Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.6% → 86.9%

Time: 12.7s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.6% 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: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 4 \cdot 10^{+124}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\ell}\right)}\\ \end{array} \end{array} \]

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 (<= (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) 4e+124)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* D (/ (* D (/ (* M (/ h d)) (/ d 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 ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 4e+124) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / 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 ((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))) <= 4d+124) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (d * ((d * ((m * (h / d_1)) / (d_1 / 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 ((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 4e+124) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / 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 (1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 4e+124:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / 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 (Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 4e+124)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(D * Float64(Float64(D * Float64(Float64(M * Float64(h / d)) / Float64(d / 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 ((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))) <= 4e+124)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / 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[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], 4e+124], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(D * N[(N[(D * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 3.99999999999999979e124

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. times-frac 97.1%

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

   3. Simplified 97.1%

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

   if 3.99999999999999979e124 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 42.5%

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

      1. times-frac 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in w0 around 0 43.0%

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

   6. Simplified 53.2%

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

      1. associate-*l/ 56.7%

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

      2. associate-/l* 60.1%

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

   8. Applied egg-rr 60.1%

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

      1. associate-*l/ 61.3%

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

   10. Applied egg-rr 61.3%

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

       1. associate-/r/ 61.3%

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

   12. Applied egg-rr 61.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 4 \cdot 10^{+124}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\ell}\right)}\\ \end{array} \]

Alternative 2: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\ell}\right)}\\ \end{array} \end{array} \]

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 (<= (/ h l) -1e-119)
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* D (/ (* M 0.5) d)) 2.0) l)))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* D (/ (* D (/ (* M (/ h d)) (/ d 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 ((h / l) <= -1e-119) {
		tmp = w0 * sqrt((1.0 - (h * (pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / 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 ((h / l) <= (-1d-119)) then
        tmp = w0 * sqrt((1.0d0 - (h * (((d * ((m * 0.5d0) / d_1)) ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * (d * ((d * ((m * (h / d_1)) / (d_1 / 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 ((h / l) <= -1e-119) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((D * ((M * 0.5) / d)), 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / 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 (h / l) <= -1e-119:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((D * ((M * 0.5) / d)), 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / 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 (Float64(h / l) <= -1e-119)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(D * Float64(Float64(D * Float64(Float64(M * Float64(h / d)) / Float64(d / 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 ((h / l) <= -1e-119)
		tmp = w0 * sqrt((1.0 - (h * (((D * ((M * 0.5) / d)) ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / 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[N[(h / l), $MachinePrecision], -1e-119], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(D * N[(N[(D * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.00000000000000001e-119

   1. Initial program 72.2%

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

      1. times-frac 73.1%

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

   3. Simplified 73.1%

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

      1. clear-num 73.1%

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

      2. un-div-inv 75.6%

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

      3. div-inv 75.6%

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

      4. metadata-eval 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. associate-/r/ 78.0%

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

      2. associate-*r/ 77.1%

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

      3. associate-/l* 78.0%

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

      4. associate-/l* 78.0%

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

      5. associate-/r* 78.0%

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

      6. associate-/r/ 77.0%

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

      7. *-commutative 77.0%

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

      8. associate-/r/ 78.0%

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

      9. *-commutative 78.0%

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

      10. associate-/l* 77.1%

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

      11. associate-*r* 77.1%

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

      12. associate-*r/ 78.0%

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

      13. associate-*r/ 77.1%

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

      14. associate-*l/ 77.0%

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

      15. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   if -1.00000000000000001e-119 < (/.f64 h l)

   1. Initial program 84.5%

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

      1. times-frac 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in w0 around 0 44.3%

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

   6. Simplified 62.1%

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

      1. associate-*l/ 71.2%

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

      2. associate-/l* 79.8%

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

   8. Applied egg-rr 79.8%

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

      1. associate-*l/ 85.0%

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

   10. Applied egg-rr 85.0%

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

       1. associate-/r/ 85.2%

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

   12. Applied egg-rr 85.2%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(D \cdot \frac{D \cdot \frac{M \cdot \frac{h}{d}}{\frac{d}{M}}}{\ell}\right)}\\ \end{array} \]

