Henrywood and Agarwal, Equation (9a)

Average Accuracy: 78.1% → 85.2%

Time: 18.6s

Precision: binary64

Cost: 28296

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{-d}\right)\right) \cdot \left(D \cdot \frac{D}{-\ell}\right)\right) \cdot -0.25}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\frac{d \cdot \frac{\ell}{D}}{D} \cdot \frac{d}{M \cdot h}} \cdot -0.25}\\ \end{array} \]

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
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (*
      w0
      (sqrt
       (+ 1.0 (* (* (* h (* (/ M d) (/ M (- d)))) (* D (/ D (- l)))) -0.25))))
     (if (<= t_0 2e-54)
       (* w0 (sqrt (- 1.0 t_0)))
       (*
        w0
        (sqrt
         (+ 1.0 (* (/ M (* (/ (* d (/ l D)) D) (/ d (* M h)))) -0.25))))))))

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 t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 + (((h * ((M / d) * (M / -d))) * (D * (D / -l))) * -0.25)));
	} else if (t_0 <= 2e-54) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0 * sqrt((1.0 + ((M / (((d * (l / D)) / D) * (d / (M * h)))) * -0.25)));
	}
	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 t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 + (((h * ((M / d) * (M / -d))) * (D * (D / -l))) * -0.25)));
	} else if (t_0 <= 2e-54) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((M / (((d * (l / D)) / D) * (d / (M * h)))) * -0.25)));
	}
	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):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = w0 * math.sqrt((1.0 + (((h * ((M / d) * (M / -d))) * (D * (D / -l))) * -0.25)))
	elif t_0 <= 2e-54:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0 * math.sqrt((1.0 + ((M / (((d * (l / D)) / D) * (d / (M * h)))) * -0.25)))
	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)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(Float64(M / d) * Float64(M / Float64(-d)))) * Float64(D * Float64(D / Float64(-l)))) * -0.25))));
	elseif (t_0 <= 2e-54)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(M / Float64(Float64(Float64(d * Float64(l / D)) / D) * Float64(d / Float64(M * h)))) * -0.25))));
	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)
	t_0 = (((M * D) / (2.0 * d)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = w0 * sqrt((1.0 + (((h * ((M / d) * (M / -d))) * (D * (D / -l))) * -0.25)));
	elseif (t_0 <= 2e-54)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0 * sqrt((1.0 + ((M / (((d * (l / D)) / D) * (d / (M * h)))) * -0.25)));
	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_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(N[(M / d), $MachinePrecision] * N[(M / (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * N[(D / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-54], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(M / N[(N[(N[(d * N[(l / D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] * N[(d / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 0.0%

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

   2. Simplified 3.7%

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

      [Start] 0.0	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 3.7	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in M around 0 2.8%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]

   4. Simplified 2.4%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}} \]
      Proof

      [Start] 2.8	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}} \]
      times-frac [=>] 2.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}} \]
      unpow2 [=>] 2.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]
      unpow2 [=>] 2.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)} \]
      unpow2 [=>] 2.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)} \]

   5. Applied egg-rr 4.1%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{\left(M \cdot \left(M \cdot \left(-h\right)\right)\right) \cdot \left(-D \cdot D\right)}{\left(d \cdot \left(-d\right)\right) \cdot \left(-\ell\right)}}} \]

   6. Simplified 9.3%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\left(\left(-h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{-d}\right)\right) \cdot \left(\left(-D\right) \cdot \frac{D}{-\ell}\right)\right)}} \]
      Proof

      [Start] 4.1	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(M \cdot \left(M \cdot \left(-h\right)\right)\right) \cdot \left(-D \cdot D\right)}{\left(d \cdot \left(-d\right)\right) \cdot \left(-\ell\right)}} \]
      times-frac [=>] 4.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{M \cdot \left(M \cdot \left(-h\right)\right)}{d \cdot \left(-d\right)} \cdot \frac{-D \cdot D}{-\ell}\right)}} \]
      associate-*r* [=>] 2.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\color{blue}{\left(M \cdot M\right) \cdot \left(-h\right)}}{d \cdot \left(-d\right)} \cdot \frac{-D \cdot D}{-\ell}\right)} \]
      associate-*l/ [<=] 2.7	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\frac{M \cdot M}{d \cdot \left(-d\right)} \cdot \left(-h\right)\right)} \cdot \frac{-D \cdot D}{-\ell}\right)} \]
      *-commutative [=>] 2.7	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\color{blue}{\left(\left(-h\right) \cdot \frac{M \cdot M}{d \cdot \left(-d\right)}\right)} \cdot \frac{-D \cdot D}{-\ell}\right)} \]
      times-frac [=>] 6.5	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\left(-h\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{M}{-d}\right)}\right) \cdot \frac{-D \cdot D}{-\ell}\right)} \]
      distribute-lft-neg-in [=>] 6.5	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\left(-h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{-d}\right)\right) \cdot \frac{\color{blue}{\left(-D\right) \cdot D}}{-\ell}\right)} \]
      associate-*r/ [<=] 9.3	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\left(-h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{-d}\right)\right) \cdot \color{blue}{\left(\left(-D\right) \cdot \frac{D}{-\ell}\right)}\right)} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 2.0000000000000001e-54

   1. Initial program 99.8%

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

   if 2.0000000000000001e-54 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   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 17.7%

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

      [Start] 14.5	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 17.7	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in M around 0 42.5%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]

   4. Simplified 40.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)}} \]
      Proof

      [Start] 42.5	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}} \]
      times-frac [=>] 40.9	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\left(\frac{{D}^{2}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}} \]
      unpow2 [=>] 40.9	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{\color{blue}{D \cdot D}}{\ell} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]
      unpow2 [=>] 40.9	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{{d}^{2}}\right)} \]
      unpow2 [=>] 40.9	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d \cdot d}}\right)} \]

   5. Applied egg-rr 66.9%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \color{blue}{\frac{M}{\left(\frac{\frac{\ell}{D}}{D} \cdot d\right) \cdot \frac{d}{M \cdot h}}}} \]

   6. Applied egg-rr 72.0%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{M}{\color{blue}{\frac{\frac{\ell}{D} \cdot d}{D}} \cdot \frac{d}{M \cdot h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{-d}\right)\right) \cdot \left(D \cdot \frac{D}{-\ell}\right)\right) \cdot -0.25}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\frac{d \cdot \frac{\ell}{D}}{D} \cdot \frac{d}{M \cdot h}} \cdot -0.25}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.5%
Cost	14344

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{M \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{\ell}{D} \cdot \frac{d}{D}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -4 \cdot 10^{-173}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Accuracy	84.5%
Cost	8264

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{M \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{\ell}{D} \cdot \frac{d}{D}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -4 \cdot 10^{-173}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot t_0}{\frac{\ell}{h} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Accuracy	79.0%
Cost	8004

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+245}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \frac{M \cdot \left(h \cdot \frac{M}{d}\right)}{\frac{\ell}{D} \cdot \frac{d}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Accuracy	77.5%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;w0 \leq -2.2 \cdot 10^{-217}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\frac{d \cdot \frac{\ell}{D}}{D} \cdot \frac{d}{M \cdot h}} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Accuracy	78.2%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;D \leq 2 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\frac{d \cdot \frac{\ell}{D}}{D} \cdot \frac{d}{M \cdot h}} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot d}\right) \cdot \frac{M \cdot D}{4}}\\ \end{array} \]

Alternative 6
Accuracy	78.8%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023122 
(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))))))