Henrywood and Agarwal, Equation (9a)

Average Accuracy: 78.8% → 85.9%

Time: 19.9s

Precision: binary64

Cost: 8392

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-267}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

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) -5e-267)
     (*
      w0
      (sqrt
       (+ 1.0 (/ (* M (* (/ D d) -0.5)) (* (/ l h) (/ 2.0 (* M (/ D d))))))))
     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) <= -5e-267) {
		tmp = w0 * sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l / h) * (2.0 / (M * (D / d)))))));
	} 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) <= -5e-267) {
		tmp = w0 * Math.sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l / h) * (2.0 / (M * (D / d)))))));
	} 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) <= -5e-267:
		tmp = w0 * math.sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l / h) * (2.0 / (M * (D / d)))))))
	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) <= -5e-267)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(M * Float64(Float64(D / d) * -0.5)) / Float64(Float64(l / h) * Float64(2.0 / Float64(M * Float64(D / d))))))));
	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) <= -5e-267)
		tmp = w0 * sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l / h) * (2.0 / (M * (D / d)))))));
	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], -5e-267], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(M * N[(N[(D / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(2.0 / N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -inf.0 or -4.9999999999999999e-267 < (/.f64 h l)

   1. Initial program 78.7%

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

   2. Simplified 78.5%

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

      [Start] 78.7	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 78.5	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in M around 0 89.4%

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

   if -inf.0 < (/.f64 h l) < -4.9999999999999999e-267

   1. Initial program 78.9%

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

   2. Simplified 79.2%

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

      [Start] 78.9	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-/l* [=>] 79.2	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 81.5%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}} \]
      Proof

      [Start] 79.2	\[ w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}} \]
      associate-*r/ [=>] 77.3	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}} \]
      associate-/l* [=>] 79.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}{\frac{\ell}{h}}}} \]
      unpow2 [=>] 79.2	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h}}} \]
      clear-num [=>] 79.2	\[ w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{2 \cdot d}{D}} \cdot \color{blue}{\frac{1}{\frac{\frac{2 \cdot d}{D}}{M}}}}{\frac{\ell}{h}}} \]
      un-div-inv [=>] 79.2	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\frac{\frac{2 \cdot d}{D}}{M}}}}{\frac{\ell}{h}}} \]
      associate-/l/ [=>] 81.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}}} \]
      div-inv [=>] 81.5	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{M \cdot \frac{1}{\frac{2 \cdot d}{D}}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/l* [=>] 81.5	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \frac{1}{\color{blue}{\frac{2}{\frac{D}{d}}}}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/r/ [=>] 81.5	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{D}{d}\right)}}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      metadata-eval [=>] 81.5	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(\color{blue}{0.5} \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{\frac{2 \cdot d}{D}}{M}}} \]
      associate-/l* [=>] 81.5	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \frac{\color{blue}{\frac{2}{\frac{D}{d}}}}{M}}} \]
      associate-/l/ [=>] 81.5	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(0.5 \cdot \frac{D}{d}\right)}{\frac{\ell}{h} \cdot \color{blue}{\frac{2}{M \cdot \frac{D}{d}}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-267}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.6%
Cost	8264

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-239}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(t_0 \cdot \frac{\frac{h}{\ell}}{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Accuracy	85.6%
Cost	8264

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -5 \cdot 10^{-239}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{t_0}{\frac{\ell}{h} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Accuracy	78.6%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;M \leq -1.68 \cdot 10^{+131}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq -6.8 \cdot 10^{-86}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{M \cdot \frac{M}{\ell}}{4 \cdot \frac{\frac{d}{D}}{\frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Accuracy	78.3%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023137 
(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))))))