Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 89.2%

Time: 20.7s

Alternatives: 8

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

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (* (/ D (/ l (/ M d))) h) (* 0.25 (* D (/ M d))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (((D / (l / (M / d))) * h) * (0.25 * (D * (M / d))))));
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 - (((d / (l / (m / d_1))) * h) * (0.25d0 * (d * (m / d_1))))))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (((D / (l / (M / d))) * h) * (0.25 * (D * (M / d))))));
}

Python

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

Julia

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

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (((D / (l / (M / d))) * h) * (0.25 * (D * (M / d))))));
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / N[(l / N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(0.25 * N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Simplified 79.8%

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

5. Applied egg-rr 80.2%

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

   1. associate-/r/ 85.4%

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

   2. *-commutative 85.4%

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

   3. associate-*l/ 85.4%

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

   4. associate-*r/ 85.4%

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

7. Simplified 85.4%

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

   1. div-inv 85.4%

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

   2. associate-*r/ 85.4%

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

   3. associate-*l/ 85.4%

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

   4. unpow2 85.4%

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

   5. associate-*l* 88.0%

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

   6. *-commutative 88.0%

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

   7. associate-*l/ 88.0%

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

   8. associate-*r/ 88.0%

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

   9. associate-*l* 88.0%

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

   10. *-commutative 88.0%

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

   11. associate-*l/ 88.0%

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

   12. associate-*r/ 88.0%

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

   13. associate-*l* 88.0%

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

9. Applied egg-rr 88.0%

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

    1. associate-*r* 89.5%

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

    2. un-div-inv 89.5%

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

    3. associate-*r/ 88.7%

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

    4. associate-*r* 88.7%

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

    5. associate-*l/ 86.5%

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

    6. associate-/l* 88.1%

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

    7. associate-*l/ 88.2%

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

    8. associate-/l* 90.4%

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

11. Applied egg-rr 90.4%

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

    1. *-commutative 90.4%

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

    2. associate-/l* 91.9%

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

    3. associate-*l* 91.9%

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

    4. *-commutative 91.9%

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

    5. associate-/l/ 85.3%

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

    6. associate-/r/ 82.7%

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

    7. *-commutative 82.7%

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

    8. div-inv 82.7%

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

    9. times-frac 82.7%

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

    10. div-inv 82.5%

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

    11. clear-num 82.6%

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

    12. associate-/r/ 85.2%

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

13. Applied egg-rr 85.2%

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

    1. associate-/r/ 91.4%

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

    2. associate-/l* 90.9%

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

    3. associate-/r/ 90.9%

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

    4. metadata-eval 90.9%

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

15. Simplified 90.9%

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

16. Final simplification 90.9%

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

Alternative 2: 81.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+281}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(\frac{M}{\ell} \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-317}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -5e+281)
   (+ w0 (* -0.125 (* (/ D (* d (/ d D))) (* (/ M l) (* h (* w0 M))))))
   (if (<= (/ h l) -1e-317)
     (+ w0 (* -0.125 (* (/ h l) (* w0 (pow (* D (/ M d)) 2.0)))))
     w0)))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e+281) {
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	} else if ((h / l) <= -1e-317) {
		tmp = w0 + (-0.125 * ((h / l) * (w0 * pow((D * (M / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 ((h / l) <= (-5d+281)) then
        tmp = w0 + ((-0.125d0) * ((d / (d_1 * (d_1 / d))) * ((m / l) * (h * (w0 * m)))))
    else if ((h / l) <= (-1d-317)) then
        tmp = w0 + ((-0.125d0) * ((h / l) * (w0 * ((d * (m / d_1)) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e+281) {
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	} else if ((h / l) <= -1e-317) {
		tmp = w0 + (-0.125 * ((h / l) * (w0 * Math.pow((D * (M / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -5e+281:
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))))
	elif (h / l) <= -1e-317:
		tmp = w0 + (-0.125 * ((h / l) * (w0 * math.pow((D * (M / d)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e+281)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(D / Float64(d * Float64(d / D))) * Float64(Float64(M / l) * Float64(h * Float64(w0 * M))))));
	elseif (Float64(h / l) <= -1e-317)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h / l) * Float64(w0 * (Float64(D * Float64(M / d)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e+281)
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	elseif ((h / l) <= -1e-317)
		tmp = w0 + (-0.125 * ((h / l) * (w0 * ((D * (M / d)) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e+281], N[(w0 + N[(-0.125 * N[(N[(D / N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] * N[(h * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-317], N[(w0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(w0 * N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -5.00000000000000016e281

