Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 86.1%

Time: 14.3s

Alternatives: 10

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: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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 (* h (/ (pow (* M (* D (/ 0.5 d))) 2.0) l))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (h * (pow((M * (D * (0.5 / d))), 2.0) / l))));
}

Fortran

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 - (h * (((m * (d * (0.5d0 / d_1))) ** 2.0d0) / l))))
end function

Java

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 - (h * (Math.pow((M * (D * (0.5 / d))), 2.0) / l))));
}

Python

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

Julia

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

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (h * (((M * (D * (0.5 / d))) ^ 2.0) / l))));
end

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 80.1%

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

3. Simplified 80.1%

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

   1. associate-/l* 79.3%

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

   2. clear-num 79.3%

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

   3. un-div-inv 79.3%

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

   4. associate-*l/ 77.8%

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

   5. *-commutative 77.8%

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

   6. associate-/r* 77.8%

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

   7. div-inv 77.8%

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

   8. metadata-eval 77.8%

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

5. Applied egg-rr 77.8%

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

   1. associate-/r/ 83.5%

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

   2. *-commutative 83.5%

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

   3. *-commutative 83.5%

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

   4. associate-*l/ 83.5%

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

   5. associate-*l* 85.8%

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

   6. *-commutative 85.8%

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

7. Simplified 85.8%

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

8. Final simplification 85.8%

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

Alternative 2: 83.6% accurate, 1.8× speedup

Math

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

FPCore

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 (* h (* 0.25 (* (/ (* M D) (* d l)) (* M (/ D d)))))))))

C

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

Fortran

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 - (h * (0.25d0 * (((m * d) / (d_1 * l)) * (m * (d / d_1)))))))
end function

Java

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 - (h * (0.25 * (((M * D) / (d * l)) * (M * (D / d)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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[(h * N[(0.25 * N[(N[(N[(M * D), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 80.1%

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

3. Simplified 80.1%

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

   1. associate-/l* 79.3%

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

   2. clear-num 79.3%

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

   3. un-div-inv 79.3%

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

   4. associate-*l/ 77.8%

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

   5. *-commutative 77.8%

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

   6. associate-/r* 77.8%

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

   7. div-inv 77.8%

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

   8. metadata-eval 77.8%

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

5. Applied egg-rr 77.8%

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

   1. associate-/r/ 83.5%

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

   2. *-commutative 83.5%

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

   3. *-commutative 83.5%

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

   4. associate-*l/ 83.5%

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

   5. associate-*l* 85.8%

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

   6. *-commutative 85.8%

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

7. Simplified 85.8%

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

8. Taylor expanded in M around 0 59.2%

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

   1. unpow2 59.2%

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

   2. unpow2 59.2%

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

   3. unswap-sqr 71.7%

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

   4. unpow2 71.7%

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

   5. *-commutative 71.7%

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

   6. associate-*l* 76.0%

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

10. Simplified 76.0%

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

    1. times-frac 83.7%

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

12. Applied egg-rr 83.7%

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

    1. div-inv 83.7%

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

14. Applied egg-rr 83.7%

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

    1. associate-*r/ 83.7%

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

    2. *-rgt-identity 83.7%

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

    3. associate-*l/ 83.0%

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

    4. *-commutative 83.0%

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

16. Simplified 83.0%

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

17. Final simplification 83.0%

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

Alternative 3: 84.8% accurate, 1.8× speedup

Math

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

FPCore

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 (* h (* 0.25 (* (/ (* M D) d) (/ (* M D) (* d l)))))))))

C

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

Fortran

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 - (h * (0.25d0 * (((m * d) / d_1) * ((m * d) / (d_1 * l)))))))
end function

Java

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 - (h * (0.25 * (((M * D) / d) * ((M * D) / (d * l)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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[(h * N[(0.25 * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 80.1%

