Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 86.2%

Time: 15.6s

Alternatives: 5

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: 80.8% 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: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - \frac{h}{\frac{\ell}{0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}}}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (/ h (/ l (* 0.25 (pow (/ (* D M) d) 2.0))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h / (l / (0.25 * pow(((D * M) / d), 2.0))))));
}

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 / (l / (0.25d0 * (((d * m) / d_1) ** 2.0d0))))))
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 / (l / (0.25 * Math.pow(((D * M) / d), 2.0))))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (h / (l / (0.25 * math.pow(((D * M) / d), 2.0))))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(l / Float64(0.25 * (Float64(Float64(D * M) / d) ^ 2.0)))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h / (l / (0.25 * (((D * M) / d) ^ 2.0))))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(l / N[(0.25 * N[Power[N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

3. Simplified 75.8%

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

   1. associate-*r/ 83.4%

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

   2. clear-num 83.4%

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

   3. associate-*l/ 84.7%

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

   4. div-inv 84.7%

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

   5. associate-*l* 83.8%

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

   6. associate-/r* 83.8%

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

   7. metadata-eval 83.8%

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

5. Applied egg-rr 83.8%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\frac{\ell}{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}}}} \]
6. Step-by-step derivation

   1. associate-/r/ 83.8%

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

   2. *-commutative 83.8%

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

   3. *-commutative 83.8%

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

   4. associate-*r* 84.8%

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

   5. *-commutative 84.8%

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

   6. associate-*l/ 84.7%

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

   7. *-commutative 84.7%

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

   8. associate-*r/ 84.7%

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

   9. associate-*r/ 83.4%

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

7. Simplified 83.4%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right) \cdot \frac{1}{\ell}}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 83.1%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right) \cdot \frac{1}{\ell}}\right)\right)} \]

   2. expm1-udef 83.1%

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

   3. un-div-inv 83.1%

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

   4. unpow-prod-down 83.1%

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

   5. metadata-eval 83.1%

      \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{h \cdot \left(\color{blue}{0.25} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}}\right)} - 1\right) \]

   6. associate-*r/ 84.4%

      \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)}{\ell}}\right)} - 1\right) \]

9. Applied egg-rr 84.4%

   \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}{\ell}}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 84.4%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}{\ell}}\right)\right)} \]

    2. expm1-log1p 84.7%

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

    3. associate-/l* 84.7%

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

    4. associate-/l* 84.1%

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

11. Simplified 84.1%

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

12. Taylor expanded in D around 0 84.7%

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

13. Final simplification 84.7%

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

Alternative 2: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 2 \cdot 10^{+27}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 2e+27)
   w0
   (* w0 (fma -0.125 (* (pow (* D (/ M d)) 2.0) (/ h l)) 1.0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2e+27) {
		tmp = w0;
	} else {
		tmp = w0 * fma(-0.125, (pow((D * (M / d)), 2.0) * (h / l)), 1.0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 2e+27)
		tmp = w0;
	else
		tmp = Float64(w0 * fma(-0.125, Float64((Float64(D * Float64(M / d)) ^ 2.0) * Float64(h / l)), 1.0));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 2e+27], w0, N[(w0 * N[(-0.125 * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2e27

   1. Initial program 78.5%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in M around 0 71.7%

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

   if 2e27 < D

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

      1. associate-*r/ 76.2%

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

      2. clear-num 76.2%

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

      3. associate-*l/ 76.2%

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

      4. div-inv 76.2%

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

      5. associate-*l* 74.1%

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

      6. associate-/r* 74.1%

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

      7. metadata-eval 74.1%

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

   5. Applied egg-rr 74.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\frac{\ell}{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*r* 76.2%

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

      5. *-commutative 76.2%

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

      6. associate-*l/ 76.2%

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

      7. *-commutative 76.2%

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

      8. associate-*r/ 76.2%

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

      9. associate-*r/ 76.2%

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

   7. Simplified 76.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right) \cdot \frac{1}{\ell}}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 76.0%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{2}\right) \cdot \frac{1}{\ell}}\right)\right)} \]

      2. expm1-udef 76.0%

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

      3. un-div-inv 76.0%

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

      4. unpow-prod-down 76.0%

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

      5. metadata-eval 76.0%

         \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{h \cdot \left(\color{blue}{0.25} \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}}\right)} - 1\right) \]

      6. associate-*r/ 76.0%

         \[\leadsto w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2}\right)}{\ell}}\right)} - 1\right) \]

   9. Applied egg-rr 76.0%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}{\ell}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 76.0%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{h \cdot \left(0.25 \cdot {\left(\frac{D \cdot M}{d}\right)}^{2}\right)}{\ell}}\right)\right)} \]

       2. expm1-log1p 76.2%

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

       3. associate-/l* 76.2%

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

       4. associate-/l* 76.2%

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

   11. Simplified 76.2%

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

   12. Taylor expanded in D around 0 76.2%

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

   13. Taylor expanded in h around 0 53.3%

       \[\leadsto w0 \cdot \color{blue}{\left(1 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   14. Step-by-step derivation