Alternative 3: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{-44}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.1e-44)
   w0
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (/ D (/ l D)) (* (/ h d) (/ (* M M) d)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e-44) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (l / D)) * ((h / d) * ((M * M) / d))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.1d-44) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / (l / d)) * ((h / d_1) * ((m * m) / d_1))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e-44) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / (l / D)) * ((h / d) * ((M * M) / d))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.1e-44:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / (l / D)) * ((h / d) * ((M * M) / d))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.1e-44)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(h / d) * Float64(Float64(M * M) / d)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.1e-44)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (l / D)) * ((h / d) * ((M * M) / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.1e-44], w0, N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.10000000000000001e-44

   1. Initial program 81.1%

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

      1. times-frac 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in M around 0 73.9%

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

   if 2.10000000000000001e-44 < M

   1. Initial program 75.0%

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

      1. times-frac 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in w0 around 0 40.1%

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

   6. Simplified 49.2%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{-44}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(\frac{h}{d} \cdot \frac{M \cdot M}{d}\right)\right)}\\ \end{array} \]

Alternative 4: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 6.1 \cdot 10^{-149}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+27}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{h}} \cdot \frac{D}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 6.1e-149)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (/ D (/ l D)) (/ (* h (/ M (/ d M))) d))))))
   (if (<= d 2e+27)
     (* w0 (+ 1.0 (* -0.125 (* (/ D (/ l h)) (/ D (pow (/ d M) 2.0))))))
     w0)))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6.1e-149) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (l / D)) * ((h * (M / (d / M))) / d)))));
	} else if (d <= 2e+27) {
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / h)) * (D / pow((d / M), 2.0)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 6.1d-149) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d / (l / d)) * ((h * (m / (d_1 / m))) / d_1)))))
    else if (d_1 <= 2d+27) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / (l / h)) * (d / ((d_1 / m) ** 2.0d0)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 6.1e-149) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D / (l / D)) * ((h * (M / (d / M))) / d)))));
	} else if (d <= 2e+27) {
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / h)) * (D / Math.pow((d / M), 2.0)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 6.1e-149:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D / (l / D)) * ((h * (M / (d / M))) / d)))))
	elif d <= 2e+27:
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / h)) * (D / math.pow((d / M), 2.0)))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 6.1e-149)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(h * Float64(M / Float64(d / M))) / d))))));
	elseif (d <= 2e+27)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / Float64(l / h)) * Float64(D / (Float64(d / M) ^ 2.0))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 6.1e-149)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D / (l / D)) * ((h * (M / (d / M))) / d)))));
	elseif (d <= 2e+27)
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / h)) * (D / ((d / M) ^ 2.0)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 6.1e-149], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e+27], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / N[(l / h), $MachinePrecision]), $MachinePrecision] * N[(D / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if d < 6.09999999999999959e-149

   1. Initial program 79.2%

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

      1. times-frac 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in w0 around 0 41.9%

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

   6. Simplified 59.3%

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

      1. associate-*r/ 58.3%

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

      2. associate-/l* 66.2%

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

   8. Applied egg-rr 66.2%

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

   if 6.09999999999999959e-149 < d < 2e27

   1. Initial program 88.7%

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

      1. times-frac 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in M around 0 58.5%

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

      1. *-commutative 58.5%

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

      2. associate-/l* 58.3%

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

      3. unpow2 58.3%

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

      4. unpow2 58.3%

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

      5. *-commutative 58.3%

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

      6. unpow2 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in l around 0 58.3%

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

      1. *-commutative 58.3%

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

      2. times-frac 64.1%

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

      3. unpow2 64.1%

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

      4. unpow2 64.1%

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

      5. times-frac 69.4%

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

   9. Simplified 69.4%

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

       1. times-frac 73.7%

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

       2. pow2 73.7%

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

   11. Applied egg-rr 73.7%

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

   if 2e27 < d

   1. Initial program 76.4%

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

      1. times-frac 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in M around 0 77.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.1 \cdot 10^{-149}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+27}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{h}} \cdot \frac{D}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 82.9% accurate, 1.8× speedup