   1. Initial program 45.5%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in M around 0 44.3%

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

      1. times-frac 44.1%

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

      2. unpow2 44.1%

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

      3. unpow2 44.1%

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

      4. times-frac 56.3%

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

      5. unpow2 56.3%

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

      6. associate-/l* 48.3%

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

   6. Simplified 48.3%

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

      1. unpow2 48.3%

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

      2. clear-num 48.3%

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

      3. frac-times 48.3%

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

      4. *-un-lft-identity 48.3%

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

   8. Applied egg-rr 48.3%

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

      1. unpow2 48.3%

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

      2. div-inv 48.3%

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

      3. times-frac 57.2%

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

   10. Applied egg-rr 57.2%

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

       1. associate-/r/ 57.2%

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

       2. *-commutative 57.2%

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

       3. associate-*r* 61.2%

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

       4. /-rgt-identity 61.2%

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

   12. Applied egg-rr 61.2%

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

   if -5.00000000000000016e281 < (/.f64 h l) < -1.00000023e-317

   1. Initial program 83.5%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in M around 0 45.4%

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

      1. times-frac 47.7%

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

      2. unpow2 47.7%

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

      3. unpow2 47.7%

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

      4. times-frac 57.4%

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

      5. unpow2 57.4%

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

      6. associate-/l* 56.6%

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

   6. Simplified 56.6%

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

      1. div-inv 56.6%

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

      2. unpow2 56.6%

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

      3. associate-*l* 60.0%

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

      4. associate-/r* 61.8%

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

      5. associate-/r/ 61.8%

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

      6. clear-num 62.5%

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

   8. Applied egg-rr 62.5%

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

   9. Taylor expanded in D around 0 45.4%

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

   11. Simplified 70.2%

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

   if -1.00000023e-317 < (/.f64 h l)

   1. Initial program 86.4%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in M around 0 94.8%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+281}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(\frac{M}{\ell} \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-317}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(\frac{M}{\ell} \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-289}:\\ \;\;\;\;w0 + -0.125 \cdot \left(w0 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -2e+295)
   (+ w0 (* -0.125 (* (/ D (* d (/ d D))) (* (/ M l) (* h (* w0 M))))))
   (if (<= (/ h l) -1e-289)
     (+ w0 (* -0.125 (* w0 (/ (pow (* D (/ M d)) 2.0) (/ l h)))))
     w0)))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e+295) {
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	} else if ((h / l) <= -1e-289) {
		tmp = w0 + (-0.125 * (w0 * (pow((D * (M / d)), 2.0) / (l / h))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 ((h / l) <= (-2d+295)) then
        tmp = w0 + ((-0.125d0) * ((d / (d_1 * (d_1 / d))) * ((m / l) * (h * (w0 * m)))))
    else if ((h / l) <= (-1d-289)) then
        tmp = w0 + ((-0.125d0) * (w0 * (((d * (m / d_1)) ** 2.0d0) / (l / h))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e+295) {
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	} else if ((h / l) <= -1e-289) {
		tmp = w0 + (-0.125 * (w0 * (Math.pow((D * (M / d)), 2.0) / (l / h))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -2e+295:
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))))
	elif (h / l) <= -1e-289:
		tmp = w0 + (-0.125 * (w0 * (math.pow((D * (M / d)), 2.0) / (l / h))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -2e+295)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(D / Float64(d * Float64(d / D))) * Float64(Float64(M / l) * Float64(h * Float64(w0 * M))))));
	elseif (Float64(h / l) <= -1e-289)
		tmp = Float64(w0 + Float64(-0.125 * Float64(w0 * Float64((Float64(D * Float64(M / d)) ^ 2.0) / Float64(l / h)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -2e+295)
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	elseif ((h / l) <= -1e-289)
		tmp = w0 + (-0.125 * (w0 * (((D * (M / d)) ^ 2.0) / (l / h))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e+295], N[(w0 + N[(-0.125 * N[(N[(D / N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] * N[(h * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-289], N[(w0 + N[(-0.125 * N[(w0 * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -2e295

   1. Initial program 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in M around 0 43.8%

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

      1. times-frac 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. times-frac 56.9%

