Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 86.3%

Time: 21.5s

Alternatives: 12

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (/ (pow (* D (* M (/ 0.5 d))) 2.0) l) h)))))

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((d * (m * (0.5d0 / d_1))) ** 2.0d0) / l) * h)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

3. Simplified 82.2%

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

   1. frac-times 81.5%

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

   2. *-commutative 81.5%

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

   3. clear-num 81.5%

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

   4. un-div-inv 82.0%

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

   5. div-inv 82.0%

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

   6. associate-*l* 83.5%

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

   7. associate-/r* 83.5%

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

   8. metadata-eval 83.5%

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

5. Applied egg-rr 83.5%

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

   1. associate-/r/ 85.8%

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

   2. *-commutative 85.8%

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

   3. associate-*r* 85.4%

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

   4. *-commutative 85.4%

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

7. Simplified 85.4%

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

8. Final simplification 85.4%

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

Alternative 2: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \cdot M \leq 5 \cdot 10^{-139}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right)\\ \mathbf{elif}\;D \cdot M \leq 5 \cdot 10^{+131}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{0.25 \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* D M) 5e-139)
   (* w0 (+ 1.0 (* (* h (/ (pow (* M (/ D d)) 2.0) l)) -0.125)))
   (if (<= (* D M) 5e+131)
     (* w0 (sqrt (- 1.0 (* h (/ (* 0.25 (* (* D M) (* D M))) (* d (* d l)))))))
     (*
      w0
      (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (* h (/ M (/ l M)))))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 5e-139) {
		tmp = w0 * (1.0 + ((h * (pow((M * (D / d)), 2.0) / l)) * -0.125));
	} else if ((D * M) <= 5e+131) {
		tmp = w0 * sqrt((1.0 - (h * ((0.25 * ((D * M) * (D * M))) / (d * (d * l))))));
	} else {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M / (l / M)))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((d * m) <= 5d-139) then
        tmp = w0 * (1.0d0 + ((h * (((m * (d / d_1)) ** 2.0d0) / l)) * (-0.125d0)))
    else if ((d * m) <= 5d+131) then
        tmp = w0 * sqrt((1.0d0 - (h * ((0.25d0 * ((d * m) * (d * m))) / (d_1 * (d_1 * l))))))
    else
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * (h * (m / (l / m)))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 5e-139) {
		tmp = w0 * (1.0 + ((h * (Math.pow((M * (D / d)), 2.0) / l)) * -0.125));
	} else if ((D * M) <= 5e+131) {
		tmp = w0 * Math.sqrt((1.0 - (h * ((0.25 * ((D * M) * (D * M))) / (d * (d * l))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M / (l / M)))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (D * M) <= 5e-139:
		tmp = w0 * (1.0 + ((h * (math.pow((M * (D / d)), 2.0) / l)) * -0.125))
	elif (D * M) <= 5e+131:
		tmp = w0 * math.sqrt((1.0 - (h * ((0.25 * ((D * M) * (D * M))) / (d * (d * l))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M / (l / M)))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(D * M) <= 5e-139)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64((Float64(M * Float64(D / d)) ^ 2.0) / l)) * -0.125)));
	elseif (Float64(D * M) <= 5e+131)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(0.25 * Float64(Float64(D * M) * Float64(D * M))) / Float64(d * Float64(d * l)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M / Float64(l / M))))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D * M) <= 5e-139)
		tmp = w0 * (1.0 + ((h * (((M * (D / d)) ^ 2.0) / l)) * -0.125));
	elseif ((D * M) <= 5e+131)
		tmp = w0 * sqrt((1.0 - (h * ((0.25 * ((D * M) * (D * M))) / (d * (d * l))))));
	else
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M / (l / M)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(D * M), $MachinePrecision], 5e-139], N[(w0 * N[(1.0 + N[(N[(h * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * M), $MachinePrecision], 5e+131], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(0.25 * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 5.00000000000000034e-139