Math

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

FPCore

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

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 + ((-0.25d0) * (d * ((d * (m * ((h / d_1) / (d_1 / m)))) / l)))))
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 + (-0.25 * (D * ((D * (M * ((h / d) / (d / M)))) / l)))));
}

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

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(-0.25 * Float64(D * Float64(Float64(D * Float64(M * Float64(Float64(h / d) / Float64(d / M)))) / l))))))
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 + (-0.25 * (D * ((D * (M * ((h / d) / (d / M)))) / l)))));
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[(-0.25 * N[(D * N[(N[(D * N[(M * N[(N[(h / d), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. times-frac 79.9%

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

3. Simplified 79.9%

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

4. Taylor expanded in w0 around 0 46.0%

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

6. Simplified 59.4%

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

   1. associate-*l/ 66.7%

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

   2. associate-/l* 73.6%

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

8. Applied egg-rr 73.6%

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

   1. associate-/r/ 74.2%

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

   2. frac-times 72.2%

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

10. Applied egg-rr 72.2%

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

11. Taylor expanded in M around 0 58.3%

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

    1. unpow2 58.3%

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

    2. times-frac 67.4%

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

    3. unpow2 67.4%

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

    4. associate-*l/ 74.2%

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

    5. associate-/r/ 74.2%

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

    6. associate-*l/ 78.5%

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

    7. associate-/r/ 78.1%

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

    8. *-commutative 78.1%

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

    9. *-commutative 78.1%

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

    10. associate-/l* 78.1%

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

13. Simplified 78.1%

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

14. Final simplification 78.1%

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

Alternative 6: 83.5% accurate, 1.8× speedup

Math

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

FPCore

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

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 + ((-0.25d0) * (d * ((d * ((m * (h / d_1)) / (d_1 / m))) / l)))))
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 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / M))) / l)))));
}

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

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(-0.25 * Float64(D * Float64(Float64(D * Float64(Float64(M * Float64(h / d)) / Float64(d / M))) / l))))))
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 + (-0.25 * (D * ((D * ((M * (h / d)) / (d / M))) / l)))));
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[(-0.25 * N[(D * N[(N[(D * N[(N[(M * N[(h / d), $MachinePrecision]), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. times-frac 79.9%

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

3. Simplified 79.9%

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

4. Taylor expanded in w0 around 0 46.0%

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

6. Simplified 59.4%

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

   1. associate-*l/ 66.7%

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

   2. associate-/l* 73.6%

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

8. Applied egg-rr 73.6%

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

   1. associate-*l/ 77.9%

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

10. Applied egg-rr 77.9%

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

    1. associate-/r/ 78.5%

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

12. Applied egg-rr 78.5%

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

13. Final simplification 78.5%

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

Alternative 7: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.25e-87)
   w0
   (* w0 (+ 1.0 (* -0.125 (* h (* (* M M) (/ (pow (/ D d) 2.0) 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-87) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * ((M * M) * (pow((D / d), 2.0) / 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-87) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (h * ((m * m) * (((d / d_1) ** 2.0d0) / 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-87) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (h * ((M * M) * (Math.pow((D / d), 2.0) / 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-87:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (h * ((M * M) * (math.pow((D / d), 2.0) / 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-87)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(h * Float64(Float64(M * M) * Float64((Float64(D / d) ^ 2.0) / 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-87)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (h * ((M * M) * (((D / d) ^ 2.0) / 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-87], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(h * N[(N[(M * M), $MachinePrecision] * N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.25000000000000011e-87

   1. Initial program 81.2%

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

      1. times-frac 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in M around 0 73.1%

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

   if 1.25000000000000011e-87 < M

   1. Initial program 75.4%

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

      1. times-frac 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in M around 0 42.2%