       1. +-commutative 53.3%

          \[\leadsto w0 \cdot \color{blue}{\left(-0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} + 1\right)} \]

       2. fma-def 53.3%

          \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}, 1\right)} \]

       3. associate-*r* 53.3%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}, 1\right) \]

       4. times-frac 51.3%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \color{blue}{\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}}, 1\right) \]

       5. unpow2 51.3%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h}{\ell}, 1\right) \]

       6. unpow2 51.3%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{{d}^{2}} \cdot \frac{h}{\ell}, 1\right) \]

       7. swap-sqr 64.0%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{{d}^{2}} \cdot \frac{h}{\ell}, 1\right) \]

       8. unpow2 64.0%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{d \cdot d}} \cdot \frac{h}{\ell}, 1\right) \]

       9. times-frac 68.1%

          \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)} \cdot \frac{h}{\ell}, 1\right) \]

       10. associate-*l/ 66.0%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \frac{D \cdot M}{d}\right) \cdot \frac{h}{\ell}, 1\right) \]

       11. associate-/r/ 68.1%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \left(\color{blue}{\frac{D}{\frac{d}{M}}} \cdot \frac{D \cdot M}{d}\right) \cdot \frac{h}{\ell}, 1\right) \]

       12. associate-*l/ 66.0%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\left(\frac{D}{d} \cdot M\right)}\right) \cdot \frac{h}{\ell}, 1\right) \]

       13. associate-/r/ 68.1%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right) \cdot \frac{h}{\ell}, 1\right) \]

       14. unpow2 68.1%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, \color{blue}{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}} \cdot \frac{h}{\ell}, 1\right) \]

       15. associate-/r/ 66.0%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, {\color{blue}{\left(\frac{D}{d} \cdot M\right)}}^{2} \cdot \frac{h}{\ell}, 1\right) \]

       16. associate-*l/ 68.1%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, {\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2} \cdot \frac{h}{\ell}, 1\right) \]

       17. *-commutative 68.1%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, {\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right) \]

       18. associate-*l/ 68.1%

           \[\leadsto w0 \cdot \mathsf{fma}\left(-0.125, {\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}, 1\right) \]

   15. Simplified 68.1%

       \[\leadsto w0 \cdot \color{blue}{\mathsf{fma}\left(-0.125, {\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\ell}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2 \cdot 10^{+27}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, {\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}, 1\right)\\ \end{array} \]

Alternative 3: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 - h \cdot \frac{{\left(\frac{M}{\frac{d}{D \cdot 0.5}}\right)}^{2}}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ (pow (/ M (/ d (* D 0.5))) 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((M / (d / (D * 0.5))), 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 * (((m / (d_1 / (d * 0.5d0))) ** 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((M / (d / (D * 0.5))), 2.0) / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (h * (math.pow((M / (d / (D * 0.5))), 2.0) / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M / Float64(d / Float64(D * 0.5))) ^ 2.0) / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M / (d / (D * 0.5))) ^ 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[(M / N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

3. Simplified 75.8%

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

   1. associate-*r/ 83.4%

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

   2. associate-*l/ 84.7%

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

   3. div-inv 84.7%

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

   4. associate-*l* 83.8%

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

   5. associate-/r* 83.8%

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

   6. metadata-eval 83.8%

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

5. Applied egg-rr 83.8%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 83.4%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2} \cdot h}{\ell}}\right)\right)} \]

   2. expm1-udef 83.4%

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

   3. associate-/l* 77.3%

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

   4. associate-*r/ 77.3%

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

7. Applied egg-rr 77.3%

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

   1. expm1-def 77.3%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \frac{{\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2}}{\frac{\ell}{h}}}\right)\right)} \]

   2. expm1-log1p 77.6%

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

   3. associate-/r/ 83.8%

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

   4. associate-*r/ 84.7%

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

   5. associate-/l* 84.1%

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

9. Simplified 84.1%

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

10. Final simplification 84.1%

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

Alternative 4: 72.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 2.7 \cdot 10^{+84}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 2.7e+84)
   w0
   (+ w0 (* -0.125 (* (pow (* D (/ M d)) 2.0) (/ h (/ l w0)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2.7e+84) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow((D * (M / d)), 2.0) * (h / (l / w0))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= 2.7d+84) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d * (m / d_1)) ** 2.0d0) * (h / (l / w0))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2.7e+84) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow((D * (M / d)), 2.0) * (h / (l / w0))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 2.7e+84:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow((D * (M / d)), 2.0) * (h / (l / w0))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 2.7e+84)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(D * Float64(M / d)) ^ 2.0) * Float64(h / Float64(l / w0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 2.7e+84)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D * (M / d)) ^ 2.0) * (h / (l / w0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 2.7e+84], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / N[(l / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.7e84

   1. Initial program 77.8%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in M around 0 69.7%

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

   if 2.7e84 < D

   1. Initial program 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in M around 0 49.7%

      \[\leadsto \color{blue}{w0 + -0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 28.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right)}}{{d}^{2} \cdot \ell} \]