   1. Initial program 83.8%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in D around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. times-frac 66.6%

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

      3. unpow2 66.6%

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

      4. unpow2 66.6%

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

      5. unpow2 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in D around 0 64.4%

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

      1. unpow2 64.4%

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

      2. unpow2 64.4%

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

      3. associate-*r* 66.7%

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

      4. associate-*r* 64.4%

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

      5. associate-*r* 63.9%

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

      6. associate-/l* 62.6%

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

      7. associate-/r/ 63.9%

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

      8. unpow2 63.9%

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

      9. unpow2 63.9%

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

   9. Simplified 83.0%

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

   if 5.00000000000000034e-139 < (*.f64 M D) < 4.99999999999999995e131

   1. Initial program 78.1%

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

   3. Simplified 74.0%

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

      1. frac-times 78.1%

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

      2. *-commutative 78.1%

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

      3. clear-num 78.1%

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

      4. un-div-inv 78.1%

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

      5. div-inv 78.1%

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

      6. associate-*l* 78.1%

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

      7. associate-/r* 78.1%

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

      8. metadata-eval 78.1%

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

   5. Applied egg-rr 78.1%

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

      1. associate-/r/ 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*r* 72.1%

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

      4. *-commutative 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in D around 0 45.2%

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

      1. associate-*r/ 45.2%

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

      2. unpow2 45.2%

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

      3. associate-*l* 47.3%

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

      4. unpow2 47.3%

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

      5. unpow2 47.3%

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

      6. unswap-sqr 74.8%

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

   10. Simplified 74.8%

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

   if 4.99999999999999995e131 < (*.f64 M D)

   1. Initial program 72.5%

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

   3. Simplified 76.0%

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

      1. frac-times 72.5%

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

      2. *-commutative 72.5%

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

      3. clear-num 72.5%

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

      4. un-div-inv 72.5%

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

      5. div-inv 72.5%

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

      6. associate-*l* 75.9%

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

      7. associate-/r* 75.9%

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

      8. metadata-eval 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. associate-/r/ 82.7%

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

      2. *-commutative 82.7%

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

      3. associate-*r* 82.8%

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

      4. *-commutative 82.8%

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

   7. Simplified 82.8%

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

   8. Taylor expanded in D around 0 50.4%

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

      1. unpow2 50.4%

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

      2. unpow2 50.4%

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

      3. times-frac 50.6%

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

      4. unpow2 50.6%

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

      5. associate-*r* 57.6%

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

      6. associate-*r* 50.6%

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

      7. associate-*l/ 50.6%

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

      8. *-commutative 50.6%

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

      9. associate-/l* 54.0%

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

   10. Simplified 54.0%

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

   11. Taylor expanded in D around 0 54.0%

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

       1. unpow2 39.8%

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

       2. unpow2 39.8%

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

       3. times-frac 47.6%

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

   13. Simplified 72.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 5 \cdot 10^{-139}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right)\\ \mathbf{elif}\;D \cdot M \leq 5 \cdot 10^{+131}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{0.25 \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)}\\ \end{array} \]