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

3. Simplified 80.1%

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

   1. associate-/l* 79.3%

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

   2. clear-num 79.3%

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

   3. un-div-inv 79.3%

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

   4. associate-*l/ 77.8%

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

   5. *-commutative 77.8%

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

   6. associate-/r* 77.8%

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

   7. div-inv 77.8%

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

   8. metadata-eval 77.8%

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

5. Applied egg-rr 77.8%

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

   1. associate-/r/ 83.5%

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

   2. *-commutative 83.5%

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

   3. *-commutative 83.5%

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

   4. associate-*l/ 83.5%

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

   5. associate-*l* 85.8%

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

   6. *-commutative 85.8%

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

7. Simplified 85.8%

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

8. Taylor expanded in M around 0 59.2%

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

   1. unpow2 59.2%

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

   2. unpow2 59.2%

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

   3. unswap-sqr 71.7%

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

   4. unpow2 71.7%

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

   5. *-commutative 71.7%

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

   6. associate-*l* 76.0%

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

10. Simplified 76.0%

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

    1. times-frac 83.7%

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

12. Applied egg-rr 83.7%

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

13. Final simplification 83.7%

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

Alternative 4: 76.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{-80} \lor \neg \left(M \leq 1.5 \cdot 10^{-76}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(M \cdot \frac{h \cdot D}{\frac{\ell}{\frac{D}{d \cdot d}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (or (<= M -2.9e-80) (not (<= M 1.5e-76)))
   (* w0 (+ 1.0 (* -0.125 (* M (* M (/ (* h D) (/ l (/ D (* d d)))))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M <= -2.9e-80) || !(M <= 1.5e-76)) {
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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 <= (-2.9d-80)) .or. (.not. (m <= 1.5d-76))) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (m * (m * ((h * d) / (l / (d / (d_1 * d_1))))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M <= -2.9e-80) || !(M <= 1.5e-76)) {
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (M <= -2.9e-80) or not (M <= 1.5e-76):
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((M <= -2.9e-80) || !(M <= 1.5e-76))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(M * Float64(M * Float64(Float64(h * D) / Float64(l / Float64(D / Float64(d * d)))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((M <= -2.9e-80) || ~((M <= 1.5e-76)))
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[Or[LessEqual[M, -2.9e-80], N[Not[LessEqual[M, 1.5e-76]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(M * N[(M * N[(N[(h * D), $MachinePrecision] / N[(l / N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if M < -2.89999999999999998e-80 or 1.50000000000000012e-76 < M

   1. Initial program 74.9%

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

      1. associate-/l* 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in M around 0 47.5%

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

      1. *-commutative 47.5%

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

      2. associate-/l* 48.2%

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

      3. unpow2 48.2%

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

      4. unpow2 48.2%

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

      5. *-commutative 48.2%

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

      6. unpow2 48.2%

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

   6. Simplified 48.2%

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

   7. Taylor expanded in D around 0 47.5%

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

      1. unpow2 47.5%

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

      2. associate-*r* 52.4%

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

      3. unpow2 52.4%

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

      4. *-commutative 52.4%

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

      5. associate-*l/ 53.2%

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

      6. unpow2 53.2%

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

      7. *-commutative 53.2%

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

      8. associate-*l* 60.9%

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

      9. *-commutative 60.9%

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

      10. associate-*l* 63.1%

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

   9. Simplified 63.1%

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

   10. Taylor expanded in M around 0 58.7%

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

       1. *-commutative 58.7%

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

       2. *-commutative 58.7%

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

       3. associate-*l* 64.9%

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

       4. *-commutative 64.9%

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

       5. associate-*r/ 65.7%

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

       6. associate-*r/ 67.9%

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

       7. unpow2 67.9%

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

       8. associate-/l* 73.3%

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

       9. associate-*r/ 71.9%

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

       10. *-commutative 71.9%

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

       11. associate-/l* 72.6%

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

       12. unpow2 72.6%

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

   12. Simplified 72.6%

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

   if -2.89999999999999998e-80 < M < 1.50000000000000012e-76

   1. Initial program 85.0%

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

      1. associate-/l* 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in M around 0 84.9%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{-80} \lor \neg \left(M \leq 1.5 \cdot 10^{-76}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(M \cdot \frac{h \cdot D}{\frac{\ell}{\frac{D}{d \cdot d}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 78.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq -1.5 \cdot 10^{-152} \lor \neg \left(D \leq 1.05 \cdot 10^{-46}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{M}} \cdot \frac{D}{\frac{d}{M} \cdot \frac{d}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (or (<= D -1.5e-152) (not (<= D 1.05e-46)))
   (* w0 (+ 1.0 (* -0.125 (* (/ D (/ l M)) (/ D (* (/ d M) (/ d h)))))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D <= -1.5e-152) || !(D <= 1.05e-46)) {
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / M)) * (D / ((d / M) * (d / h))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

