Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 86.0%

Time: 15.9s

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: 80.5% 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: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+236}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\right)}{\ell}}\\ \end{array} \end{array} \]

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 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 5e+236)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt (- 1.0 (/ (* 0.25 (* M (* (/ M d) (/ (* D (* D h)) d)))) l)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+236) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+236) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * (m * ((m / d_1) * ((d * (d * h)) / 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 t_0 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+236) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+236:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+236)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(M * Float64(Float64(M / d) * Float64(Float64(D * Float64(D * h)) / d)))) / l))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+236)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / 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_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 5e+236], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(M * N[(N[(M / d), $MachinePrecision] * N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 4.9999999999999997e236

   1. Initial program 99.9%

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

   if 4.9999999999999997e236 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 40.9%

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

      1. *-commutative 40.9%

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

      2. times-frac 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in M around 0 46.3%

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

      1. associate-*r/ 46.3%

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

      2. *-commutative 46.3%

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

      3. times-frac 50.0%

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

      4. *-commutative 50.0%

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

      5. *-commutative 50.0%

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

      6. associate-*l* 49.0%

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

      7. unpow2 49.0%

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

      8. unpow2 49.0%

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

      9. unpow2 49.0%

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

   6. Simplified 49.0%

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

      1. div-inv 49.0%

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

      2. associate-*l* 49.2%

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

   8. Applied egg-rr 49.2%

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

   9. Taylor expanded in M around 0 50.0%

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

       1. associate-*r* 49.0%

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

       2. *-commutative 49.0%

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

       3. unpow2 49.0%

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

       4. *-commutative 49.0%

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

       5. unpow2 49.0%

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

       6. times-frac 50.7%

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

       7. unpow2 50.7%

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

       8. associate-*r/ 53.3%

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

       9. associate-*r* 56.8%

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

       10. *-commutative 56.8%

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

       11. *-commutative 56.8%

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

   11. Simplified 56.8%

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

       1. associate-*l/ 56.8%

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

       2. associate-*l* 57.9%

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

       3. *-commutative 57.9%

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

   13. Applied egg-rr 57.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+236}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\right)}{\ell}}\\ \end{array} \]

Alternative 2: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\right)}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-285}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* M (* (/ M d) (/ (* D (* D h)) d)))) l))))
   (if (<= (/ h l) -2e-285)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
     w0)))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)));
	} else if ((h / l) <= -2e-285) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)));
	} else if ((h / l) <= -2e-285) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)))
	elif (h / l) <= -2e-285:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(M * Float64(Float64(M / d) * Float64(Float64(D * Float64(D * h)) / d)))) / l))));
	elseif (Float64(h / l) <= -2e-285)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	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 ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)));
	elseif ((h / l) <= -2e-285)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	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[LessEqual[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(M * N[(N[(M / d), $MachinePrecision] * N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -2e-285], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. times-frac 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in M around 0 56.9%

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

      1. associate-*r/ 56.9%

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

      2. *-commutative 56.9%

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

      3. times-frac 56.8%

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

      4. *-commutative 56.8%

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

      5. *-commutative 56.8%

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

      6. associate-*l* 52.2%

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

      7. unpow2 52.2%

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

      8. unpow2 52.2%

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

      9. unpow2 52.2%

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

   6. Simplified 52.2%

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

      1. div-inv 52.2%

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

      2. associate-*l* 52.2%

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

   8. Applied egg-rr 52.2%

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

   9. Taylor expanded in M around 0 56.8%

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

       1. associate-*r* 52.2%

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

       2. *-commutative 52.2%

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

       3. unpow2 52.2%

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

       4. *-commutative 52.2%

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

       5. unpow2 52.2%

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

       6. times-frac 53.2%

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

       7. unpow2 53.2%

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

       8. associate-*r/ 53.2%

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

       9. associate-*r* 63.5%

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

       10. *-commutative 63.5%

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

       11. *-commutative 63.5%

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

   11. Simplified 63.5%

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

       1. associate-*l/ 63.5%

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

       2. associate-*l* 63.5%

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

       3. *-commutative 63.5%

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

   13. Applied egg-rr 63.5%

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

   if -inf.0 < (/.f64 h l) < -2.00000000000000015e-285

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

      2. times-frac 80.9%

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

   3. Simplified 80.9%

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

   if -2.00000000000000015e-285 < (/.f64 h l)

