Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 86.1%

Time: 16.6s

Alternatives: 6

Speedup: 1.8×

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: 81.0% 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: 86.1% accurate, 1.0× speedup

Math

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

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(Float64(h * (Float64(Float64(M * 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[(N[(h * N[Power[N[(N[(M * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.1%

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

3. Simplified 80.3%

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

   1. unpow2 80.3%

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

   2. unpow2 80.3%

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

   3. *-commutative 80.3%

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

   4. associate-*l/ 81.1%

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

   5. associate-*r/ 81.5%

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

   6. times-frac 81.1%

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

   7. associate-*r/ 86.3%

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

   8. *-commutative 86.3%

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

   9. pow1 86.3%

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

   10. associate-/l* 87.0%

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

   11. pow1 87.0%

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

   12. div-inv 87.0%

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

   13. associate-/r* 87.0%

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

   14. metadata-eval 87.0%

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

6. Applied egg-rr 87.0%

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

   1. associate-*r/ 87.0%

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

   2. associate-*r/ 86.3%

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

8. Applied egg-rr 86.3%

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

Alternative 2: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right) \cdot \frac{h}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 2.8e+17)
   w0
   (* w0 (sqrt (- 1.0 (* (* D (* (/ M d) (* 0.25 (* D (/ M d))))) (/ h l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2.8e+17) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((D * ((M / d) * (0.25 * (D * (M / d))))) * (h / l))));
	}
	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 <= 2.8d+17) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((d * ((m / d_1) * (0.25d0 * (d * (m / d_1))))) * (h / l))))
    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 <= 2.8e+17) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((D * ((M / d) * (0.25 * (D * (M / d))))) * (h / l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 2.8e+17:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((D * ((M / d) * (0.25 * (D * (M / d))))) * (h / l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 2.8e+17)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(D * Float64(Float64(M / d) * Float64(0.25 * Float64(D * Float64(M / d))))) * Float64(h / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 2.8e+17)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((D * ((M / d) * (0.25 * (D * (M / d))))) * (h / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 2.8e+17], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(D * N[(N[(M / d), $MachinePrecision] * N[(0.25 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.8e17

   1. Initial program 82.3%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in D around 0 72.4%

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

   if 2.8e17 < D

   1. Initial program 75.6%

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

   3. Simplified 75.6%

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

      1. unpow2 75.6%

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

      2. *-commutative 75.6%

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

      3. associate-*l/ 75.6%

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

      4. associate-*r/ 75.6%

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

      5. times-frac 75.6%

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

      6. clear-num 75.6%

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

      7. un-div-inv 75.6%

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

      8. *-commutative 75.6%

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

      9. associate-*l/ 75.6%

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

      10. associate-*r/ 75.6%

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

      11. times-frac 75.6%

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

      12. associate-/l* 75.6%

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

      13. div-inv 75.6%

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

      14. associate-/r* 75.6%

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

      15. metadata-eval 75.6%

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

      16. times-frac 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. associate-*r* 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*r/ 75.6%

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

      4. associate-*r* 75.6%

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

      5. associate-/r* 73.6%

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

      6. *-commutative 73.6%

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

      7. associate-/l* 73.7%

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

      8. *-commutative 73.7%

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

   8. Simplified 73.7%

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

      1. times-frac 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. associate-*l/ 75.6%