Alternative 3: 85.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \cdot M \leq 5 \cdot 10^{-139}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right)\\ \mathbf{elif}\;D \cdot M \leq 10^{+131}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{0.25 \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* D M) 5e-139)
   (* w0 (+ 1.0 (* (* h (/ (pow (* M (/ D d)) 2.0) l)) -0.125)))
   (if (<= (* D M) 1e+131)
     (* w0 (sqrt (- 1.0 (* h (/ (* 0.25 (* (* D M) (* D M))) (* d (* d l)))))))
     (*
      w0
      (sqrt (- 1.0 (/ (* 0.25 (* (* M (* M h)) (* (/ D d) (/ D d)))) l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 5e-139) {
		tmp = w0 * (1.0 + ((h * (pow((M * (D / d)), 2.0) / l)) * -0.125));
	} else if ((D * M) <= 1e+131) {
		tmp = w0 * sqrt((1.0 - (h * ((0.25 * ((D * M) * (D * M))) / (d * (d * l))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((0.25 * ((M * (M * h)) * ((D / d) * (D / d)))) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((d * m) <= 5d-139) then
        tmp = w0 * (1.0d0 + ((h * (((m * (d / d_1)) ** 2.0d0) / l)) * (-0.125d0)))
    else if ((d * m) <= 1d+131) then
        tmp = w0 * sqrt((1.0d0 - (h * ((0.25d0 * ((d * m) * (d * m))) / (d_1 * (d_1 * l))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 * ((m * (m * h)) * ((d / d_1) * (d / d_1)))) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 5e-139) {
		tmp = w0 * (1.0 + ((h * (Math.pow((M * (D / d)), 2.0) / l)) * -0.125));
	} else if ((D * M) <= 1e+131) {
		tmp = w0 * Math.sqrt((1.0 - (h * ((0.25 * ((D * M) * (D * M))) / (d * (d * l))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 * ((M * (M * h)) * ((D / d) * (D / d)))) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (D * M) <= 5e-139:
		tmp = w0 * (1.0 + ((h * (math.pow((M * (D / d)), 2.0) / l)) * -0.125))
	elif (D * M) <= 1e+131:
		tmp = w0 * math.sqrt((1.0 - (h * ((0.25 * ((D * M) * (D * M))) / (d * (d * l))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((0.25 * ((M * (M * h)) * ((D / d) * (D / d)))) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(D * M) <= 5e-139)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64((Float64(M * Float64(D / d)) ^ 2.0) / l)) * -0.125)));
	elseif (Float64(D * M) <= 1e+131)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(0.25 * Float64(Float64(D * M) * Float64(D * M))) / Float64(d * Float64(d * l)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(M * Float64(M * h)) * Float64(Float64(D / d) * Float64(D / d)))) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D * M) <= 5e-139)
		tmp = w0 * (1.0 + ((h * (((M * (D / d)) ^ 2.0) / l)) * -0.125));
	elseif ((D * M) <= 1e+131)
		tmp = w0 * sqrt((1.0 - (h * ((0.25 * ((D * M) * (D * M))) / (d * (d * l))))));
	else
		tmp = w0 * sqrt((1.0 - ((0.25 * ((M * (M * h)) * ((D / d) * (D / d)))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(D * M), $MachinePrecision], 5e-139], N[(w0 * N[(1.0 + N[(N[(h * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(D * M), $MachinePrecision], 1e+131], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(0.25 * N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 5.00000000000000034e-139

   1. Initial program 83.8%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in D around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. times-frac 66.6%

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

      3. unpow2 66.6%

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

      4. unpow2 66.6%

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

      5. unpow2 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in D around 0 64.4%

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

      1. unpow2 64.4%

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

      2. unpow2 64.4%

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

      3. associate-*r* 66.7%

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

      4. associate-*r* 64.4%

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

      5. associate-*r* 63.9%

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

      6. associate-/l* 62.6%

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

      7. associate-/r/ 63.9%

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

      8. unpow2 63.9%

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

      9. unpow2 63.9%

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

   9. Simplified 83.0%

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

   if 5.00000000000000034e-139 < (*.f64 M D) < 9.9999999999999991e130

   1. Initial program 78.1%

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

   3. Simplified 74.0%

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

      1. frac-times 78.1%

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

      2. *-commutative 78.1%

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

      3. clear-num 78.1%

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

      4. un-div-inv 78.1%

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

      5. div-inv 78.1%

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

      6. associate-*l* 78.1%

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

      7. associate-/r* 78.1%

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

      8. metadata-eval 78.1%

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

   5. Applied egg-rr 78.1%

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

      1. associate-/r/ 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*r* 72.1%

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

      4. *-commutative 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in D around 0 45.2%