   1. Initial program 87.6%

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

      1. *-commutative 87.6%

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

      2. times-frac 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in M around 0 95.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\right)}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-285}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-137}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \left(h \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) -2e-137)
   (* w0 (sqrt (- 1.0 (* (/ 0.25 l) (* (* M (/ M d)) (* h (* 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 ((h / l) <= -2e-137) {
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((M * (M / d)) * (h * (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 ((h / l) <= (-2d-137)) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 / l) * ((m * (m / d_1)) * (h * (d * (d / 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 ((h / l) <= -2e-137) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / l) * ((M * (M / d)) * (h * (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 (h / l) <= -2e-137:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / l) * ((M * (M / d)) * (h * (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 (Float64(h / l) <= -2e-137)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / l) * Float64(Float64(M * Float64(M / d)) * Float64(h * 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 ((h / l) <= -2e-137)
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((M * (M / d)) * (h * (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[LessEqual[N[(h / l), $MachinePrecision], -2e-137], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / l), $MachinePrecision] * N[(N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.99999999999999996e-137

   1. Initial program 73.5%

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

      1. *-commutative 73.5%

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

      2. times-frac 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in M around 0 50.7%

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

      1. associate-*r/ 50.7%

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

      2. *-commutative 50.7%

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

      3. times-frac 52.5%

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

      4. *-commutative 52.5%

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

      5. *-commutative 52.5%

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

      6. associate-*l* 53.8%

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

      7. unpow2 53.8%

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

      8. unpow2 53.8%

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

      9. unpow2 53.8%

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

   6. Simplified 53.8%

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

      1. div-inv 52.9%

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

      2. associate-*l* 57.4%

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

   8. Applied egg-rr 57.4%

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

   9. Taylor expanded in M around 0 52.5%

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

       1. associate-*r* 53.8%

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

       2. *-commutative 53.8%

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

       3. unpow2 53.8%

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

       4. *-commutative 53.8%

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

       5. unpow2 53.8%

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

       6. times-frac 59.1%

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

       7. unpow2 59.1%

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

       8. associate-*r/ 62.6%

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

       9. associate-*r* 66.8%

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

       10. *-commutative 66.8%

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

       11. *-commutative 66.8%

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

   11. Simplified 66.8%

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

   12. Taylor expanded in D around 0 62.6%

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

       1. associate-/l* 60.7%

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

       2. associate-/r/ 64.3%

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

       3. unpow2 64.3%

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

       4. associate-*l/ 66.1%

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

       5. *-commutative 66.1%

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

   14. Simplified 66.1%

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

   if -1.99999999999999996e-137 < (/.f64 h l)

   1. Initial program 87.3%

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

      1. *-commutative 87.3%

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

      2. times-frac 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in M around 0 91.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-137}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\left(M \cdot \frac{M}{d}\right) \cdot \left(h \cdot \left(D \cdot \frac{D}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \leq 5 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 + D \cdot \left(w0 \cdot \left(\frac{M}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{h}\right)} \cdot \left(D \cdot -0.125\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= D 5e-12)
   (* w0 (sqrt (- 1.0 (/ (* 0.25 (* M (* (/ M d) (/ (* D (* D h)) d)))) l))))
   (+ w0 (* D (* w0 (* (/ M (* l (* (/ d M) (/ d h)))) (* D -0.125)))))))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 5e-12) {
		tmp = w0 * sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)));
	} else {
		tmp = w0 + (D * (w0 * ((M / (l * ((d / M) * (d / h)))) * (D * -0.125))));
	}
	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 <= 5d-12) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * (m * ((m / d_1) * ((d * (d * h)) / d_1)))) / l)))
    else
        tmp = w0 + (d * (w0 * ((m / (l * ((d_1 / m) * (d_1 / h)))) * (d * (-0.125d0)))))
    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 <= 5e-12) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)));
	} else {
		tmp = w0 + (D * (w0 * ((M / (l * ((d / M) * (d / h)))) * (D * -0.125))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 5e-12:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)))
	else:
		tmp = w0 + (D * (w0 * ((M / (l * ((d / M) * (d / h)))) * (D * -0.125))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 5e-12)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(M * Float64(Float64(M / d) * Float64(Float64(D * Float64(D * h)) / d)))) / l))));
	else
		tmp = Float64(w0 + Float64(D * Float64(w0 * Float64(Float64(M / Float64(l * Float64(Float64(d / M) * Float64(d / h)))) * Float64(D * -0.125)))));
	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 <= 5e-12)
		tmp = w0 * sqrt((1.0 - ((0.25 * (M * ((M / d) * ((D * (D * h)) / d)))) / l)));
	else
		tmp = w0 + (D * (w0 * ((M / (l * ((d / M) * (d / h)))) * (D * -0.125))));
	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[D, 5e-12], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(M * N[(N[(M / d), $MachinePrecision] * N[(N[(D * N[(D * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 + N[(D * N[(w0 * N[(N[(M / N[(l * N[(N[(d / M), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 4.9999999999999997e-12