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 <= (-1.5d-152)) .or. (.not. (d <= 1.05d-46))) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / (l / m)) * (d / ((d_1 / m) * (d_1 / h))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D <= -1.5e-152) || !(D <= 1.05e-46)) {
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / M)) * (D / ((d / M) * (d / h))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (D <= -1.5e-152) or not (D <= 1.05e-46):
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / M)) * (D / ((d / M) * (d / h))))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((D <= -1.5e-152) || !(D <= 1.05e-46))
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / Float64(l / M)) * Float64(D / Float64(Float64(d / M) * Float64(d / h)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D <= -1.5e-152) || ~((D <= 1.05e-46)))
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / M)) * (D / ((d / M) * (d / h))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[Or[LessEqual[D, -1.5e-152], N[Not[LessEqual[D, 1.05e-46]], $MachinePrecision]], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / N[(l / M), $MachinePrecision]), $MachinePrecision] * N[(D / N[(N[(d / M), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if D < -1.5e-152 or 1.04999999999999994e-46 < D

   1. Initial program 70.4%

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

      1. associate-/l* 71.7%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in M around 0 46.7%

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

      1. *-commutative 46.7%

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

      2. associate-/l* 48.1%

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

      3. unpow2 48.1%

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

      4. unpow2 48.1%

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

      5. *-commutative 48.1%

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

      6. unpow2 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in l around 0 48.1%

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

      1. unpow2 48.1%

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

      2. associate-*l* 50.9%

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

   9. Simplified 50.9%

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

       1. associate-*r* 48.1%

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

       2. *-commutative 48.1%

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

       3. associate-*l* 48.8%

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

       4. frac-times 53.6%

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

       5. times-frac 63.7%

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

       6. times-frac 70.7%

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

   11. Applied egg-rr 70.7%

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

   if -1.5e-152 < D < 1.04999999999999994e-46

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in M around 0 94.1%

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -1.5 \cdot 10^{-152} \lor \neg \left(D \leq 1.05 \cdot 10^{-46}\right):\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{M}} \cdot \frac{D}{\frac{d}{M} \cdot \frac{d}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 77.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -5.2 \cdot 10^{-81}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(M \cdot \frac{h \cdot D}{\frac{\ell}{\frac{D}{d \cdot d}}}\right)\right)\right)\\ \mathbf{elif}\;M \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d \cdot \ell}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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 -5.2e-81)
   (* w0 (+ 1.0 (* -0.125 (* M (* M (/ (* h D) (/ l (/ D (* d d)))))))))
   (if (<= M 6.8e-161)
     w0
     (* w0 (+ 1.0 (* -0.125 (* M (* (* h M) (* (/ D d) (/ D (* d l)))))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -5.2e-81) {
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	} else if (M <= 6.8e-161) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (M * ((h * M) * ((D / d) * (D / (d * l)))))));
	}
	return tmp;
}

Fortran

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 <= (-5.2d-81)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (m * (m * ((h * d) / (l / (d / (d_1 * d_1))))))))
    else if (m <= 6.8d-161) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (m * ((h * m) * ((d / d_1) * (d / (d_1 * l)))))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -5.2e-81) {
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	} else if (M <= 6.8e-161) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (M * ((h * M) * ((D / d) * (D / (d * l)))))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -5.2e-81:
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))))
	elif M <= 6.8e-161:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (M * ((h * M) * ((D / d) * (D / (d * l)))))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -5.2e-81)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(M * Float64(M * Float64(Float64(h * D) / Float64(l / Float64(D / Float64(d * d)))))))));
	elseif (M <= 6.8e-161)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(M * Float64(Float64(h * M) * Float64(Float64(D / d) * Float64(D / Float64(d * l))))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= -5.2e-81)
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	elseif (M <= 6.8e-161)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (M * ((h * M) * ((D / d) * (D / (d * l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, -5.2e-81], N[(w0 * N[(1.0 + N[(-0.125 * N[(M * N[(M * N[(N[(h * D), $MachinePrecision] / N[(l / N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 6.8e-161], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(M * N[(N[(h * M), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -5.1999999999999998e-81