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

      1. associate-*r/ 45.2%

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

      2. unpow2 45.2%

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

      3. associate-*l* 47.3%

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

      4. unpow2 47.3%

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

      5. unpow2 47.3%

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

      6. unswap-sqr 74.8%

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

   10. Simplified 74.8%

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

   if 9.9999999999999991e130 < (*.f64 M D)

   1. Initial program 72.5%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in D around 0 50.4%

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

      1. associate-*r/ 50.4%

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

      2. *-commutative 50.4%

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

      3. times-frac 50.6%

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

      4. *-commutative 50.6%

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

      5. associate-/l* 50.6%

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

      6. unpow2 50.6%

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

      7. associate-*l* 57.6%

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

      8. unpow2 57.6%

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

      9. unpow2 57.6%

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

   6. Simplified 57.6%

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

      1. associate-*l/ 57.6%

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

      2. div-inv 57.6%

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

      3. clear-num 57.6%

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

      4. times-frac 75.5%

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

   8. Applied egg-rr 75.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 5 \cdot 10^{-139}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right)\\ \mathbf{elif}\;D \cdot M \leq 10^{+131}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{0.25 \cdot \left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)}{d \cdot \left(d \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)\right)}{\ell}}\\ \end{array} \]

Alternative 4: 81.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -50:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -50.0)
   (* w0 (sqrt (- 1.0 (* 0.25 (* (* (/ D d) (/ D d)) (* h (/ M (/ l M))))))))
   (* w0 (+ 1.0 (* (* h (/ (pow (* M (/ D d)) 2.0) l)) -0.125)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -50.0) {
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M / (l / M)))))));
	} else {
		tmp = w0 * (1.0 + ((h * (pow((M * (D / d)), 2.0) / l)) * -0.125));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-50.0d0)) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * (((d / d_1) * (d / d_1)) * (h * (m / (l / m)))))))
    else
        tmp = w0 * (1.0d0 + ((h * (((m * (d / d_1)) ** 2.0d0) / l)) * (-0.125d0)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -50.0) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M / (l / M)))))));
	} else {
		tmp = w0 * (1.0 + ((h * (Math.pow((M * (D / d)), 2.0) / l)) * -0.125));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (h / l) <= -50.0:
		tmp = w0 * math.sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M / (l / M)))))))
	else:
		tmp = w0 * (1.0 + ((h * (math.pow((M * (D / d)), 2.0) / l)) * -0.125))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -50.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h * Float64(M / Float64(l / M))))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64((Float64(M * Float64(D / d)) ^ 2.0) / l)) * -0.125)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -50.0)
		tmp = w0 * sqrt((1.0 - (0.25 * (((D / d) * (D / d)) * (h * (M / (l / M)))))));
	else
		tmp = w0 * (1.0 + ((h * (((M * (D / d)) ^ 2.0) / l)) * -0.125));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -50.0], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h * N[(M / N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(h * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -50

   1. Initial program 68.8%

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

   3. Simplified 70.7%

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

      1. frac-times 68.8%

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

      2. *-commutative 68.8%

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

      3. clear-num 68.7%

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

      4. un-div-inv 70.3%

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

      5. div-inv 70.3%

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

      6. associate-*l* 73.3%

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

      7. associate-/r* 73.3%

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

      8. metadata-eval 73.3%

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

   5. Applied egg-rr 73.3%

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

      1. associate-/r/ 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-*r* 76.2%

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

      4. *-commutative 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in D around 0 47.6%

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

      1. unpow2 47.6%

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

      2. unpow2 47.6%

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

      3. times-frac 49.6%

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

      4. unpow2 49.6%

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

      5. associate-*r* 52.1%

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

      6. associate-*r* 49.6%

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

      7. associate-*l/ 48.5%

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

      8. *-commutative 48.5%

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

      9. associate-/l* 50.9%

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

   10. Simplified 50.9%

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

   11. Taylor expanded in D around 0 50.9%

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

       1. unpow2 24.8%

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

       2. unpow2 24.8%

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

       3. times-frac 27.6%

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

   13. Simplified 63.9%

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

   if -50 < (/.f64 h l)