   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. *-commutative 80.2%

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

      2. times-frac 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in M around 0 52.8%

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

      1. associate-*r/ 52.8%

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

      2. *-commutative 52.8%

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

      3. times-frac 55.3%

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

      4. *-commutative 55.3%

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

      5. *-commutative 55.3%

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

      6. associate-*l* 56.0%

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

      7. unpow2 56.0%

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

      8. unpow2 56.0%

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

      9. unpow2 56.0%

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

   6. Simplified 56.0%

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

      1. div-inv 55.0%

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

      2. associate-*l* 62.9%

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

   8. Applied egg-rr 62.9%

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

   9. Taylor expanded in M around 0 55.3%

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

       1. associate-*r* 56.0%

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

       2. *-commutative 56.0%

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

       3. unpow2 56.0%

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

       4. *-commutative 56.0%

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

       5. unpow2 56.0%

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

       6. times-frac 64.3%

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

       7. unpow2 64.3%

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

       8. associate-*r/ 68.7%

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

       9. associate-*r* 73.9%

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

       10. *-commutative 73.9%

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

       11. *-commutative 73.9%

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

   11. Simplified 73.9%

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

       1. associate-*l/ 73.9%

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

       2. associate-*l* 79.3%

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

       3. *-commutative 79.3%

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

   13. Applied egg-rr 79.3%

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

   if 4.9999999999999997e-12 < D

   1. Initial program 84.7%

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

      1. *-commutative 84.7%

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

      2. times-frac 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in M around 0 52.6%

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

      1. *-commutative 52.6%

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

      2. unpow2 52.6%

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

      3. unpow2 52.6%

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

      4. unpow2 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in D around 0 52.6%

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

      1. unpow2 52.6%

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

      2. *-commutative 52.6%

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

      3. times-frac 56.6%

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

      4. unpow2 56.6%

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

      5. times-frac 58.6%

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

      6. associate-*l/ 62.8%

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

      7. associate-/r/ 62.8%

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

      8. *-commutative 62.8%

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

      9. unpow2 62.8%

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

      10. associate-/l* 68.8%

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

      11. associate-*l/ 72.8%

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

      12. associate-/r/ 82.9%

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

      13. *-commutative 82.9%

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

      14. *-commutative 82.9%

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

      15. associate-/l* 72.8%

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

   9. Simplified 66.5%

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

       1. distribute-rgt-in 66.6%

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

       2. *-un-lft-identity 66.6%

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

       3. associate-*l* 66.6%

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

       4. associate-/r/ 74.9%

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

       5. associate-/l* 79.3%

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

       6. div-inv 79.3%

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

       7. clear-num 79.3%

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

   11. Applied egg-rr 79.3%

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

       1. pow1 79.3%

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

       2. associate-*l* 79.3%

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

       3. associate-*l* 79.3%

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

       4. associate-/l/ 79.3%

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

       5. associate-/l* 81.3%

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

   13. Applied egg-rr 81.3%

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

       1. unpow1 81.3%

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

       2. *-commutative 81.3%

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

       3. associate-/r/ 81.3%

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

       4. *-commutative 81.3%

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

   15. Simplified 81.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(M \cdot \left(\frac{M}{d} \cdot \frac{D \cdot \left(D \cdot h\right)}{d}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 + D \cdot \left(w0 \cdot \left(\frac{M}{\ell \cdot \left(\frac{d}{M} \cdot \frac{d}{h}\right)} \cdot \left(D \cdot -0.125\right)\right)\right)\\ \end{array} \]

Alternative 5: 69.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -4.8 \cdot 10^{+269} \lor \neg \left(M \leq 5.9 \cdot 10^{+62}\right):\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\left(\frac{M}{d} \cdot \left(w0 \cdot \frac{M}{d}\right)\right) \cdot \left(h \cdot \frac{D}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 -4.8e+269) (not (<= M 5.9e+62)))
   (* -0.125 (* D (* (* (/ M d) (* w0 (/ M d))) (* h (/ D l)))))
   w0))

