Henrywood and Agarwal, Equation (9a)

Average Error: 13.9 → 9.9

Time: 16.0s

Precision: binary64

Cost: 14344

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -3.6 \cdot 10^{-176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

↓

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) (- INFINITY))
   w0
   (if (<= (/ h l) -3.6e-176)
     (* w0 (sqrt (- 1.0 (* (pow (/ (* M 0.5) (/ d D)) 2.0) (/ h l)))))
     w0)))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0;
	} else if ((h / l) <= -3.6e-176) {
		tmp = w0 * sqrt((1.0 - (pow(((M * 0.5) / (d / D)), 2.0) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0;
	} else if ((h / l) <= -3.6e-176) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(((M * 0.5) / (d / D)), 2.0) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

↓

def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0
	elif (h / l) <= -3.6e-176:
		tmp = w0 * math.sqrt((1.0 - (math.pow(((M * 0.5) / (d / D)), 2.0) * (h / l))))
	else:
		tmp = w0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

↓

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = w0;
	elseif (Float64(h / l) <= -3.6e-176)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0) * Float64(h / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

↓

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -Inf)
		tmp = w0;
	elseif ((h / l) <= -3.6e-176)
		tmp = w0 * sqrt((1.0 - ((((M * 0.5) / (d / D)) ^ 2.0) * (h / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], w0, If[LessEqual[N[(h / l), $MachinePrecision], -3.6e-176], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -inf.0 or -3.6000000000000003e-176 < (/.f64 h l)

   1. Initial program 13.6

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Simplified 13.4

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
      Proof

      [Start] 13.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational.json-simplify-43 [=>] 13.4	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in D around 0 7.3

      \[\leadsto \color{blue}{w0} \]

   if -inf.0 < (/.f64 h l) < -3.6000000000000003e-176

   1. Initial program 14.5

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Simplified 14.4

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
      Proof

      [Start] 14.5	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational.json-simplify-43 [=>] 14.4	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 14.2

      \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -3.6 \cdot 10^{-176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	27588

\[\begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;w0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	10.0
Cost	14344

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -8.5 \cdot 10^{-176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	13.7
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))