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

       2. associate-/l* 75.6%

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

       3. associate-/r* 75.6%

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

   12. Simplified 75.6%

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

   13. Taylor expanded in d around 0 75.6%

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

       1. associate-*r/ 75.6%

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

   15. Simplified 75.6%

       \[\leadsto w0 \cdot \sqrt{1 - \left(D \cdot \left(\frac{M}{d} \cdot \color{blue}{\left(0.25 \cdot \left(D \cdot \frac{M}{d}\right)\right)}\right)\right) \cdot \frac{h}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M}{d} \cdot \left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 1e+17)
   w0
   (* w0 (sqrt (- 1.0 (* D (* (/ h l) (* (/ M d) (* (* M D) (/ 0.25 d))))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1e+17) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (D * ((h / l) * ((M / d) * ((M * D) * (0.25 / 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 <= 1d+17) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (d * ((h / l) * ((m / d_1) * ((m * d) * (0.25d0 / 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 <= 1e+17) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (D * ((h / l) * ((M / d) * ((M * D) * (0.25 / d)))))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 1e+17:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (D * ((h / l) * ((M / d) * ((M * D) * (0.25 / d)))))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 1e+17)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(D * Float64(Float64(h / l) * Float64(Float64(M / d) * Float64(Float64(M * D) * Float64(0.25 / d))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 1e+17)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (D * ((h / l) * ((M / d) * ((M * D) * (0.25 / d)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 1e+17], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(D * N[(N[(h / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * N[(0.25 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1e17

   1. Initial program 82.3%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in D around 0 72.4%

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

   if 1e17 < D

   1. Initial program 75.6%

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

   3. Simplified 75.6%

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

      1. unpow2 75.6%

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

      2. *-commutative 75.6%

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

      3. associate-*l/ 75.6%

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

      4. associate-*r/ 75.6%

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

      5. times-frac 75.6%

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

      6. clear-num 75.6%

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

      7. un-div-inv 75.6%

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

      8. *-commutative 75.6%

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

      9. associate-*l/ 75.6%

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

      10. associate-*r/ 75.6%

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

      11. times-frac 75.6%

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

      12. associate-/l* 75.6%

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

      13. div-inv 75.6%

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

      14. associate-/r* 75.6%

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

      15. metadata-eval 75.6%

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

      16. times-frac 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. associate-*r* 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*r/ 75.6%

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

      4. associate-*r* 75.6%

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

      5. associate-/r* 73.6%

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

      6. *-commutative 73.6%

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

      7. associate-/l* 73.7%

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

      8. *-commutative 73.7%

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

   8. Simplified 73.7%

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

      1. times-frac 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. associate-*l/ 75.6%

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

       2. associate-/l* 75.6%

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

       3. associate-/r* 75.6%

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

   12. Simplified 75.6%

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

       1. associate-*r/ 75.9%

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

       2. associate-/r* 75.9%

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

       3. metadata-eval 75.9%

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

       4. *-commutative 75.9%

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

   14. Applied egg-rr 75.9%

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

       1. associate-*r/ 75.6%

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

       2. associate-*l* 77.6%

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

       3. associate-/r/ 77.6%

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

       4. *-commutative 77.6%

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

   16. Simplified 77.6%

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{D \cdot \left(\left(\frac{M}{d} \cdot \left(\frac{0.25}{d} \cdot \left(D \cdot M\right)\right)\right) \cdot \frac{h}{\ell}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 10^{+17}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M}{d} \cdot \left(\left(M \cdot D\right) \cdot \frac{0.25}{d}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 1.18 \cdot 10^{+145}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{1}{{\left(\frac{\frac{d}{M}}{D}\right)}^{2} \cdot \frac{\ell}{w0 \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 1.18e+145)
   w0
   (+ w0 (* -0.125 (/ 1.0 (* (pow (/ (/ d M) D) 2.0) (/ l (* w0 h))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 1.18e+145) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (1.0 / (pow(((d / M) / D), 2.0) * (l / (w0 * h)))));
	}
	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 <= 1.18d+145) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (1.0d0 / ((((d_1 / m) / d) ** 2.0d0) * (l / (w0 * h)))))
    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 <= 1.18e+145) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (1.0 / (Math.pow(((d / M) / D), 2.0) * (l / (w0 * h)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 1.18e+145:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (1.0 / (math.pow(((d / M) / D), 2.0) * (l / (w0 * h)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 1.18e+145)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(1.0 / Float64((Float64(Float64(d / M) / D) ^ 2.0) * Float64(l / Float64(w0 * h))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 1.18e+145)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (1.0 / ((((d / M) / D) ^ 2.0) * (l / (w0 * h)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 1.18e+145], w0, N[(w0 + N[(-0.125 * N[(1.0 / N[(N[Power[N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision], 2.0], $MachinePrecision] * N[(l / N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.17999999999999998e145