   1. Initial program 70.5%

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

      1. associate-/l* 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in M around 0 37.7%

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

      1. *-commutative 37.7%

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

      2. associate-/l* 37.7%

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

      3. unpow2 37.7%

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

      4. unpow2 37.7%

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

      5. *-commutative 37.7%

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

      6. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in D around 0 37.7%

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

      1. unpow2 37.7%

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

      2. associate-*r* 43.8%

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

      3. unpow2 43.8%

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

      4. *-commutative 43.8%

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

      5. associate-*l/ 43.8%

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

      6. unpow2 43.8%

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

      7. *-commutative 43.8%

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

      8. associate-*l* 53.1%

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

      9. *-commutative 53.1%

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

      10. associate-*l* 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in M around 0 51.6%

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

       1. *-commutative 51.6%

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

       2. *-commutative 51.6%

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

       3. associate-*l* 57.5%

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

       4. *-commutative 57.5%

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

       5. associate-*r/ 59.0%

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

       6. associate-*r/ 60.8%

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

       7. unpow2 60.8%

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

       8. associate-/l* 66.1%

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

       9. associate-*r/ 64.6%

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

       10. *-commutative 64.6%

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

       11. associate-/l* 66.0%

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

       12. unpow2 66.0%

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

   12. Simplified 66.0%

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

   if -5.1999999999999998e-81 < M < 6.79999999999999964e-161

   1. Initial program 88.1%

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

      1. associate-/l* 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in M around 0 83.5%

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

   if 6.79999999999999964e-161 < M

   1. Initial program 77.2%

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

      1. associate-/l* 78.2%

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

   3. Simplified 78.2%

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

   4. Taylor expanded in M around 0 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-/l* 59.3%

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

      3. unpow2 59.3%

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

      4. unpow2 59.3%

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

      5. *-commutative 59.3%

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

      6. unpow2 59.3%

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

   6. Simplified 59.3%

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

   7. Taylor expanded in D around 0 57.2%

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

      1. unpow2 57.2%

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

      2. associate-*r* 60.2%

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

      3. unpow2 60.2%

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

      4. *-commutative 60.2%

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

      5. associate-*l/ 62.3%

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

      6. unpow2 62.3%

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

      7. *-commutative 62.3%

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

      8. associate-*l* 67.4%

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

      9. *-commutative 67.4%

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

      10. associate-*l* 69.6%

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

   9. Simplified 69.6%

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

   10. Taylor expanded in D around 0 67.4%

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

       1. unpow2 67.4%

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

       2. associate-*r* 69.6%

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

       3. unpow2 69.6%

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

       4. times-frac 77.8%

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

   12. Simplified 77.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -5.2 \cdot 10^{-81}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(M \cdot \frac{h \cdot D}{\frac{\ell}{\frac{D}{d \cdot d}}}\right)\right)\right)\\ \mathbf{elif}\;M \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(\left(h \cdot M\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d \cdot \ell}\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 76.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -2.7 \cdot 10^{-80}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(M \cdot \frac{h \cdot D}{\frac{\ell}{\frac{D}{d \cdot d}}}\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.8 \cdot 10^{-118}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(h \cdot M\right) \cdot \left(M \cdot \left(\frac{D}{\ell} \cdot \frac{\frac{D}{d}}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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 -2.7e-80)
   (* w0 (+ 1.0 (* -0.125 (* M (* M (/ (* h D) (/ l (/ D (* d d)))))))))
   (if (<= M 2.8e-118)
     w0
     (* w0 (+ 1.0 (* -0.125 (* (* h M) (* M (* (/ D l) (/ (/ D d) d))))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -2.7e-80) {
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	} else if (M <= 2.8e-118) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h * M) * (M * ((D / l) * ((D / d) / d))))));
	}
	return tmp;
}