C

assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((M <= -4.8e+269) || !(M <= 5.9e+62)) {
		tmp = -0.125 * (D * (((M / d) * (w0 * (M / d))) * (h * (D / l))));
	} 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 <= (-4.8d+269)) .or. (.not. (m <= 5.9d+62))) then
        tmp = (-0.125d0) * (d * (((m / d_1) * (w0 * (m / d_1))) * (h * (d / l))))
    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 <= -4.8e+269) || !(M <= 5.9e+62)) {
		tmp = -0.125 * (D * (((M / d) * (w0 * (M / d))) * (h * (D / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (M <= -4.8e+269) or not (M <= 5.9e+62):
		tmp = -0.125 * (D * (((M / d) * (w0 * (M / d))) * (h * (D / l))))
	else:
		tmp = w0
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((M <= -4.8e+269) || !(M <= 5.9e+62))
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(Float64(M / d) * Float64(w0 * Float64(M / d))) * Float64(h * Float64(D / l)))));
	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 <= -4.8e+269) || ~((M <= 5.9e+62)))
		tmp = -0.125 * (D * (((M / d) * (w0 * (M / d))) * (h * (D / l))));
	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, -4.8e+269], N[Not[LessEqual[M, 5.9e+62]], $MachinePrecision]], N[(-0.125 * N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(w0 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if M < -4.79999999999999987e269 or 5.9000000000000003e62 < M

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

      2. times-frac 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in M around 0 40.5%

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

      1. *-commutative 40.5%

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

      2. unpow2 40.5%

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

      3. unpow2 40.5%

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

      4. unpow2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in D around 0 40.5%

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

      1. unpow2 40.5%

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

      2. *-commutative 40.5%

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

      3. times-frac 42.2%

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

      4. unpow2 42.2%

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

      5. times-frac 44.2%

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

      6. associate-*l/ 54.1%

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

      7. associate-/r/ 54.1%

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

      8. *-commutative 54.1%

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

      9. unpow2 54.1%

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

      10. associate-/l* 54.3%

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

      11. associate-*l/ 54.5%

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

      12. associate-/r/ 54.6%

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

      13. *-commutative 54.6%

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

      14. *-commutative 54.6%

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

      15. associate-/l* 54.5%

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

   9. Simplified 44.3%

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

   10. Taylor expanded in D around inf 29.4%

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

       1. *-commutative 29.4%

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

       2. times-frac 32.8%

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

       3. associate-*r* 31.3%

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

       4. unpow2 31.3%

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

       5. times-frac 31.3%

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

       6. associate-*l/ 32.9%

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

       7. unpow2 32.9%

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

       8. associate-*l/ 33.4%

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

       9. unpow2 33.4%

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

       10. associate-/l* 33.8%

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

       11. associate-*l/ 33.9%

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

       12. associate-/r/ 34.0%

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

       13. *-commutative 34.0%

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

       14. associate-/l* 33.9%

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

       15. associate-/r/ 33.9%

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

   12. Simplified 34.6%

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

   13. Taylor expanded in D around 0 29.9%

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

       1. *-commutative 29.9%

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

       2. times-frac 33.1%

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

       3. associate-/l* 33.2%

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

       4. *-commutative 33.2%

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

       5. associate-/l* 33.1%

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

       6. *-commutative 33.1%

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

       7. unpow2 33.1%

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

       8. associate-*r* 31.5%

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

       9. times-frac 31.8%

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

       10. unpow2 31.8%

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

       11. associate-/l* 32.2%

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

       12. times-frac 32.7%

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

       13. associate-*r* 34.2%

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

       14. *-commutative 34.2%

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

       15. associate-*r* 34.4%

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

       16. *-commutative 34.4%

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

       17. *-commutative 34.4%

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

   15. Simplified 32.9%

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

   if -4.79999999999999987e269 < M < 5.9000000000000003e62

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in M around 0 78.7%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4.8 \cdot 10^{+269} \lor \neg \left(M \leq 5.9 \cdot 10^{+62}\right):\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\left(\frac{M}{d} \cdot \left(w0 \cdot \frac{M}{d}\right)\right) \cdot \left(h \cdot \frac{D}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 69.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{+269} \lor \neg \left(M \leq 5.9 \cdot 10^{+62}\right):\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(w0 \cdot \frac{M}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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+269) (not (<= M 5.9e+62)))
   (* -0.125 (* D (* (/ D l) (* h (* (/ M d) (* w0 (/ M 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+269) || !(M <= 5.9e+62)) {
		tmp = -0.125 * (D * ((D / l) * (h * ((M / d) * (w0 * (M / 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+269)) .or. (.not. (m <= 5.9d+62))) then
        tmp = (-0.125d0) * (d * ((d / l) * (h * ((m / d_1) * (w0 * (m / 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+269) || !(M <= 5.9e+62)) {
		tmp = -0.125 * (D * ((D / l) * (h * ((M / d) * (w0 * (M / 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+269) or not (M <= 5.9e+62):
		tmp = -0.125 * (D * ((D / l) * (h * ((M / d) * (w0 * (M / 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+269) || !(M <= 5.9e+62))
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(h * Float64(Float64(M / d) * Float64(w0 * Float64(M / 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+269) || ~((M <= 5.9e+62)))
		tmp = -0.125 * (D * ((D / l) * (h * ((M / d) * (w0 * (M / 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+269], N[Not[LessEqual[M, 5.9e+62]], $MachinePrecision]], N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(w0 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if M < -2.90000000000000025e269 or 5.9000000000000003e62 < M