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

      5. unpow2 56.9%

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

      6. associate-/l* 48.2%

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

   6. Simplified 48.2%

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

      1. unpow2 48.2%

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

      2. clear-num 48.2%

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

      3. frac-times 48.2%

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

      4. *-un-lft-identity 48.2%

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

   8. Applied egg-rr 48.2%

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

      1. unpow2 48.2%

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

      2. div-inv 48.2%

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

      3. times-frac 57.8%

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

   10. Applied egg-rr 57.8%

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

       1. associate-/r/ 57.8%

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

       2. *-commutative 57.8%

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

       3. associate-*r* 62.2%

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

       4. /-rgt-identity 62.2%

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

   12. Applied egg-rr 62.2%

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

   if -2e295 < (/.f64 h l) < -1e-289

   1. Initial program 84.0%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in M around 0 46.1%

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

      1. times-frac 48.4%

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

      2. unpow2 48.4%

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

      3. unpow2 48.4%

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

      4. times-frac 57.5%

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

      5. unpow2 57.5%

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

      6. associate-/l* 56.7%

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

   6. Simplified 56.7%

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

      1. div-inv 56.7%

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

      2. unpow2 56.7%

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

      3. associate-*l* 59.3%

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

      4. associate-/r* 61.9%

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

      5. associate-/r/ 61.9%

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

      6. clear-num 61.9%

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

   8. Applied egg-rr 61.9%

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

      1. unpow2 61.9%

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

   10. Applied egg-rr 61.9%

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

       1. clear-num 61.9%

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

       2. times-frac 61.9%

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

       3. *-un-lft-identity 61.9%

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

       4. associate-*r* 59.4%

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

       5. unpow2 59.4%

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

       6. associate-*l/ 56.7%

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

       7. clear-num 56.7%

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

       8. div-inv 56.7%

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

       9. associate-*r/ 57.7%

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

   12. Applied egg-rr 67.2%

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

       1. associate-/r/ 70.4%

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

   14. Simplified 70.4%

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

   if -1e-289 < (/.f64 h l)

   1. Initial program 86.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in M around 0 94.1%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(\frac{M}{\ell} \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-289}:\\ \;\;\;\;w0 + -0.125 \cdot \left(w0 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4: 78.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5.3 \cdot 10^{+93}:\\ \;\;\;\;w0 + -0.125 \cdot \left(w0 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(w0 \cdot \frac{{\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M}{\ell}\right)}{\frac{1}{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 5.3e+93)
   (+ w0 (* -0.125 (* w0 (/ (pow (* D (/ M d)) 2.0) (/ l h)))))
   (+ w0 (* -0.125 (* w0 (/ (* (pow (/ D d) 2.0) (* M (/ M l))) (/ 1.0 h)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.3e+93) {
		tmp = w0 + (-0.125 * (w0 * (pow((D * (M / d)), 2.0) / (l / h))));
	} else {
		tmp = w0 + (-0.125 * (w0 * ((pow((D / d), 2.0) * (M * (M / l))) / (1.0 / h))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 <= 5.3d+93) then
        tmp = w0 + ((-0.125d0) * (w0 * (((d * (m / d_1)) ** 2.0d0) / (l / h))))
    else
        tmp = w0 + ((-0.125d0) * (w0 * ((((d / d_1) ** 2.0d0) * (m * (m / l))) / (1.0d0 / h))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.3e+93) {
		tmp = w0 + (-0.125 * (w0 * (Math.pow((D * (M / d)), 2.0) / (l / h))));
	} else {
		tmp = w0 + (-0.125 * (w0 * ((Math.pow((D / d), 2.0) * (M * (M / l))) / (1.0 / h))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 5.3e+93:
		tmp = w0 + (-0.125 * (w0 * (math.pow((D * (M / d)), 2.0) / (l / h))))
	else:
		tmp = w0 + (-0.125 * (w0 * ((math.pow((D / d), 2.0) * (M * (M / l))) / (1.0 / h))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 5.3e+93)
		tmp = Float64(w0 + Float64(-0.125 * Float64(w0 * Float64((Float64(D * Float64(M / d)) ^ 2.0) / Float64(l / h)))));
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(w0 * Float64(Float64((Float64(D / d) ^ 2.0) * Float64(M * Float64(M / l))) / Float64(1.0 / h)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 5.3e+93)
		tmp = w0 + (-0.125 * (w0 * (((D * (M / d)) ^ 2.0) / (l / h))));
	else
		tmp = w0 + (-0.125 * (w0 * ((((D / d) ^ 2.0) * (M * (M / l))) / (1.0 / h))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 5.3e+93], N[(w0 + N[(-0.125 * N[(w0 * N[(N[Power[N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 + N[(-0.125 * N[(w0 * N[(N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(M * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.3000000000000004e93