   1. Initial program 88.7%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in D around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. times-frac 66.4%

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

      3. unpow2 66.4%

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

      4. unpow2 66.4%

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

      5. unpow2 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in D around 0 64.9%

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

      1. unpow2 64.9%

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

      2. unpow2 64.9%

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

      3. associate-*r* 68.6%

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

      4. associate-*r* 64.9%

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

      5. associate-*r* 65.0%

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

      6. associate-/l* 64.5%

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

      7. associate-/r/ 65.2%

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

      8. unpow2 65.2%

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

      9. unpow2 65.2%

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

   9. Simplified 88.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -50:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(h \cdot \frac{M}{\frac{\ell}{M}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 5: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \left(1 + \left(h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right) \cdot -0.125\right) \end{array} \]

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * (1.0d0 + ((h * (((m * (d / d_1)) ** 2.0d0) / l)) * (-0.125d0)))
end function

Java

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

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 * (1.0 + ((h * (math.pow((M * (D / d)), 2.0) / l)) * -0.125))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * Float64(1.0 + Float64(Float64(h * Float64((Float64(M * Float64(D / d)) ^ 2.0) / l)) * -0.125)))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * (1.0 + ((h * (((M * (D / d)) ^ 2.0) / l)) * -0.125));
end

Wolfram

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

Derivation

1. Initial program 81.5%

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

3. Simplified 82.2%

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

4. Taylor expanded in D around 0 57.6%

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

   1. *-commutative 57.6%

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

   2. times-frac 59.3%

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

   3. unpow2 59.3%

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

   4. unpow2 59.3%

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

   5. unpow2 59.3%

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

6. Simplified 59.3%

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

7. Taylor expanded in D around 0 57.6%

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

   1. unpow2 57.6%

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

   2. unpow2 57.6%

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

   3. associate-*r* 60.9%

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

   4. associate-*r* 57.6%

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

   5. associate-*r* 57.3%

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

   6. associate-/l* 56.5%

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

   7. associate-/r/ 57.5%

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

   8. unpow2 57.5%

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

   9. unpow2 57.5%

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

9. Simplified 78.3%

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

10. Final simplification 78.3%

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

Alternative 6: 72.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{-75}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M}{\frac{\frac{\ell}{h}}{M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(M \cdot M\right) \cdot \left(w0 \cdot h\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.2e-75)
   w0
   (if (<= M 1.9e+155)
     (* w0 (+ 1.0 (* -0.125 (* (/ (* D D) (* d d)) (/ M (/ (/ l h) M))))))
     (* -0.125 (* (* (/ D d) (/ D d)) (/ (* (* M M) (* w0 h)) l))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.2e-75) {
		tmp = w0;
	} else if (M <= 1.9e+155) {
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / (d * d)) * (M / ((l / h) / M)))));
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * (((M * M) * (w0 * h)) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 3.2d-75) then
        tmp = w0
    else if (m <= 1.9d+155) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d * d) / (d_1 * d_1)) * (m / ((l / h) / m)))))
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * (((m * m) * (w0 * h)) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.2e-75) {
		tmp = w0;
	} else if (M <= 1.9e+155) {
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / (d * d)) * (M / ((l / h) / M)))));
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * (((M * M) * (w0 * h)) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.2e-75:
		tmp = w0
	elif M <= 1.9e+155:
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / (d * d)) * (M / ((l / h) / M)))))
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * (((M * M) * (w0 * h)) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.2e-75)
		tmp = w0;
	elseif (M <= 1.9e+155)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(M / Float64(Float64(l / h) / M))))));
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(Float64(M * M) * Float64(w0 * h)) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.2e-75)
		tmp = w0;
	elseif (M <= 1.9e+155)
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / (d * d)) * (M / ((l / h) / M)))));
	else
		tmp = -0.125 * (((D / d) * (D / d)) * (((M * M) * (w0 * h)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.2e-75], w0, If[LessEqual[M, 1.9e+155], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(M / N[(N[(l / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 3.19999999999999977e-75

