Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 89.8%

Time: 19.7s

Alternatives: 3

Speedup: 1.0×

Specification

Math

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

FPCore

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

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))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

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))));
}

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))))

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

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

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]

Initial Program: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

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))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

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))));
}

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))))

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

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

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]

Alternative 1: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+145}:\\ \;\;\;\;t\_0 \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot e^{\mathsf{log1p}\left(h \cdot \left(\frac{D \cdot \frac{M}{2 \cdot d}}{\ell} \cdot \left(D \cdot \frac{M}{d \cdot \left(-2\right)}\right)\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
   (if (<= t_0 5e+145)
     (* t_0 w0)
     (*
      w0
      (exp
       (*
        (log1p (* h (* (/ (* D (/ M (* 2.0 d))) l) (* D (/ M (* d (- 2.0)))))))
        0.5))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+145) {
		tmp = t_0 * w0;
	} else {
		tmp = w0 * exp((log1p((h * (((D * (M / (2.0 * d))) / l) * (D * (M / (d * -2.0)))))) * 0.5));
	}
	return tmp;
}

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 5e+145) {
		tmp = t_0 * w0;
	} else {
		tmp = w0 * Math.exp((Math.log1p((h * (((D * (M / (2.0 * d))) / l) * (D * (M / (d * -2.0)))))) * 0.5));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 5e+145:
		tmp = t_0 * w0
	else:
		tmp = w0 * math.exp((math.log1p((h * (((D * (M / (2.0 * d))) / l) * (D * (M / (d * -2.0)))))) * 0.5))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 5e+145)
		tmp = Float64(t_0 * w0);
	else
		tmp = Float64(w0 * exp(Float64(log1p(Float64(h * Float64(Float64(Float64(D * Float64(M / Float64(2.0 * d))) / l) * Float64(D * Float64(M / Float64(d * Float64(-2.0))))))) * 0.5)));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 5e+145], N[(t$95$0 * w0), $MachinePrecision], N[(w0 * N[Exp[N[(N[Log[1 + N[(h * N[(N[(N[(D * N[(M / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D * N[(M / N[(d * (-2.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))) < 4.99999999999999967e145

   1. Initial program 99.9%

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

   if 4.99999999999999967e145 < (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))

   1. Initial program 38.5%

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

   3. Simplified 43.1%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 43.1%

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

      2. un-div-inv 43.1%

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

      3. associate-/l* 43.1%

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

   6. Applied egg-rr 43.1%

      \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]
   7. Step-by-step derivation

      1. associate-*r/ 63.7%

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

      2. *-un-lft-identity 63.7%

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

      3. *-commutative 63.7%

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

      4. times-frac 63.7%

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

      5. clear-num 63.7%

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

      6. times-frac 60.2%

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

   8. Applied egg-rr 60.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}} \]
   9. Step-by-step derivation

      1. *-commutative 60.2%

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

      2. associate-/l* 60.2%

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

      3. associate-/l* 64.8%

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

      4. *-commutative 64.8%

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

   10. Simplified 64.8%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{h \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}} \]
   11. Step-by-step derivation

       1. pow1/2 64.9%

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

       2. pow-to-exp 64.4%

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

       3. cancel-sign-sub-inv 64.4%

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

       4. log1p-define 64.4%

          \[\leadsto w0 \cdot e^{\color{blue}{\mathsf{log1p}\left(\left(-h\right) \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}\right)} \cdot 0.5} \]

       5. associate-*r/ 59.9%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{2 \cdot d}\right)}}^{2}}{\ell}\right) \cdot 0.5} \]

   12. Applied egg-rr 59.9%

       \[\leadsto w0 \cdot \color{blue}{e^{\mathsf{log1p}\left(\left(-h\right) \cdot \frac{{\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2}}{\ell}\right) \cdot 0.5}} \]
   13. Step-by-step derivation

       1. associate-/l* 64.4%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \frac{{\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2}}{\ell}\right) \cdot 0.5} \]

       2. *-commutative 64.4%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \frac{{\left(D \cdot \frac{M}{\color{blue}{d \cdot 2}}\right)}^{2}}{\ell}\right) \cdot 0.5} \]

       3. associate-/l* 59.9%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \frac{{\color{blue}{\left(\frac{D \cdot M}{d \cdot 2}\right)}}^{2}}{\ell}\right) \cdot 0.5} \]

       4. unpow2 59.9%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \frac{\color{blue}{\frac{D \cdot M}{d \cdot 2} \cdot \frac{D \cdot M}{d \cdot 2}}}{\ell}\right) \cdot 0.5} \]

       5. *-un-lft-identity 59.9%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \frac{\frac{D \cdot M}{d \cdot 2} \cdot \frac{D \cdot M}{d \cdot 2}}{\color{blue}{1 \cdot \ell}}\right) \cdot 0.5} \]

       6. times-frac 63.4%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \color{blue}{\left(\frac{\frac{D \cdot M}{d \cdot 2}}{1} \cdot \frac{\frac{D \cdot M}{d \cdot 2}}{\ell}\right)}\right) \cdot 0.5} \]

       7. associate-/l* 63.4%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \left(\frac{\color{blue}{D \cdot \frac{M}{d \cdot 2}}}{1} \cdot \frac{\frac{D \cdot M}{d \cdot 2}}{\ell}\right)\right) \cdot 0.5} \]

       8. associate-/l* 70.2%

          \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \left(\frac{D \cdot \frac{M}{d \cdot 2}}{1} \cdot \frac{\color{blue}{D \cdot \frac{M}{d \cdot 2}}}{\ell}\right)\right) \cdot 0.5} \]

   14. Applied egg-rr 70.2%

       \[\leadsto w0 \cdot e^{\mathsf{log1p}\left(\left(-h\right) \cdot \color{blue}{\left(\frac{D \cdot \frac{M}{d \cdot 2}}{1} \cdot \frac{D \cdot \frac{M}{d \cdot 2}}{\ell}\right)}\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot e^{\mathsf{log1p}\left(h \cdot \left(\frac{D \cdot \frac{M}{2 \cdot d}}{\ell} \cdot \left(D \cdot \frac{M}{d \cdot \left(-2\right)}\right)\right)\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - (h * (((d * (m / (2.0d0 * d_1))) ** 2.0d0) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Simplified 80.9%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 80.9%

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

   2. un-div-inv 81.7%

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

   3. associate-/l* 81.7%

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

6. Applied egg-rr 81.7%

   \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]
7. Step-by-step derivation

   1. associate-*r/ 88.0%

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

   2. *-un-lft-identity 88.0%

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

   3. *-commutative 88.0%

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

   4. times-frac 87.2%

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

   5. clear-num 87.2%

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

   6. times-frac 86.9%

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

8. Applied egg-rr 86.9%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot h}{\ell}}} \]
9. Step-by-step derivation

   1. *-commutative 86.9%

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

   2. associate-/l* 86.5%

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

   3. associate-/l* 86.1%

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

   4. *-commutative 86.1%

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

10. Simplified 86.1%

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

11. Final simplification 86.1%

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

Alternative 3: 68.7% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

def code(w0, M, D, h, l, d):
	return w0

Julia

function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 80.2%

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

3. Simplified 80.9%

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

5. Taylor expanded in M around 0 64.2%

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

6. Final simplification 64.2%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024061 
(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))))))