   1. Initial program 81.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in M around 0 50.8%

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

      1. times-frac 51.7%

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

      2. unpow2 51.7%

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

      3. unpow2 51.7%

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

      4. times-frac 59.1%

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

      5. unpow2 59.1%

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

      6. associate-/l* 56.5%

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

   6. Simplified 56.5%

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

      1. div-inv 56.5%

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

      2. unpow2 56.5%

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

      3. associate-*l* 58.8%

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

      4. associate-/r* 61.1%

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

      5. associate-/r/ 60.6%

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

      6. clear-num 61.1%

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

   8. Applied egg-rr 61.1%

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

      1. unpow2 61.1%

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

   10. Applied egg-rr 61.1%

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

       1. clear-num 61.1%

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

       2. times-frac 61.1%

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

       3. *-un-lft-identity 61.1%

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

       4. associate-*r* 58.8%

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

       5. unpow2 58.8%

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

       6. associate-*l/ 56.5%

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

       7. clear-num 56.5%

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

       8. div-inv 56.5%

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

       9. associate-*r/ 57.1%

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

   12. Applied egg-rr 70.7%

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

       1. associate-/r/ 73.7%

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

   14. Simplified 73.7%

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

   if 5.3000000000000004e93 < d

   1. Initial program 79.6%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in M around 0 54.2%

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

      1. times-frac 55.9%

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

      2. unpow2 55.9%

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

      3. unpow2 55.9%

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

      4. times-frac 72.2%

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

      5. unpow2 72.2%

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

      6. associate-/l* 70.6%

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

   6. Simplified 70.6%

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

      1. div-inv 69.1%

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

      2. unpow2 69.1%

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

      3. associate-*l* 74.0%

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

      4. associate-/r* 69.2%

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

      5. associate-/r/ 69.2%

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

      6. clear-num 69.2%

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

   8. Applied egg-rr 69.2%

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

      1. unpow2 69.2%

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

   10. Applied egg-rr 69.2%

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

       1. clear-num 69.2%

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

       2. times-frac 69.2%

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

       3. *-un-lft-identity 69.2%

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

       4. associate-*r* 64.3%

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

       5. unpow2 64.3%

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

       6. associate-*l/ 69.1%

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

       7. clear-num 69.1%

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

       8. div-inv 70.6%

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

       9. unpow2 70.6%

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

       10. div-inv 70.6%

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

       11. frac-times 78.7%

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

       12. associate-*r* 81.9%

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

       13. associate-*r/ 84.8%

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

   12. Applied egg-rr 84.8%

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

       1. associate-/r/ 89.6%

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

       2. associate-*l* 84.8%

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

   14. Simplified 84.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.3 \cdot 10^{+93}:\\ \;\;\;\;w0 + -0.125 \cdot \left(w0 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\frac{\ell}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(w0 \cdot \frac{{\left(\frac{D}{d}\right)}^{2} \cdot \left(M \cdot \frac{M}{\ell}\right)}{\frac{1}{h}}\right)\\ \end{array} \]

Alternative 5: 76.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{-58}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.35e-58)
   w0
   (+ w0 (* -0.125 (* (* (/ D d) (/ D d)) (* M (* M (* w0 (/ h l)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.35e-58) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * (M * (M * (w0 * (h / l))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 (m <= 1.35d-58) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d / d_1) * (d / d_1)) * (m * (m * (w0 * (h / l))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.35e-58) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * (M * (M * (w0 * (h / l))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.35e-58:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * (M * (M * (w0 * (h / l))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.35e-58)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * Float64(M * Float64(w0 * Float64(h / l)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.35e-58)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D / d) * (D / d)) * (M * (M * (w0 * (h / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.35e-58], w0, N[(w0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * N[(w0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.3499999999999999e-58