Fortran

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 <= (-2.7d-80)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (m * (m * ((h * d) / (l / (d / (d_1 * d_1))))))))
    else if (m <= 2.8d-118) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h * m) * (m * ((d / l) * ((d / d_1) / d_1))))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= -2.7e-80) {
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	} else if (M <= 2.8e-118) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h * M) * (M * ((D / l) * ((D / d) / d))))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -2.7e-80:
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))))
	elif M <= 2.8e-118:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((h * M) * (M * ((D / l) * ((D / d) / d))))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -2.7e-80)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(M * Float64(M * Float64(Float64(h * D) / Float64(l / Float64(D / Float64(d * d)))))))));
	elseif (M <= 2.8e-118)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h * M) * Float64(M * Float64(Float64(D / l) * Float64(Float64(D / d) / d)))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= -2.7e-80)
		tmp = w0 * (1.0 + (-0.125 * (M * (M * ((h * D) / (l / (D / (d * d))))))));
	elseif (M <= 2.8e-118)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((h * M) * (M * ((D / l) * ((D / d) / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, -2.7e-80], N[(w0 * N[(1.0 + N[(-0.125 * N[(M * N[(M * N[(N[(h * D), $MachinePrecision] / N[(l / N[(D / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 2.8e-118], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h * M), $MachinePrecision] * N[(M * N[(N[(D / l), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.7000000000000002e-80

   1. Initial program 70.5%

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

      1. associate-/l* 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in M around 0 37.7%

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

      1. *-commutative 37.7%

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

      2. associate-/l* 37.7%

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

      3. unpow2 37.7%

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

      4. unpow2 37.7%

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

      5. *-commutative 37.7%

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

      6. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in D around 0 37.7%

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

      1. unpow2 37.7%

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

      2. associate-*r* 43.8%

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

      3. unpow2 43.8%

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

      4. *-commutative 43.8%

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

      5. associate-*l/ 43.8%

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

      6. unpow2 43.8%

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

      7. *-commutative 43.8%

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

      8. associate-*l* 53.1%

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

      9. *-commutative 53.1%

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

      10. associate-*l* 56.2%

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

   9. Simplified 56.2%

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

   10. Taylor expanded in M around 0 51.6%

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

       1. *-commutative 51.6%

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

       2. *-commutative 51.6%

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

       3. associate-*l* 57.5%

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

       4. *-commutative 57.5%

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

       5. associate-*r/ 59.0%

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

       6. associate-*r/ 60.8%

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

       7. unpow2 60.8%

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

       8. associate-/l* 66.1%

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

       9. associate-*r/ 64.6%

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

       10. *-commutative 64.6%

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

       11. associate-/l* 66.0%

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

       12. unpow2 66.0%

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

   12. Simplified 66.0%

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

   if -2.7000000000000002e-80 < M < 2.8e-118

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

   3. Simplified 86.0%

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

   4. Taylor expanded in M around 0 83.9%

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

   if 2.8e-118 < M

   1. Initial program 78.4%

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

      1. associate-/l* 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in M around 0 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/l* 61.1%

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

      3. unpow2 61.1%

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

      4. unpow2 61.1%

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

      5. *-commutative 61.1%

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

      6. unpow2 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in l around 0 61.1%