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

      2. times-frac 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in M around 0 40.5%

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

      1. *-commutative 40.5%

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

      2. unpow2 40.5%

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

      3. unpow2 40.5%

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

      4. unpow2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in D around 0 40.5%

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

      1. unpow2 40.5%

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

      2. *-commutative 40.5%

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

      3. times-frac 42.2%

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

      4. unpow2 42.2%

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

      5. times-frac 44.2%

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

      6. associate-*l/ 54.1%

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

      7. associate-/r/ 54.1%

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

      8. *-commutative 54.1%

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

      9. unpow2 54.1%

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

      10. associate-/l* 54.3%

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

      11. associate-*l/ 54.5%

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

      12. associate-/r/ 54.6%

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

      13. *-commutative 54.6%

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

      14. *-commutative 54.6%

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

      15. associate-/l* 54.5%

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

   9. Simplified 44.3%

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

   10. Taylor expanded in D around inf 29.4%

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

       1. *-commutative 29.4%

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

       2. times-frac 32.8%

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

       3. associate-*r* 31.3%

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

       4. unpow2 31.3%

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

       5. times-frac 31.3%

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

       6. associate-*l/ 32.9%

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

       7. unpow2 32.9%

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

       8. associate-*l/ 33.4%

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

       9. unpow2 33.4%

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

       10. associate-/l* 33.8%

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

       11. associate-*l/ 33.9%

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

       12. associate-/r/ 34.0%

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

       13. *-commutative 34.0%

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

       14. associate-/l* 33.9%

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

       15. associate-/r/ 33.9%

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

   12. Simplified 34.6%

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

   13. Taylor expanded in h around 0 33.1%

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

       1. *-commutative 33.1%

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

       2. unpow2 33.1%

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

       3. associate-*r* 31.5%

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

       4. times-frac 31.8%

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

       5. unpow2 31.8%

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

       6. associate-/l* 32.2%

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

       7. times-frac 32.7%

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

       8. associate-*r* 34.2%

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

       9. *-commutative 34.2%

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

       10. associate-*r* 34.4%

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

       11. *-commutative 34.4%

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

       12. *-commutative 34.4%

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

       13. associate-*r/ 34.5%

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

       14. times-frac 34.6%

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

       15. associate-/l* 34.6%

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

       16. associate-*r/ 34.6%

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

   15. Simplified 34.6%

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

   if -2.90000000000000025e269 < M < 5.9000000000000003e62

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in M around 0 78.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.9 \cdot 10^{+269} \lor \neg \left(M \leq 5.9 \cdot 10^{+62}\right):\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(w0 \cdot \frac{M}{d}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7: 69.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -1.35 \cdot 10^{+269}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(w0 \cdot \frac{M}{d}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;M \leq 5.9 \cdot 10^{+62}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h \cdot \left(\frac{M}{d} \cdot \left(M \cdot w0\right)\right)}{d}\right)\right)\\ \end{array} \end{array} \]

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+269)
   (* -0.125 (* D (* (/ D l) (* h (* (/ M d) (* w0 (/ M d)))))))
   (if (<= M 5.9e+62)
     w0
     (* -0.125 (* D (* (/ D l) (/ (* h (* (/ M d) (* M w0))) 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+269) {
		tmp = -0.125 * (D * ((D / l) * (h * ((M / d) * (w0 * (M / d))))));
	} else if (M <= 5.9e+62) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / l) * ((h * ((M / d) * (M * w0))) / 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+269)) then
        tmp = (-0.125d0) * (d * ((d / l) * (h * ((m / d_1) * (w0 * (m / d_1))))))
    else if (m <= 5.9d+62) then
        tmp = w0
    else
        tmp = (-0.125d0) * (d * ((d / l) * ((h * ((m / d_1) * (m * w0))) / 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+269) {
		tmp = -0.125 * (D * ((D / l) * (h * ((M / d) * (w0 * (M / d))))));
	} else if (M <= 5.9e+62) {
		tmp = w0;
	} else {
		tmp = -0.125 * (D * ((D / l) * ((h * ((M / d) * (M * w0))) / d)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -1.35e+269:
		tmp = -0.125 * (D * ((D / l) * (h * ((M / d) * (w0 * (M / d))))))
	elif M <= 5.9e+62:
		tmp = w0
	else:
		tmp = -0.125 * (D * ((D / l) * ((h * ((M / d) * (M * w0))) / d)))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -1.35e+269)
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(h * Float64(Float64(M / d) * Float64(w0 * Float64(M / d)))))));
	elseif (M <= 5.9e+62)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(Float64(h * Float64(Float64(M / d) * Float64(M * w0))) / 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+269)
		tmp = -0.125 * (D * ((D / l) * (h * ((M / d) * (w0 * (M / d))))));
	elseif (M <= 5.9e+62)
		tmp = w0;
	else
		tmp = -0.125 * (D * ((D / l) * ((h * ((M / d) * (M * w0))) / 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+269], N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(w0 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 5.9e+62], w0, N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(N[(h * N[(N[(M / d), $MachinePrecision] * N[(M * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.3499999999999999e269