   1. Initial program 82.8%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in M around 0 71.3%

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

   if 1.3499999999999999e-58 < M

   1. Initial program 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in M around 0 45.4%

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

      1. times-frac 46.9%

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

      2. unpow2 46.9%

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

      3. unpow2 46.9%

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

      4. times-frac 56.8%

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

      5. unpow2 56.8%

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

      6. associate-/l* 56.7%

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

   6. Simplified 56.7%

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

      1. div-inv 56.7%

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

      2. unpow2 56.7%

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

      3. associate-*l* 62.7%

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

      4. associate-/r* 60.3%

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

      5. associate-/r/ 60.3%

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

      6. clear-num 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. unpow2 61.8%

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

   10. Applied egg-rr 61.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{-58}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 76.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 7.7 \cdot 10^{-60}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(\frac{M}{\ell} \cdot \left(M \cdot \left(w0 \cdot h\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 7.7e-60)
   w0
   (+ w0 (* -0.125 (* (/ D (* d (/ d D))) (* (/ M l) (* M (* w0 h))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7.7e-60) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (M * (w0 * h)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 (m <= 7.7d-60) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((d / (d_1 * (d_1 / d))) * ((m / l) * (m * (w0 * h)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 7.7e-60) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (M * (w0 * h)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 7.7e-60:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (M * (w0 * h)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 7.7e-60)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(D / Float64(d * Float64(d / D))) * Float64(Float64(M / l) * Float64(M * Float64(w0 * h))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 7.7e-60)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (M * (w0 * h)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 7.7e-60], w0, N[(w0 + N[(-0.125 * N[(N[(D / N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] * N[(M * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.6999999999999996e-60

   1. Initial program 82.8%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in M around 0 71.3%

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

   if 7.6999999999999996e-60 < M

   1. Initial program 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in M around 0 45.4%

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

      1. times-frac 46.9%

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

      2. unpow2 46.9%

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

      3. unpow2 46.9%

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

      4. times-frac 56.8%

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

      5. unpow2 56.8%

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

      6. associate-/l* 56.7%

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

   6. Simplified 56.7%

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

      1. unpow2 56.7%

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

      2. clear-num 56.7%

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

      3. frac-times 56.7%

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

      4. *-un-lft-identity 56.7%

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

   8. Applied egg-rr 56.7%

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

      1. unpow2 56.7%

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

      2. div-inv 56.7%

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

      3. times-frac 64.0%

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

   10. Applied egg-rr 64.0%

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

       1. associate-/r/ 64.0%

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

       2. /-rgt-identity 64.0%

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

   12. Simplified 64.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 7.7 \cdot 10^{-60}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(\frac{M}{\ell} \cdot \left(M \cdot \left(w0 \cdot h\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 76.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.65 \cdot 10^{-61}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(\frac{M}{\ell} \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.65e-61)
   w0
   (+ w0 (* -0.125 (* (/ D (* d (/ d D))) (* (/ M l) (* h (* w0 M))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.65e-61) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 (m <= 2.65d-61) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((d / (d_1 * (d_1 / d))) * ((m / l) * (h * (w0 * m)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.65e-61) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.65e-61:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.65e-61)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(D / Float64(d * Float64(d / D))) * Float64(Float64(M / l) * Float64(h * Float64(w0 * M))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.65e-61)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((D / (d * (d / D))) * ((M / l) * (h * (w0 * M)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.65e-61], w0, N[(w0 + N[(-0.125 * N[(N[(D / N[(d * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] * N[(h * N[(w0 * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.65e-61

   1. Initial program 82.7%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in M around 0 71.2%

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

   if 2.65e-61 < M

   1. Initial program 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in M around 0 46.2%

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

      1. times-frac 47.6%

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

      2. unpow2 47.6%

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

      3. unpow2 47.6%

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

      4. times-frac 57.4%

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

      5. unpow2 57.4%

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

      6. associate-/l* 57.3%

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

   6. Simplified 57.3%

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

      1. unpow2 57.3%

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

      2. clear-num 57.3%

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

      3. frac-times 57.3%

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

      4. *-un-lft-identity 57.3%

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

   8. Applied egg-rr 57.3%

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

      1. unpow2 57.3%

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

      2. div-inv 57.3%

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

      3. times-frac 64.5%

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

   10. Applied egg-rr 64.5%

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

       1. associate-/r/ 64.6%

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

       2. *-commutative 64.6%

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

       3. associate-*r* 62.5%

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

       4. /-rgt-identity 62.5%

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

   12. Applied egg-rr 62.5%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.65 \cdot 10^{-61}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{D}{d \cdot \frac{d}{D}} \cdot \left(\frac{M}{\ell} \cdot \left(h \cdot \left(w0 \cdot M\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 68.4% accurate, 216.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d) :precision binary64 w0)

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 81.0%

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

3. Simplified 79.8%

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

4. Taylor expanded in M around 0 67.4%

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

5. Final simplification 67.4%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023301 
(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))))))