Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 89.2%

Time: 18.0s

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: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot \left(D \cdot M\right)\\ w0 \cdot \sqrt{1 - \frac{h \cdot t\_0}{\frac{\ell}{t\_0}}} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) (* D M))))
   (* w0 (sqrt (- 1.0 (/ (* h t_0) (/ l t_0)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (0.5 / d) * (D * M);
	return w0 * sqrt((1.0 - ((h * t_0) / (l / t_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
    real(8) :: t_0
    t_0 = (0.5d0 / d_1) * (d * m)
    code = w0 * sqrt((1.0d0 - ((h * t_0) / (l / t_0))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (0.5 / d) * (D * M);
	return w0 * Math.sqrt((1.0 - ((h * t_0) / (l / t_0))));
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (0.5 / d) * (D * M)
	return w0 * math.sqrt((1.0 - ((h * t_0) / (l / t_0))))

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(0.5 / d) * Float64(D * M))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * t_0) / Float64(l / t_0)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	t_0 = (0.5 / d) * (D * M);
	tmp = w0 * sqrt((1.0 - ((h * t_0) / (l / t_0))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * t$95$0), $MachinePrecision] / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.2%

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

3. Simplified 76.5%

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

   1. clear-num 76.5%

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

   2. un-div-inv 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*l/ 77.3%

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

   5. associate-*r/ 76.5%

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

   6. div-inv 76.5%

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

   7. metadata-eval 76.5%

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

6. Applied egg-rr 76.5%

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

   1. associate-/r/ 84.7%

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

   2. associate-*l/ 84.4%

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

   3. *-commutative 84.4%

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

   4. associate-/l* 84.7%

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

   5. associate-*r/ 84.8%

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

   6. *-commutative 84.8%

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

   7. associate-/l* 84.4%

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

   8. associate-*r/ 84.4%

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

8. Simplified 84.4%

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

   1. unpow2 84.4%

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

   2. add-sqr-sqrt 41.8%

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

   3. times-frac 42.5%

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

10. Applied egg-rr 42.5%

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

    1. unpow2 42.5%

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

12. Simplified 42.5%

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

    1. unpow2 42.5%

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

    2. frac-times 41.8%

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

    3. add-sqr-sqrt 84.4%

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

    4. *-un-lft-identity 84.4%

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

    5. frac-times 86.2%

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

    6. clear-num 86.2%

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

    7. frac-times 86.2%

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

    8. metadata-eval 86.2%

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

    9. div-inv 86.2%

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

    10. /-rgt-identity 86.2%

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

    11. metadata-eval 86.2%

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

    12. times-frac 86.2%

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

    13. *-un-lft-identity 86.2%

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

    14. *-un-lft-identity 86.2%

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

14. Applied egg-rr 86.2%

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

    1. div-inv 86.2%

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

    2. associate-*r* 85.1%

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

    3. clear-num 85.1%

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

    4. associate-*r* 86.2%

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

    5. clear-num 86.2%

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

    6. div-inv 86.2%

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

    7. associate-*r* 85.1%

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

    8. associate-*r/ 86.4%

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

    9. *-commutative 86.4%

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

    10. associate-*r* 87.5%

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

    11. *-commutative 87.5%

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

16. Applied egg-rr 87.5%

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

17. Final simplification 87.5%

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

Alternative 2: 71.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d}\\ \mathbf{if}\;D \leq 1.12 \cdot 10^{+132}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(t\_0 \cdot t\_0\right) \cdot \frac{w0 \cdot h}{\ell}, w0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* D M) d)))
   (if (<= D 1.12e+132) w0 (fma -0.125 (* (* t_0 t_0) (/ (* w0 h) l)) w0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D * M) / d;
	double tmp;
	if (D <= 1.12e+132) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, ((t_0 * t_0) * ((w0 * h) / l)), w0);
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(D * M) / d)
	tmp = 0.0
	if (D <= 1.12e+132)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(Float64(t_0 * t_0) * Float64(Float64(w0 * h) / l)), w0);
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[D, 1.12e+132], w0, N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.12e132

   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 77.6%

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

   5. Taylor expanded in D around 0 72.1%

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

   if 1.12e132 < D

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in D around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. fma-define 37.1%