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

      1. unpow2 61.1%

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

      2. associate-*l* 62.3%

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

   9. Simplified 62.3%

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

   10. Taylor expanded in D around 0 58.9%

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

       1. unpow2 58.9%

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

       2. unpow2 58.9%

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

       3. associate-*r* 62.2%

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

       4. associate-*l/ 64.5%

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

       5. unpow2 64.5%

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

       6. associate-*r* 70.0%

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

       7. *-commutative 70.0%

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

       8. *-commutative 70.0%

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

       9. unpow2 70.0%

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

       10. times-frac 74.4%

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

       11. unpow2 74.4%

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

       12. associate-/r* 76.6%

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

   12. Simplified 76.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.7 \cdot 10^{-80}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(M \cdot \frac{h \cdot D}{\frac{\ell}{\frac{D}{d \cdot d}}}\right)\right)\right)\\ \mathbf{elif}\;M \leq 2.8 \cdot 10^{-118}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(h \cdot M\right) \cdot \left(M \cdot \left(\frac{D}{\ell} \cdot \frac{\frac{D}{d}}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 78.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{M} \cdot \frac{d}{h}\\ \mathbf{if}\;D \leq -5 \cdot 10^{-150}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{\frac{\ell}{M} \cdot t_0}\right)\right)\\ \mathbf{elif}\;D \leq 1.45 \cdot 10^{-46}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{M}} \cdot \frac{D}{t_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ d M) (/ d h))))
   (if (<= D -5e-150)
     (* w0 (+ 1.0 (* -0.125 (* (* D D) (/ 1.0 (* (/ l M) t_0))))))
     (if (<= D 1.45e-46)
       w0
       (* w0 (+ 1.0 (* -0.125 (* (/ D (/ l M)) (/ D t_0)))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (d / M) * (d / h);
	double tmp;
	if (D <= -5e-150) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) * (1.0 / ((l / M) * t_0)))));
	} else if (D <= 1.45e-46) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / M)) * (D / t_0))));
	}
	return tmp;
}

Fortran

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) :: t_0
    real(8) :: tmp
    t_0 = (d_1 / m) * (d_1 / h)
    if (d <= (-5d-150)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * d) * (1.0d0 / ((l / m) * t_0)))))
    else if (d <= 1.45d-46) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / (l / m)) * (d / t_0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (d / M) * (d / h);
	double tmp;
	if (D <= -5e-150) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) * (1.0 / ((l / M) * t_0)))));
	} else if (D <= 1.45e-46) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / M)) * (D / t_0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = (d / M) * (d / h)
	tmp = 0
	if D <= -5e-150:
		tmp = w0 * (1.0 + (-0.125 * ((D * D) * (1.0 / ((l / M) * t_0)))))
	elif D <= 1.45e-46:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / M)) * (D / t_0))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(d / M) * Float64(d / h))
	tmp = 0.0
	if (D <= -5e-150)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * D) * Float64(1.0 / Float64(Float64(l / M) * t_0))))));
	elseif (D <= 1.45e-46)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / Float64(l / M)) * Float64(D / t_0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (d / M) * (d / h);
	tmp = 0.0;
	if (D <= -5e-150)
		tmp = w0 * (1.0 + (-0.125 * ((D * D) * (1.0 / ((l / M) * t_0)))));
	elseif (D <= 1.45e-46)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / M)) * (D / t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(d / M), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, -5e-150], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(1.0 / N[(N[(l / M), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 1.45e-46], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / N[(l / M), $MachinePrecision]), $MachinePrecision] * N[(D / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if D < -4.9999999999999999e-150

   1. Initial program 70.5%

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

      1. associate-/l* 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in M around 0 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-/l* 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

      5. *-commutative 42.1%

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

      6. unpow2 42.1%

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

   6. Simplified 42.1%

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

   7. Taylor expanded in l around 0 42.1%

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

      1. unpow2 42.1%

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

      2. associate-*l* 45.9%

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

   9. Simplified 45.9%

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

       1. div-inv 45.9%

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

       2. associate-*r* 42.1%

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

       3. *-commutative 42.1%

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

       4. associate-*l* 43.3%

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

       5. frac-times 50.5%

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

       6. times-frac 58.0%

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

   11. Applied egg-rr 58.0%

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

   if -4.9999999999999999e-150 < D < 1.45000000000000002e-46

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in M around 0 94.1%

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

   if 1.45000000000000002e-46 < D

   1. Initial program 70.1%

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

      1. associate-/l* 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in M around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-/l* 56.0%

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

      3. unpow2 56.0%

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

      4. unpow2 56.0%

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

      5. *-commutative 56.0%

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

      6. unpow2 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in l around 0 56.0%

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

      1. unpow2 56.0%

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

      2. associate-*l* 57.6%

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

   9. Simplified 57.6%

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

       1. associate-*r* 56.0%

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

       2. *-commutative 56.0%

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

       3. associate-*l* 56.1%

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

       4. frac-times 57.6%

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

       5. times-frac 66.5%

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

       6. times-frac 73.0%

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

   11. Applied egg-rr 73.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -5 \cdot 10^{-150}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{1}{\frac{\ell}{M} \cdot \left(\frac{d}{M} \cdot \frac{d}{h}\right)}\right)\right)\\ \mathbf{elif}\;D \leq 1.45 \cdot 10^{-46}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{M}} \cdot \frac{D}{\frac{d}{M} \cdot \frac{d}{h}}\right)\right)\\ \end{array} \]