      2. expm1-udef 28.0%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)} - 1}}{{d}^{2} \cdot \ell} \]

      3. associate-*r* 28.0%

         \[\leadsto w0 + -0.125 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}\right)} - 1}{{d}^{2} \cdot \ell} \]

      4. unpow-prod-down 37.4%

         \[\leadsto w0 + -0.125 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot \left(h \cdot w0\right)\right)} - 1}{{d}^{2} \cdot \ell} \]

   6. Applied egg-rr 37.4%

      \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)} - 1}}{{d}^{2} \cdot \ell} \]
   7. Step-by-step derivation

      1. expm1-def 37.5%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)\right)\right)}}{{d}^{2} \cdot \ell} \]

      2. expm1-log1p 62.2%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

      3. *-commutative 62.2%

         \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left(h \cdot w0\right) \cdot {\left(D \cdot M\right)}^{2}}}{{d}^{2} \cdot \ell} \]

   8. Simplified 62.2%

      \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left(h \cdot w0\right) \cdot {\left(D \cdot M\right)}^{2}}}{{d}^{2} \cdot \ell} \]
   9. Step-by-step derivation

      1. unpow2 62.2%

         \[\leadsto w0 + -0.125 \cdot \frac{\left(h \cdot w0\right) \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}}{{d}^{2} \cdot \ell} \]

   10. Applied egg-rr 62.2%

       \[\leadsto w0 + -0.125 \cdot \frac{\left(h \cdot w0\right) \cdot \color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}}{{d}^{2} \cdot \ell} \]

   11. Taylor expanded in h around 0 49.7%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   12. Step-by-step derivation

       1. associate-*r* 49.7%

          \[\leadsto w0 + -0.125 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot \left(h \cdot w0\right)}}{{d}^{2} \cdot \ell} \]

       2. times-frac 49.4%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}\right)} \]

       3. unpow2 49.4%

          \[\leadsto w0 + -0.125 \cdot \left(\frac{\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}\right) \]

       4. unpow2 49.4%

          \[\leadsto w0 + -0.125 \cdot \left(\frac{\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}\right) \]

       5. swap-sqr 61.9%

          \[\leadsto w0 + -0.125 \cdot \left(\frac{\color{blue}{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}}{{d}^{2}} \cdot \frac{h \cdot w0}{\ell}\right) \]

       6. unpow2 61.9%

          \[\leadsto w0 + -0.125 \cdot \left(\frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{\color{blue}{d \cdot d}} \cdot \frac{h \cdot w0}{\ell}\right) \]

       7. times-frac 65.0%

          \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)} \cdot \frac{h \cdot w0}{\ell}\right) \]

       8. associate-*l/ 61.9%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\color{blue}{\left(\frac{D}{d} \cdot M\right)} \cdot \frac{D \cdot M}{d}\right) \cdot \frac{h \cdot w0}{\ell}\right) \]

       9. associate-/r/ 64.8%

          \[\leadsto w0 + -0.125 \cdot \left(\left(\color{blue}{\frac{D}{\frac{d}{M}}} \cdot \frac{D \cdot M}{d}\right) \cdot \frac{h \cdot w0}{\ell}\right) \]

       10. associate-*l/ 61.8%

           \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\left(\frac{D}{d} \cdot M\right)}\right) \cdot \frac{h \cdot w0}{\ell}\right) \]

       11. associate-/r/ 64.8%

           \[\leadsto w0 + -0.125 \cdot \left(\left(\frac{D}{\frac{d}{M}} \cdot \color{blue}{\frac{D}{\frac{d}{M}}}\right) \cdot \frac{h \cdot w0}{\ell}\right) \]

       12. unpow2 64.8%

           \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}} \cdot \frac{h \cdot w0}{\ell}\right) \]

       13. associate-/r/ 61.9%

           \[\leadsto w0 + -0.125 \cdot \left({\color{blue}{\left(\frac{D}{d} \cdot M\right)}}^{2} \cdot \frac{h \cdot w0}{\ell}\right) \]

       14. associate-*l/ 65.0%

           \[\leadsto w0 + -0.125 \cdot \left({\color{blue}{\left(\frac{D \cdot M}{d}\right)}}^{2} \cdot \frac{h \cdot w0}{\ell}\right) \]

       15. *-commutative 65.0%

           \[\leadsto w0 + -0.125 \cdot \left({\left(\frac{\color{blue}{M \cdot D}}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right) \]

       16. associate-*l/ 64.8%

           \[\leadsto w0 + -0.125 \cdot \left({\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot \frac{h \cdot w0}{\ell}\right) \]

       17. associate-/l* 67.8%

           \[\leadsto w0 + -0.125 \cdot \left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \color{blue}{\frac{h}{\frac{\ell}{w0}}}\right) \]

   13. Simplified 67.8%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.7 \cdot 10^{+84}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\\ \end{array} \]

Alternative 5: 68.2% 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 77.3%

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

3. Simplified 75.8%

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

4. Taylor expanded in M around 0 66.1%

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

5. Final simplification 66.1%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023338 
(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))))))