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

      3. associate-*r* 33.3%

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

      4. unpow2 33.3%

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

      5. unpow2 33.3%

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

      6. swap-sqr 53.8%

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

      7. unpow2 53.8%

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

   7. Simplified 53.8%

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

      1. times-frac 57.0%

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

   9. Applied egg-rr 57.0%

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

       1. add-sqr-sqrt 57.0%

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

       2. sqrt-div 57.0%

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

       3. sqrt-pow1 36.9%

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

       4. metadata-eval 36.9%

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

       5. pow1 36.9%

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

       6. sqrt-pow1 44.9%

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

       7. metadata-eval 44.9%

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

       8. pow1 44.9%

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

       9. sqrt-div 44.9%

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

       10. sqrt-pow1 49.1%

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

       11. metadata-eval 49.1%

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

       12. pow1 49.1%

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

       13. sqrt-pow1 65.1%

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

       14. metadata-eval 65.1%

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

       15. pow1 65.1%

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

   11. Applied egg-rr 65.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.12 \cdot 10^{+132}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right) \cdot \frac{w0 \cdot h}{\ell}, w0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (*
  w0
  (sqrt (- 1.0 (* h (/ (* D (* (/ 0.5 d) M)) (/ l (* 0.5 (/ (* D M) d)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * ((D * ((0.5 / d) * M)) / (l / (0.5 * ((D * M) / d)))))));
}

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 * ((0.5d0 / d_1) * m)) / (l / (0.5d0 * ((d * m) / d_1)))))))
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 * ((0.5 / d) * M)) / (l / (0.5 * ((D * M) / d)))))));
}

Python

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

Julia

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

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * ((D * ((0.5 / d) * M)) / (l / (0.5 * ((D * M) / d)))))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(D * N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] / N[(l / N[(0.5 * N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

3. Simplified 76.5%

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

   1. clear-num 76.5%

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

   2. un-div-inv 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*l/ 77.3%

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

   5. associate-*r/ 76.5%

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

   6. div-inv 76.5%

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

   7. metadata-eval 76.5%

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

6. Applied egg-rr 76.5%

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

   1. associate-/r/ 84.7%

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

   2. associate-*l/ 84.4%

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

   3. *-commutative 84.4%

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

   4. associate-/l* 84.7%

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

   5. associate-*r/ 84.8%

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

   6. *-commutative 84.8%

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

   7. associate-/l* 84.4%

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

   8. associate-*r/ 84.4%

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

8. Simplified 84.4%

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

   1. unpow2 84.4%

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

   2. add-sqr-sqrt 41.8%

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

   3. times-frac 42.5%

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

10. Applied egg-rr 42.5%

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

    1. unpow2 42.5%

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

12. Simplified 42.5%

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

    1. unpow2 42.5%

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

    2. frac-times 41.8%

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

    3. add-sqr-sqrt 84.4%

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

    4. *-un-lft-identity 84.4%

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

    5. frac-times 86.2%

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

    6. clear-num 86.2%

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

    7. frac-times 86.2%

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

    8. metadata-eval 86.2%

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

    9. div-inv 86.2%

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

    10. /-rgt-identity 86.2%

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

    11. metadata-eval 86.2%

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

    12. times-frac 86.2%

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

    13. *-un-lft-identity 86.2%

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

    14. *-un-lft-identity 86.2%

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

14. Applied egg-rr 86.2%

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

15. Taylor expanded in D around 0 85.1%

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

16. Final simplification 85.1%

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

Alternative 4: 88.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := D \cdot \left(\frac{0.5}{d} \cdot M\right)\\ w0 \cdot \sqrt{1 - h \cdot \frac{t\_0}{\frac{\ell}{t\_0}}} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* D (* (/ 0.5 d) M))))
   (* w0 (sqrt (- 1.0 (* h (/ t_0 (/ l t_0))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * ((0.5 / d) * M);
	return w0 * sqrt((1.0 - (h * (t_0 / (l / t_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
    real(8) :: t_0
    t_0 = d * ((0.5d0 / d_1) * m)
    code = w0 * sqrt((1.0d0 - (h * (t_0 / (l / t_0)))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * ((0.5 / d) * M);
	return w0 * Math.sqrt((1.0 - (h * (t_0 / (l / t_0)))));
}

Python

def code(w0, M, D, h, l, d):
	t_0 = D * ((0.5 / d) * M)
	return w0 * math.sqrt((1.0 - (h * (t_0 / (l / t_0)))))

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(D * Float64(Float64(0.5 / d) * M))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 / Float64(l / t_0))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	t_0 = D * ((0.5 / d) * M);
	tmp = w0 * sqrt((1.0 - (h * (t_0 / (l / t_0)))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(D * N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$0 / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.2%