   1. Initial program 83.5%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in D around 0 70.9%

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

   if 3.19999999999999977e-75 < M < 1.9e155

   1. Initial program 83.8%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in D around 0 63.0%

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

      1. *-commutative 63.0%

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

      2. times-frac 67.7%

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

      3. unpow2 67.7%

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

      4. unpow2 67.7%

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

      5. unpow2 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in M around 0 67.7%

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

      1. unpow2 67.7%

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

      2. associate-/l* 69.9%

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

      3. associate-/l* 72.3%

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

   9. Simplified 72.3%

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

   if 1.9e155 < M

   1. Initial program 56.9%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in D around 0 16.8%

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

      1. +-commutative 16.8%

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

      2. *-commutative 16.8%

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

      3. fma-def 16.8%

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

      4. times-frac 16.7%

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

      5. unpow2 16.7%

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

      6. unpow2 16.7%

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

      7. associate-/l* 11.5%

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

      8. unpow2 11.5%

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

   6. Simplified 11.5%

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

   7. Taylor expanded in D around inf 11.6%

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

      1. unpow2 11.6%

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

      2. unpow2 11.6%

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

      3. times-frac 11.4%

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

      4. unpow2 11.4%

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

      5. *-commutative 11.4%

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

   9. Simplified 11.4%

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

   10. Taylor expanded in D around 0 11.4%

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

       1. unpow2 11.4%

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

       2. unpow2 11.4%

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

       3. times-frac 17.1%

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

   12. Simplified 17.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.2 \cdot 10^{-75}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 1.9 \cdot 10^{+155}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{M}{\frac{\frac{\ell}{h}}{M}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{\left(M \cdot M\right) \cdot \left(w0 \cdot h\right)}{\ell}\right)\\ \end{array} \]

Alternative 7: 72.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;D \cdot M \leq 8 \cdot 10^{+202}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (* D M) 8e+202)
   w0
   (* -0.125 (* (* (/ D d) (* (/ D d) (* M M))) (* h (/ w0 l))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 8e+202) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * ((D / d) * (M * M))) * (h * (w0 / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((d * m) <= 8d+202) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * ((d / d_1) * (m * m))) * (h * (w0 / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((D * M) <= 8e+202) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * ((D / d) * (M * M))) * (h * (w0 / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (D * M) <= 8e+202:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * ((D / d) * (M * M))) * (h * (w0 / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(D * M) <= 8e+202)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(Float64(D / d) * Float64(M * M))) * Float64(h * Float64(w0 / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if ((D * M) <= 8e+202)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * ((D / d) * (M * M))) * (h * (w0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(D * M), $MachinePrecision], 8e+202], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 7.9999999999999992e202

   1. Initial program 83.2%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in D around 0 70.9%

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

   if 7.9999999999999992e202 < (*.f64 M D)