   1. Initial program 82.9%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in D around 0 71.2%

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

   if 1.17999999999999998e145 < D

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in D around 0 26.5%

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

      1. unpow2 26.5%

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

   7. Applied egg-rr 26.5%

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

      1. expm1-log1p-u 13.5%

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

      2. associate-*r* 13.8%

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

      3. pow2 13.8%

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

      4. pow-prod-down 22.9%

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

      5. *-commutative 22.9%

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

   9. Applied egg-rr 22.9%

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

       1. clear-num 22.9%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\frac{1}{\frac{{d}^{2} \cdot \ell}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w0\right)\right)\right)}}} \]

       2. inv-pow 22.9%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{{\left(\frac{{d}^{2} \cdot \ell}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(M \cdot D\right)}^{2} \cdot \left(h \cdot w0\right)\right)\right)}\right)}^{-1}} \]

       3. expm1-log1p-u 40.5%

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

   11. Applied egg-rr 40.5%

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

       1. unpow-1 40.5%

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

       2. times-frac 40.2%

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

       3. unpow2 40.2%

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

       4. unpow2 40.2%

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

       5. times-frac 57.6%

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

       6. unpow2 57.6%

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

       7. associate-/r* 57.6%

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

   13. Simplified 57.6%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.18 \cdot 10^{+145}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{1}{{\left(\frac{\frac{d}{M}}{D}\right)}^{2} \cdot \frac{\ell}{w0 \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - ((h * (D * ((M / d) * (0.25 / (d / (M * D)))))) / 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 / d_1) * (0.25d0 / (d_1 / (m * d)))))) / 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 * (D * ((M / d) * (0.25 / (d / (M * D)))))) / l)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.1%

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

3. Simplified 80.3%

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

   1. unpow2 80.3%

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

   2. *-commutative 80.3%

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

   3. associate-*l/ 79.9%

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

   4. associate-*r/ 80.4%

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

   5. times-frac 79.9%

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

   6. clear-num 79.9%

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

   7. un-div-inv 79.9%

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

   8. *-commutative 79.9%

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

   9. associate-*l/ 81.0%

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

   10. associate-*r/ 81.1%

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

   11. times-frac 81.0%

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

   12. associate-/l* 81.1%

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

   13. div-inv 81.1%

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

   14. associate-/r* 81.1%

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

   15. metadata-eval 81.1%

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

   16. times-frac 81.5%

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

6. Applied egg-rr 81.5%

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

   1. associate-*r* 81.0%

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

   2. *-commutative 81.0%

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

   3. associate-*r/ 81.0%

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

   4. associate-*r* 81.0%

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

   5. associate-/r* 79.2%

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

   6. *-commutative 79.2%

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

   7. associate-/l* 78.5%

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

   8. *-commutative 78.5%

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

8. Simplified 78.5%

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

   1. times-frac 79.6%

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

10. Applied egg-rr 79.6%

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

    1. associate-*l/ 79.6%

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

    2. associate-/l* 79.6%

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

    3. associate-/r* 79.1%

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

12. Simplified 79.1%

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

    1. associate-*r/ 84.4%

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

    2. associate-/r* 84.4%

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

    3. metadata-eval 84.4%

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

    4. *-commutative 84.4%

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

14. Applied egg-rr 84.4%

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

15. Final simplification 84.4%

    \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot \left(D \cdot \left(\frac{M}{d} \cdot \frac{0.25}{\frac{d}{M \cdot D}}\right)\right)}{\ell}} \]
16. Add Preprocessing

Alternative 6: 67.6% 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 81.1%

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

3. Simplified 80.3%

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

5. Taylor expanded in D around 0 67.7%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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))))))