Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.4% → 85.6%

Time: 15.8s

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: 80.4% 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: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (h * (math.pow((M * ((D * 0.5) / d)), 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(Float64(D * 0.5) / d)) ^ 2.0) / l)))))
end

MATLAB

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

Derivation

1. Initial program 78.0%

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

3. Simplified 78.0%

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

   1. associate-/r* 78.0%

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

   2. associate-*r/ 78.0%

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

   3. *-commutative 78.0%

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

   4. associate-*r/ 78.4%

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

   5. unpow2 78.4%

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

   6. unpow2 78.4%

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

   7. associate-*r/ 85.1%

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

   8. clear-num 85.1%

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

   9. unpow2 85.1%

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

   10. unpow2 85.1%

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

   11. associate-*r/ 84.3%

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

   12. times-frac 85.1%

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

   13. div-inv 85.1%

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

   14. metadata-eval 85.1%

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

6. Applied egg-rr 85.1%

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

   1. associate-/r/ 85.1%

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

   2. *-commutative 85.1%

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

   3. associate-*r/ 84.3%

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

   4. *-commutative 84.3%

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

   5. associate-/l* 84.3%

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

   6. associate-*r/ 84.3%

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

8. Simplified 84.3%

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

   1. *-un-lft-identity 84.3%

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

   2. associate-*l/ 84.3%

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

   3. *-un-lft-identity 84.3%

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

10. Applied egg-rr 84.3%

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

    1. *-lft-identity 84.3%

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

    2. associate-/l* 85.0%

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

    3. associate-*r/ 85.1%

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

    4. associate-*r/ 84.7%

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

    5. *-commutative 84.7%

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

    6. associate-/l* 85.1%

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

    7. associate-*r* 85.1%

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

    8. *-commutative 85.1%

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

    9. associate-*l/ 85.1%

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

12. Simplified 85.1%

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

13. Final simplification 85.1%

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

Alternative 2: 69.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 5.4 \cdot 10^{+143}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(M \cdot D\right)}^{2} \cdot \left(\frac{\frac{h}{\ell}}{d} \cdot \frac{w0}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 5.4e+143)
   w0
   (+ w0 (* -0.125 (* (pow (* M D) 2.0) (* (/ (/ h l) d) (/ w0 d)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 5.4e+143) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (pow((M * D), 2.0) * (((h / l) / d) * (w0 / d))));
	}
	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 <= 5.4d+143) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((m * d) ** 2.0d0) * (((h / l) / d_1) * (w0 / d_1))))
    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 <= 5.4e+143) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (Math.pow((M * D), 2.0) * (((h / l) / d) * (w0 / d))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 5.4e+143:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (math.pow((M * D), 2.0) * (((h / l) / d) * (w0 / d))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 5.4e+143)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(M * D) ^ 2.0) * Float64(Float64(Float64(h / l) / d) * Float64(w0 / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 5.4e+143)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((M * D) ^ 2.0) * (((h / l) / d) * (w0 / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 5.4e+143], w0, N[(w0 + N[(-0.125 * N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] / d), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5.4000000000000003e143

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in D around 0 72.3%

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

   if 5.4000000000000003e143 < D

   1. Initial program 62.2%

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

   3. Simplified 62.6%

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

      1. *-commutative 62.6%

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

      2. associate-*l/ 62.2%

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

      3. associate-*r/ 65.8%

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

      4. clear-num 65.8%

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

      5. frac-times 62.6%

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

      6. *-un-lft-identity 62.6%

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

   6. Applied egg-rr 62.6%

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

   7. Taylor expanded in D around 0 33.4%

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

      1. associate-*r* 37.0%

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

      2. unpow2 37.0%

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

      3. unpow2 37.0%

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

      4. swap-sqr 55.1%

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

      5. unpow2 55.1%

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

      6. *-commutative 55.1%

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

      7. associate-*r/ 51.4%

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

   9. Simplified 51.4%

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

       1. frac-times 51.3%

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

       2. associate-*r/ 51.3%

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

       3. unpow2 51.3%

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

       4. times-frac 54.9%

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

   11. Applied egg-rr 54.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5.4 \cdot 10^{+143}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(M \cdot D\right)}^{2} \cdot \left(\frac{\frac{h}{\ell}}{d} \cdot \frac{w0}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% 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 78.0%

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

3. Simplified 78.0%

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

5. Taylor expanded in D around 0 67.3%

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

6. Final simplification 67.3%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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))))))