   1. Initial program 61.6%

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

   3. Simplified 66.4%

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

   4. Taylor expanded in D around 0 50.6%

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

      1. +-commutative 50.6%

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

      2. *-commutative 50.6%

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

      3. fma-def 50.6%

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

      4. times-frac 50.6%

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

      5. unpow2 50.6%

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

      6. unpow2 50.6%

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

      7. associate-/l* 45.6%

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

      8. unpow2 45.6%

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

   6. Simplified 45.6%

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

   7. Taylor expanded in D around inf 40.4%

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

      1. unpow2 40.4%

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

      2. unpow2 40.4%

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

      3. unpow2 40.4%

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

      4. associate-*r* 40.4%

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

      5. times-frac 35.4%

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

      6. unswap-sqr 35.4%

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

      7. times-frac 46.5%

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

      8. *-commutative 46.5%

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

      9. associate-*r/ 46.5%

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

      10. unpow1 46.5%

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

      11. *-commutative 46.5%

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

      12. associate-*r/ 46.7%

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

      13. pow-plus 46.7%

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

      14. metadata-eval 46.7%

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

   9. Simplified 46.7%

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

       1. unpow2 46.7%

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

       2. *-commutative 46.7%

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

       3. *-commutative 46.7%

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

       4. unswap-sqr 45.9%

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

       5. associate-*l* 46.5%

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

   11. Applied egg-rr 46.5%

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

       1. *-un-lft-identity 46.5%

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

       2. associate-/l* 51.4%

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

   13. Applied egg-rr 51.4%

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

       1. *-lft-identity 51.4%

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

       2. associate-/l* 46.5%

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

       3. *-rgt-identity 46.5%

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

       4. associate-*r/ 46.5%

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

       5. associate-*l* 51.4%

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

       6. associate-*r/ 51.4%

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

       7. *-rgt-identity 51.4%

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

   15. Simplified 51.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot M \leq 8 \cdot 10^{+202}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \left(\frac{D}{d} \cdot \left(M \cdot M\right)\right)\right) \cdot \left(h \cdot \frac{w0}{\ell}\right)\right)\\ \end{array} \]

Alternative 8: 72.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{-75}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.8e-75)
   w0
   (* w0 (+ 1.0 (* -0.125 (* (/ (* D D) (* d d)) (/ (* h (* M M)) l)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.8e-75) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((h * (M * M)) / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 3.8d-75) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((d * d) / (d_1 * d_1)) * ((h * (m * m)) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.8e-75) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((h * (M * M)) / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.8e-75:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((h * (M * M)) / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.8e-75)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(h * Float64(M * M)) / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.8e-75)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((D * D) / (d * d)) * ((h * (M * M)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.8e-75], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.79999999999999994e-75

   1. Initial program 83.5%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in D around 0 70.9%

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

   if 3.79999999999999994e-75 < M

   1. Initial program 75.1%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in D around 0 48.1%

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

      1. *-commutative 48.1%

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

      2. times-frac 51.2%

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

      3. unpow2 51.2%

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

      4. unpow2 51.2%

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

      5. unpow2 51.2%

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

   6. Simplified 51.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.8 \cdot 10^{-75}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right)\right)\\ \end{array} \]

Alternative 9: 76.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4.1 \cdot 10^{-137}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)}{\frac{\ell}{h}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4.1e-137)
   w0
   (* w0 (+ 1.0 (* -0.125 (/ (* (* (/ D d) (/ D d)) (* M M)) (/ l h)))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.1e-137) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * M)) / (l / h))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 4.1d-137) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((((d / d_1) * (d / d_1)) * (m * m)) / (l / h))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4.1e-137) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * M)) / (l / h))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4.1e-137:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * M)) / (l / h))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4.1e-137)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(M * M)) / Float64(l / h)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4.1e-137)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((((D / d) * (D / d)) * (M * M)) / (l / h))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4.1e-137], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.0999999999999999e-137

   1. Initial program 83.7%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in D around 0 71.6%

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

   if 4.0999999999999999e-137 < M

   1. Initial program 76.3%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in D around 0 51.9%

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

      1. *-commutative 51.9%

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

      2. times-frac 54.5%

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

      3. unpow2 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

   6. Simplified 54.5%

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

      1. associate-/l* 54.4%

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

      2. associate-*r/ 55.7%

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

      3. times-frac 61.2%

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

   8. Applied egg-rr 61.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.1 \cdot 10^{-137}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot M\right)}{\frac{\ell}{h}}\right)\\ \end{array} \]