   1. Initial program 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in M around 0 25.2%

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

      1. *-commutative 25.2%

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

      2. unpow2 25.2%

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

      3. unpow2 25.2%

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

      4. unpow2 25.2%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in D around 0 25.2%

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

      1. unpow2 25.2%

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

      2. *-commutative 25.2%

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

      3. times-frac 38.2%

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

      4. unpow2 38.2%

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

      5. times-frac 38.2%

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

      6. associate-*l/ 38.2%

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

      7. associate-/r/ 38.2%

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

      8. *-commutative 38.2%

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

      9. unpow2 38.2%

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

      10. associate-/l* 39.1%

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

      11. associate-*l/ 39.9%

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

      12. associate-/r/ 39.9%

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

      13. *-commutative 39.9%

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

      14. *-commutative 39.9%

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

      15. associate-/l* 39.9%

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

   9. Simplified 38.8%

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

   10. Taylor expanded in D around inf 25.2%

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

       1. *-commutative 25.2%

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

       2. times-frac 38.2%

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

       3. associate-*r* 38.2%

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

       4. unpow2 38.2%

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

       5. times-frac 38.2%

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

       6. associate-*l/ 38.2%

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

       7. unpow2 38.2%

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

       8. associate-*l/ 38.2%

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

       9. unpow2 38.2%

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

       10. associate-/l* 39.1%

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

       11. associate-*l/ 39.6%

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

       12. associate-/r/ 39.6%

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

       13. *-commutative 39.6%

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

       14. associate-/l* 39.6%

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

       15. associate-/r/ 39.6%

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

   12. Simplified 40.8%

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

   13. Taylor expanded in h around 0 38.8%

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

       1. *-commutative 38.8%

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

       2. unpow2 38.8%

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

       3. associate-*r* 38.3%

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

       4. times-frac 39.1%

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

       5. unpow2 39.1%

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

       6. associate-/l* 39.1%

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

       7. times-frac 40.4%

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

       8. associate-*r* 40.3%

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

       9. *-commutative 40.3%

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

       10. associate-*r* 40.8%

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

       11. *-commutative 40.8%

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

       12. *-commutative 40.8%

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

       13. associate-*r/ 40.8%

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

       14. times-frac 40.8%

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

       15. associate-/l* 40.8%

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

       16. associate-*r/ 40.8%

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

   15. Simplified 40.8%

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

   if -1.3499999999999999e269 < M < 5.9000000000000003e62

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in M around 0 78.7%

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

   if 5.9000000000000003e62 < M

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

      2. times-frac 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in M around 0 42.8%

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

      1. *-commutative 42.8%

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

      2. unpow2 42.8%

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

      3. unpow2 42.8%

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

      4. unpow2 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in D around 0 42.8%

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

      1. unpow2 42.8%

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

      2. *-commutative 42.8%

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

      3. times-frac 42.8%

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

      4. unpow2 42.8%

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

      5. times-frac 45.1%

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

      6. associate-*l/ 56.5%

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

      7. associate-/r/ 56.5%

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

      8. *-commutative 56.5%

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

      9. unpow2 56.5%

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

      10. associate-/l* 56.7%

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

      11. associate-*l/ 56.7%

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

      12. associate-/r/ 56.9%

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

      13. *-commutative 56.9%

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

      14. *-commutative 56.9%

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

      15. associate-/l* 56.7%

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

   9. Simplified 45.2%

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

   10. Taylor expanded in D around inf 30.1%

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

       1. *-commutative 30.1%

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

       2. times-frac 32.0%

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

       3. associate-*r* 30.2%

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

       4. unpow2 30.2%

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

       5. times-frac 30.3%

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

       6. associate-*l/ 32.2%

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

       7. unpow2 32.2%

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

       8. associate-*l/ 32.6%

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

       9. unpow2 32.6%

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

       10. associate-/l* 33.0%

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

       11. associate-*l/ 33.1%

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

       12. associate-/r/ 33.2%

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

       13. *-commutative 33.2%

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

       14. associate-/l* 33.0%

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

       15. associate-/r/ 33.1%

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

   12. Simplified 33.7%

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

       1. associate-*r/ 33.5%

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

       2. div-inv 33.5%

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

       3. clear-num 33.5%

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

   14. Applied egg-rr 33.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.35 \cdot 10^{+269}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \left(w0 \cdot \frac{M}{d}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;M \leq 5.9 \cdot 10^{+62}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \frac{h \cdot \left(\frac{M}{d} \cdot \left(M \cdot w0\right)\right)}{d}\right)\right)\\ \end{array} \]

