V))));
	end
	return tmp
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -6.6560613513075085e-183], N[(N[Sqrt[N[(N[(N[Power[A, 1/3], $MachinePrecision] * N[Power[A, 1/3], $MachinePrecision]), $MachinePrecision] / N[(V / N[(N[Power[A, 1/3], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2.23e-321], N[(c0 * N[Sqrt[N[(1.0 / N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1.0135061622844712e+304], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(1.0 / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -6.6560613513075085e-183

   1. Initial program 14.9

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Applied add-sqr-sqrt_binary64 15.1

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)} \]

   3. Applied associate-*r*_binary64 15.1

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}} \]

   4. Applied pow1_binary64 15.1

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \]

   5. Applied pow1_binary64 15.1

      \[\leadsto \left(c0 \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}}\right) \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   6. Applied pow1_binary64 15.1

      \[\leadsto \left(\color{blue}{{c0}^{1}} \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}\right) \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   7. Applied pow-prod-down_binary64 15.1

      \[\leadsto \color{blue}{{\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   8. Applied pow-prod-down_binary64 15.1

      \[\leadsto \color{blue}{{\left(\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \]

   9. Simplified 14.9

      \[\leadsto {\color{blue}{\left(\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\right)}}^{1} \]

   10. Applied add-cube-cbrt_binary64 15.3

       \[\leadsto {\left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A}}}{V \cdot \ell}} \cdot c0\right)}^{1} \]

   11. Applied associate-/l*_binary64 15.3

       \[\leadsto {\left(\sqrt{\color{blue}{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\frac{V \cdot \ell}{\sqrt[3]{A}}}}} \cdot c0\right)}^{1} \]

   12. Simplified 15.0

       \[\leadsto {\left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\color{blue}{\frac{V}{\frac{\sqrt[3]{A}}{\ell}}}}} \cdot c0\right)}^{1} \]

   if -6.6560613513075085e-183 < (*.f64 V l) < 2.22824e-321

   1. Initial program 46.7

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Applied add-sqr-sqrt_binary64 46.8

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)} \]

   3. Applied associate-*r*_binary64 46.8

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}} \]

   4. Applied pow1_binary64 46.8

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \]

   5. Applied pow1_binary64 46.8

      \[\leadsto \left(c0 \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}}\right) \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   6. Applied pow1_binary64 46.8

      \[\leadsto \left(\color{blue}{{c0}^{1}} \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}\right) \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   7. Applied pow-prod-down_binary64 46.8

      \[\leadsto \color{blue}{{\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   8. Applied pow-prod-down_binary64 46.8

      \[\leadsto \color{blue}{{\left(\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \]

   9. Simplified 46.7

      \[\leadsto {\color{blue}{\left(\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\right)}}^{1} \]

   10. Applied clear-num_binary64 46.7

       \[\leadsto {\left(\sqrt{\color{blue}{\frac{1}{\frac{V \cdot \ell}{A}}}} \cdot c0\right)}^{1} \]

   11. Simplified 30.8

       \[\leadsto {\left(\sqrt{\frac{1}{\color{blue}{\frac{V}{\frac{A}{\ell}}}}} \cdot c0\right)}^{1} \]

   if 2.22824e-321 < (*.f64 V l) < 1.01350616228447121e304

   1. Initial program 10.1

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Applied sqrt-div_binary64 0.5

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}} \]

   if 1.01350616228447121e304 < (*.f64 V l)

   1. Initial program 41.6

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Applied add-sqr-sqrt_binary64 41.6

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)} \]

   3. Applied associate-*r*_binary64 41.6

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}} \]

   4. Applied pow1_binary64 41.6

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \]

   5. Applied pow1_binary64 41.6

      \[\leadsto \left(c0 \cdot \color{blue}{{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}}\right) \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   6. Applied pow1_binary64 41.6

      \[\leadsto \left(\color{blue}{{c0}^{1}} \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}\right) \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   7. Applied pow-prod-down_binary64 41.6

      \[\leadsto \color{blue}{{\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \cdot {\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1} \]

   8. Applied pow-prod-down_binary64 41.6

      \[\leadsto \color{blue}{{\left(\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}^{1}} \]

   9. Simplified 41.6

      \[\leadsto {\color{blue}{\left(\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\right)}}^{1} \]

   10. Applied *-un-lft-identity_binary64 41.6

       \[\leadsto {\left(\sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}} \cdot c0\right)}^{1} \]

   11. Applied times-frac_binary64 22.5

       \[\leadsto {\left(\sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}} \cdot c0\right)}^{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -6.6560613513075085 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\frac{V}{\frac{\sqrt[3]{A}}{\ell}}}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 2.23 \cdot 10^{-321}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 1.0135061622844712 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))