Alternative 10: 77.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.7 \cdot 10^{-152}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.7e-152)
   w0
   (* w0 (+ 1.0 (* -0.125 (/ (* (* M (* M h)) (* (/ D d) (/ D d))) l))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.7e-152) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((M * (M * h)) * ((D / d) * (D / d))) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 3.7d-152) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (((m * (m * h)) * ((d / d_1) * (d / d_1))) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.7e-152) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (((M * (M * h)) * ((D / d) * (D / d))) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.7e-152:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (((M * (M * h)) * ((D / d) * (D / d))) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.7e-152)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(Float64(M * Float64(M * h)) * Float64(Float64(D / d) * Float64(D / d))) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.7e-152)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (((M * (M * h)) * ((D / d) * (D / d))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.7e-152], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.6999999999999998e-152

   1. Initial program 83.6%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in D around 0 71.3%

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

   if 3.6999999999999998e-152 < M

   1. Initial program 76.9%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in D around 0 51.8%

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

      1. *-commutative 51.8%

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

      2. times-frac 54.4%

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

      3. unpow2 54.4%

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

      4. unpow2 54.4%

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

      5. unpow2 54.4%

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

   6. Simplified 54.4%

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

      1. associate-*r* 58.1%

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

      2. associate-*r/ 58.3%

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

      3. times-frac 64.8%

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

   8. Applied egg-rr 64.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.7 \cdot 10^{-152}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(\frac{D}{d} \cdot \frac{D}{d}\right)}{\ell}\right)\\ \end{array} \]

Alternative 11: 72.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot \frac{w0 \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.25e+36)
   w0
   (* -0.125 (* (* (/ D d) (* M (* M (/ D d)))) (/ (* w0 h) l)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.25e+36) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (M * (M * (D / d)))) * ((w0 * h) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 1.25d+36) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (m * (m * (d / d_1)))) * ((w0 * h) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.25e+36) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (M * (M * (D / d)))) * ((w0 * h) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.25e+36:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (M * (M * (D / d)))) * ((w0 * h) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.25e+36)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(M * Float64(M * Float64(D / d)))) * Float64(Float64(w0 * h) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.25e+36)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (M * (M * (D / d)))) * ((w0 * h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.25e+36], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(M * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.24999999999999994e36

   1. Initial program 83.0%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in D around 0 70.4%

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

   if 1.24999999999999994e36 < M

   1. Initial program 73.6%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in D around 0 38.6%

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

      1. +-commutative 38.6%

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

      2. *-commutative 38.6%

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

      3. fma-def 38.6%

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

      4. times-frac 41.0%

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

      5. unpow2 41.0%

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

      6. unpow2 41.0%

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

      7. associate-/l* 40.9%

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

      8. unpow2 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in D around inf 19.1%

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

      1. unpow2 19.1%

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

      2. unpow2 19.1%

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

      3. unpow2 19.1%

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

      4. associate-*r* 19.2%

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

      5. times-frac 19.1%

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

      6. unswap-sqr 19.7%

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

      7. times-frac 23.2%

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

      8. *-commutative 23.2%

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

      9. associate-*r/ 23.2%

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

      10. unpow1 23.2%

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

      11. *-commutative 23.2%

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

      12. associate-*r/ 23.2%

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

      13. pow-plus 23.2%

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

      14. metadata-eval 23.2%

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

   9. Simplified 23.2%

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

       1. unpow2 23.2%

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

       2. *-commutative 23.2%

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

       3. *-commutative 23.2%

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

       4. unswap-sqr 22.2%

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

       5. associate-*l* 22.6%

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

   11. Applied egg-rr 22.6%

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

   12. Taylor expanded in D around 0 22.6%

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

       1. unpow2 22.6%

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

       2. associate-*l/ 22.6%

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

       3. associate-*r* 23.2%

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

       4. *-commutative 23.2%

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

       5. *-commutative 23.2%

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

   14. Simplified 23.2%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{+36}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \left(M \cdot \left(M \cdot \frac{D}{d}\right)\right)\right) \cdot \frac{w0 \cdot h}{\ell}\right)\\ \end{array} \]

Alternative 12: 68.2% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

3. Simplified 82.2%

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

4. Taylor expanded in D around 0 66.0%

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

5. Final simplification 66.0%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023285 
(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))))))