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

3. Simplified 76.5%

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

   1. clear-num 76.5%

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

   2. un-div-inv 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*l/ 77.3%

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

   5. associate-*r/ 76.5%

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

   6. div-inv 76.5%

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

   7. metadata-eval 76.5%

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

6. Applied egg-rr 76.5%

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

   1. associate-/r/ 84.7%

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

   2. associate-*l/ 84.4%

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

   3. *-commutative 84.4%

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

   4. associate-/l* 84.7%

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

   5. associate-*r/ 84.8%

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

   6. *-commutative 84.8%

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

   7. associate-/l* 84.4%

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

   8. associate-*r/ 84.4%

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

8. Simplified 84.4%

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

   1. unpow2 84.4%

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

   2. add-sqr-sqrt 41.8%

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

   3. times-frac 42.5%

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

10. Applied egg-rr 42.5%

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

    1. unpow2 42.5%

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

12. Simplified 42.5%

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

    1. unpow2 42.5%

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

    2. frac-times 41.8%

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

    3. add-sqr-sqrt 84.4%

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

    4. *-un-lft-identity 84.4%

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

    5. frac-times 86.2%

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

    6. clear-num 86.2%

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

    7. frac-times 86.2%

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

    8. metadata-eval 86.2%

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

    9. div-inv 86.2%

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

    10. /-rgt-identity 86.2%

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

    11. metadata-eval 86.2%

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

    12. times-frac 86.2%

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

    13. *-un-lft-identity 86.2%

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

    14. *-un-lft-identity 86.2%

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

14. Applied egg-rr 86.2%

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

15. Final simplification 86.2%

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

Alternative 5: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \left(\frac{0.5}{d} \cdot D\right)\\ w0 \cdot \sqrt{1 - \frac{h \cdot t\_0}{\frac{\ell}{t\_0}}} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* M (* (/ 0.5 d) D))))
   (* w0 (sqrt (- 1.0 (/ (* h t_0) (/ l t_0)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * ((0.5 / d) * D);
	return w0 * sqrt((1.0 - ((h * t_0) / (l / t_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
    real(8) :: t_0
    t_0 = m * ((0.5d0 / d_1) * d)
    code = w0 * sqrt((1.0d0 - ((h * t_0) / (l / t_0))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * ((0.5 / d) * D);
	return w0 * Math.sqrt((1.0 - ((h * t_0) / (l / t_0))));
}

Python

def code(w0, M, D, h, l, d):
	t_0 = M * ((0.5 / d) * D)
	return w0 * math.sqrt((1.0 - ((h * t_0) / (l / t_0))))

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(Float64(0.5 / d) * D))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * t_0) / Float64(l / t_0)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	t_0 = M * ((0.5 / d) * D);
	tmp = w0 * sqrt((1.0 - ((h * t_0) / (l / t_0))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * t$95$0), $MachinePrecision] / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 77.2%

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

3. Simplified 76.5%

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

   1. clear-num 76.5%

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

   2. un-div-inv 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*l/ 77.3%

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

   5. associate-*r/ 76.5%

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

   6. div-inv 76.5%

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

   7. metadata-eval 76.5%

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

6. Applied egg-rr 76.5%

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

   1. associate-/r/ 84.7%

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

   2. associate-*l/ 84.4%

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

   3. *-commutative 84.4%

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

   4. associate-/l* 84.7%

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

   5. associate-*r/ 84.8%

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

   6. *-commutative 84.8%

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

   7. associate-/l* 84.4%

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

   8. associate-*r/ 84.4%

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

8. Simplified 84.4%

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

   1. unpow2 84.4%

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

   2. add-sqr-sqrt 41.8%

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

   3. times-frac 42.5%

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

10. Applied egg-rr 42.5%

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

    1. unpow2 42.5%

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

12. Simplified 42.5%

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

    1. unpow2 42.5%

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

    2. frac-times 41.8%

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

    3. add-sqr-sqrt 84.4%

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

    4. *-un-lft-identity 84.4%

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

    5. frac-times 86.2%

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

    6. clear-num 86.2%

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

    7. frac-times 86.2%

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

    8. metadata-eval 86.2%

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

    9. div-inv 86.2%

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

    10. /-rgt-identity 86.2%

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

    11. metadata-eval 86.2%

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

    12. times-frac 86.2%

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

    13. *-un-lft-identity 86.2%

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

    14. *-un-lft-identity 86.2%

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

14. Applied egg-rr 86.2%

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

    1. div-inv 86.2%

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

    2. associate-*r* 85.1%

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

    3. clear-num 85.1%

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

    4. associate-*r* 86.2%

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

    5. clear-num 86.2%

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

    6. div-inv 86.2%

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

    7. associate-*r* 85.1%

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

    8. associate-*r/ 86.4%

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

    9. *-commutative 86.4%

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

    10. associate-*r* 87.5%

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

    11. *-commutative 87.5%

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

16. Applied egg-rr 87.5%

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

    1. associate-*r* 86.4%

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

    2. associate-*r* 87.5%

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

18. Simplified 87.5%

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

19. Final simplification 87.5%

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

Alternative 6: 68.8% 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 77.2%

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

3. Simplified 76.5%

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

5. Taylor expanded in D around 0 68.5%

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

6. Final simplification 68.5%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(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))))))