Alternative 8: 77.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+14}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{M \cdot \frac{D}{\ell}}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(h \cdot \frac{M}{d}\right) \cdot \frac{M \cdot D}{d \cdot \ell}\right)\right)\right)\\ \end{array} \end{array} \]

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 -1e+14)
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ (* M (/ D l)) d) (/ M (/ d h)))))))
   (* w0 (+ 1.0 (* -0.125 (* D (* (* h (/ M d)) (/ (* M 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 <= -1e+14) {
		tmp = w0 * (1.0 + (-0.125 * (D * (((M * (D / l)) / d) * (M / (d / h))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((h * (M / d)) * ((M * 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 <= (-1d+14)) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * (((m * (d / l)) / d_1) * (m / (d_1 / h))))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((h * (m / d_1)) * ((m * 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 <= -1e+14) {
		tmp = w0 * (1.0 + (-0.125 * (D * (((M * (D / l)) / d) * (M / (d / h))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((h * (M / d)) * ((M * D) / (d * l))))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= -1e+14:
		tmp = w0 * (1.0 + (-0.125 * (D * (((M * (D / l)) / d) * (M / (d / h))))))
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * ((h * (M / d)) * ((M * D) / (d * l))))))
	return tmp

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= -1e+14)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(Float64(M * Float64(D / l)) / d) * Float64(M / Float64(d / h)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(h * Float64(M / d)) * Float64(Float64(M * 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 <= -1e+14)
		tmp = w0 * (1.0 + (-0.125 * (D * (((M * (D / l)) / d) * (M / (d / h))))));
	else
		tmp = w0 * (1.0 + (-0.125 * (D * ((h * (M / d)) * ((M * 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, -1e+14], N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(N[(M * N[(D / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1e14

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

      2. times-frac 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in M around 0 37.2%

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

      1. *-commutative 37.2%

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

      2. unpow2 37.2%

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

      3. unpow2 37.2%

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

      4. unpow2 37.2%

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

   6. Simplified 37.2%

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

   7. Taylor expanded in D around 0 37.2%

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

      1. unpow2 37.2%

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

      2. *-commutative 37.2%

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

      3. times-frac 42.9%

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

      4. unpow2 42.9%

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

      5. times-frac 46.8%

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

      6. associate-*l/ 54.5%

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

      7. associate-/r/ 54.5%

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

      8. *-commutative 54.5%

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

      9. unpow2 54.5%

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

      10. associate-/l* 60.6%

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

      11. associate-*l/ 60.8%

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

      12. associate-/r/ 61.1%

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

      13. *-commutative 61.1%

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

      14. *-commutative 61.1%

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

      15. associate-/l* 60.8%

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

   9. Simplified 51.0%

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

       1. div-inv 51.0%

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

       2. associate-/l* 66.4%

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

       3. div-inv 66.4%

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

       4. clear-num 66.4%

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

   11. Applied egg-rr 66.4%

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

       1. pow1 66.4%

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

       2. clear-num 66.4%

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

       3. associate-*l/ 70.7%

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

       4. associate-/l* 70.8%

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

   13. Applied egg-rr 70.8%

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

       1. unpow1 70.8%

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

       2. associate-/r/ 72.7%

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

   15. Simplified 72.7%

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

   if -1e14 < M

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. times-frac 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in M around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. unpow2 55.9%

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

      3. unpow2 55.9%

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

      4. unpow2 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in D around 0 55.9%