Alternative 9: 68.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{+50}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(w0 \cdot \left(\left(h \cdot M\right) \cdot \left(M \cdot \left(\frac{D}{\ell} \cdot \frac{\frac{D}{d}}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

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+50)
   w0
   (* -0.125 (* w0 (* (* h M) (* M (* (/ D l) (/ (/ D d) d))))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.35e+50) {
		tmp = w0;
	} else {
		tmp = -0.125 * (w0 * ((h * M) * (M * ((D / l) * ((D / d) / d)))));
	}
	return tmp;
}

Fortran

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+50) then
        tmp = w0
    else
        tmp = (-0.125d0) * (w0 * ((h * m) * (m * ((d / l) * ((d / d_1) / d_1)))))
    end if
    code = tmp
end function

Java

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+50) {
		tmp = w0;
	} else {
		tmp = -0.125 * (w0 * ((h * M) * (M * ((D / l) * ((D / d) / d)))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.35e+50:
		tmp = w0
	else:
		tmp = -0.125 * (w0 * ((h * M) * (M * ((D / l) * ((D / d) / d)))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.35e+50)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(w0 * Float64(Float64(h * M) * Float64(M * Float64(Float64(D / l) * Float64(Float64(D / d) / d))))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.35e+50)
		tmp = w0;
	else
		tmp = -0.125 * (w0 * ((h * M) * (M * ((D / l) * ((D / d) / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

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+50], w0, N[(-0.125 * N[(w0 * N[(N[(h * M), $MachinePrecision] * N[(M * N[(N[(D / l), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.35e50

   1. Initial program 80.2%

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

      1. associate-/l* 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in M around 0 73.0%

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

   if 1.35e50 < M

   1. Initial program 75.1%

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

      1. associate-/l* 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in M around 0 47.4%

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

      1. *-commutative 47.4%

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

      2. associate-/l* 47.4%

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

      3. unpow2 47.4%

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

      4. unpow2 47.4%

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

      5. *-commutative 47.4%

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

      6. unpow2 47.4%

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

   6. Simplified 47.4%

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

   7. Taylor expanded in D around 0 47.4%

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

      1. unpow2 47.4%

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

      2. associate-*r* 54.3%

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

      3. unpow2 54.3%

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

      4. *-commutative 54.3%

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

      5. associate-*l/ 54.3%

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

      6. unpow2 54.3%

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

      7. *-commutative 54.3%

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

      8. associate-*l* 63.6%

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

      9. *-commutative 63.6%

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

      10. associate-*l* 63.8%

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

   9. Simplified 63.8%

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

   10. Taylor expanded in M around inf 31.4%

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

       1. *-commutative 31.4%

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

       2. times-frac 31.4%

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

       3. unpow2 31.4%

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

       4. *-commutative 31.4%

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

       5. unpow2 31.4%

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

       6. unpow2 31.4%

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

       7. *-commutative 31.4%

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

       8. unpow2 31.4%

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

       9. unpow2 31.4%

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

       10. times-frac 34.5%

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

       11. associate-/l* 34.4%

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

   12. Simplified 34.4%

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

   13. Taylor expanded in D around 0 31.4%

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

       1. unpow2 31.4%

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

       2. *-commutative 31.4%

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

       3. unpow2 31.4%

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

       4. *-commutative 31.4%

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

       5. associate-*r* 31.6%

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

       6. associate-*l* 31.8%

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

       7. *-commutative 31.8%

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

       8. unpow2 31.8%

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

       9. associate-*r* 31.6%

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

       10. unpow2 31.6%

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

       11. *-commutative 31.6%

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

       12. associate-*r/ 31.6%

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

       13. unpow2 31.6%

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

       14. unpow2 31.6%

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

       15. associate-*r* 31.8%

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

   15. Simplified 35.1%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.35 \cdot 10^{+50}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(w0 \cdot \left(\left(h \cdot M\right) \cdot \left(M \cdot \left(\frac{D}{\ell} \cdot \frac{\frac{D}{d}}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 67.7% accurate, 216.0× speedup

Math

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

FPCore

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

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 79.3%

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

   1. associate-/l* 80.1%

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

3. Simplified 80.1%

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

4. Taylor expanded in M around 0 69.0%

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

5. Final simplification 69.0%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023187 
(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))))))