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

      1. unpow2 55.9%

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

      2. *-commutative 55.9%

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

      3. times-frac 56.3%

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

      4. unpow2 56.3%

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

      5. times-frac 61.4%

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

      6. associate-*l/ 64.9%

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

      7. associate-/r/ 64.9%

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

      8. *-commutative 64.9%

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

      9. unpow2 64.9%

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

      10. associate-/l* 66.9%

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

      11. associate-*l/ 72.5%

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

      12. associate-/r/ 78.0%

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

      13. *-commutative 78.0%

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

      14. *-commutative 78.0%

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

      15. associate-/l* 72.5%

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

   9. Simplified 65.0%

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

       1. div-inv 64.5%

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

       2. associate-/l* 70.5%

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

       3. div-inv 70.5%

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

       4. clear-num 70.5%

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

   11. Applied egg-rr 70.5%

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

   12. Taylor expanded in M around 0 67.3%

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

       1. *-commutative 67.3%

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

       2. *-commutative 67.3%

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

       3. times-frac 62.9%

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

       4. *-commutative 62.9%

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

       5. associate-/l* 62.5%

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

       6. unpow2 62.5%

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

       7. unpow2 62.5%

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

       8. associate-*r/ 64.5%

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

       9. times-frac 74.4%

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

       10. associate-/r/ 73.9%

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

       11. associate-/r/ 74.4%

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

       12. associate-/l* 73.8%

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

       13. associate-/r/ 74.8%

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

       14. *-commutative 74.8%

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

       15. associate-/r/ 77.8%

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

       16. *-commutative 77.8%

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

       17. associate-*r/ 82.8%

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

       18. *-commutative 82.8%

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

       19. associate-/l/ 82.3%

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

       20. *-commutative 82.3%

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

   14. Simplified 82.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+14}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{M \cdot \frac{D}{\ell}}{d} \cdot \frac{M}{\frac{d}{h}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(h \cdot \frac{M}{d}\right) \cdot \frac{M \cdot D}{d \cdot \ell}\right)\right)\right)\\ \end{array} \]

Alternative 9: 77.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\left(h \cdot \frac{M}{d}\right) \cdot \frac{M \cdot D}{d \cdot \ell}\right)\right)\right) \end{array} \]

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 (+ 1.0 (* -0.125 (* D (* (* h (/ M 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 * (1.0 + (-0.125 * (D * ((h * (M / 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 * (1.0d0 + ((-0.125d0) * (d * ((h * (m / 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 * (1.0 + (-0.125 * (D * ((h * (M / d)) * ((M * D) / (d * l))))));
}

Python

[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + (-0.125 * (D * ((h * (M / d)) * ((M * D) / (d * l))))))

Julia

M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(h * Float64(M / 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 * (1.0 + (-0.125 * (D * ((h * (M / 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[(1.0 + N[(-0.125 * N[(D * N[(N[(h * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

   1. *-commutative 81.0%

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

   2. times-frac 81.1%

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

3. Simplified 81.1%

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

4. Taylor expanded in M around 0 52.1%

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

   1. *-commutative 52.1%

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

   2. unpow2 52.1%

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

   3. unpow2 52.1%

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

   4. unpow2 52.1%

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

6. Simplified 52.1%

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

7. Taylor expanded in D around 0 52.1%

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

   1. unpow2 52.1%

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

   2. *-commutative 52.1%

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

   3. times-frac 53.6%

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

   4. unpow2 53.6%

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

   5. times-frac 58.4%

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

   6. associate-*l/ 62.8%

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

   7. associate-/r/ 62.8%

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

   8. *-commutative 62.8%

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

   9. unpow2 62.8%

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

   10. associate-/l* 65.6%

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

   11. associate-*l/ 70.1%

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

   12. associate-/r/ 74.5%

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

   13. *-commutative 74.5%

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

   14. *-commutative 74.5%

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

   15. associate-/l* 70.1%

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

9. Simplified 62.2%

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

    1. div-inv 61.8%

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

    2. associate-/l* 69.7%

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

    3. div-inv 69.7%

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

    4. clear-num 69.7%

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

11. Applied egg-rr 69.7%

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

12. Taylor expanded in M around 0 62.1%

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

    1. *-commutative 62.1%

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

    2. *-commutative 62.1%

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

    3. times-frac 59.7%

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

    4. *-commutative 59.7%

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

    5. associate-/l* 60.2%

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

    6. unpow2 60.2%

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

    7. unpow2 60.2%

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

    8. associate-*r/ 61.8%

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

    9. times-frac 72.8%

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

    10. associate-/r/ 72.4%

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

    11. associate-/r/ 73.7%

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

    12. associate-/l* 73.2%

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

    13. associate-/r/ 74.4%

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

    14. *-commutative 74.4%

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

    15. associate-/r/ 76.4%

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

    16. *-commutative 76.4%

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

    17. associate-*r/ 79.6%

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

    18. *-commutative 79.6%

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

    19. associate-/l/ 78.1%

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

    20. *-commutative 78.1%

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

14. Simplified 78.1%

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

15. Final simplification 78.1%

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

Alternative 10: 67.4% 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 81.0%

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

   1. *-commutative 81.0%

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

   2. times-frac 81.1%

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

3. Simplified 81.1%

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

4. Taylor expanded in M around 0 68.6%

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

5. Final simplification 68.6%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023182 
(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))))))