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

      1. *-commutative 42.2%

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

      2. associate-/l* 46.2%

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

      3. unpow2 46.2%

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

      4. unpow2 46.2%

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

      5. *-commutative 46.2%

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

      6. unpow2 46.2%

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

   6. Simplified 46.2%

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

   7. Taylor expanded in D around 0 42.2%

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

      1. unpow2 42.2%

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

      2. associate-*r* 47.6%

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

      3. *-commutative 47.6%

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

      4. unpow2 47.6%

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

      5. associate-*l/ 52.8%

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

      6. unpow2 52.8%

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

      7. associate-*r* 46.2%

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

      8. unpow2 46.2%

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

      9. associate-*r* 49.0%

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

      10. unpow2 49.0%

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

      11. unpow2 49.0%

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

      12. *-commutative 49.0%

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

      13. associate-/r* 49.1%

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

      14. unpow2 49.1%

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

      15. unpow2 49.1%

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

      16. times-frac 56.0%

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

      17. unpow2 56.0%

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

      18. unpow2 56.0%

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

   9. Simplified 56.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(h \cdot \left(\left(M \cdot M\right) \cdot \frac{{\left(\frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array} \]

Alternative 8: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.05 \cdot 10^{+26}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{h}} \cdot \frac{D}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 2.05e+26)
   (* w0 (+ 1.0 (* -0.125 (* (/ D (/ l h)) (/ D (pow (/ d M) 2.0))))))
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2.05e+26) {
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / h)) * (D / pow((d / M), 2.0)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 2.05d+26) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d / (l / h)) * (d / ((d_1 / m) ** 2.0d0)))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2.05e+26) {
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / h)) * (D / Math.pow((d / M), 2.0)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2.05e+26:
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / h)) * (D / math.pow((d / M), 2.0)))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2.05e+26)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D / Float64(l / h)) * Float64(D / (Float64(d / M) ^ 2.0))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2.05e+26)
		tmp = w0 * (1.0 + (-0.125 * ((D / (l / h)) * (D / ((d / M) ^ 2.0)))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 2.05e+26], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D / N[(l / h), $MachinePrecision]), $MachinePrecision] * N[(D / N[Power[N[(d / M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 2.04999999999999992e26

   1. Initial program 80.5%

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

      1. times-frac 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in M around 0 43.7%

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

      1. *-commutative 43.7%

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

      2. associate-/l* 45.2%

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

      3. unpow2 45.2%

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

      4. unpow2 45.2%

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

      5. *-commutative 45.2%

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

      6. unpow2 45.2%

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

   6. Simplified 45.2%

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

   7. Taylor expanded in l around 0 45.2%

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

      1. *-commutative 45.2%

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

      2. times-frac 45.6%

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

      3. unpow2 45.6%

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

      4. unpow2 45.6%

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

      5. times-frac 63.0%

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

   9. Simplified 63.0%

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

       1. times-frac 67.6%

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

       2. pow2 67.6%

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

   11. Applied egg-rr 67.6%

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

   if 2.04999999999999992e26 < d

   1. Initial program 76.4%

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

      1. times-frac 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in M around 0 77.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.05 \cdot 10^{+26}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{D}{\frac{\ell}{h}} \cdot \frac{D}{{\left(\frac{d}{M}\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 9: 67.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 320000:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{\frac{\ell}{h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 320000.0)
   (* w0 (+ 1.0 (* -0.125 (/ (* D D) (* (/ l h) (* (/ d M) (/ d M)))))))
   w0))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 320000.0) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d / M) * (d / M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 320000.0d0) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * d) / ((l / h) * ((d_1 / m) * (d_1 / m))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 320000.0) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d / M) * (d / M))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 320000.0:
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d / M) * (d / M))))))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 320000.0)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * D) / Float64(Float64(l / h) * Float64(Float64(d / M) * Float64(d / M)))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 320000.0)
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / ((l / h) * ((d / M) * (d / M))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 320000.0], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * D), $MachinePrecision] / N[(N[(l / h), $MachinePrecision] * N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 3.2e5

   1. Initial program 80.2%

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

      1. times-frac 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in M around 0 43.3%

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

      1. *-commutative 43.3%

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

      2. associate-/l* 44.8%

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

      3. unpow2 44.8%

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

      4. unpow2 44.8%

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

      5. *-commutative 44.8%

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

      6. unpow2 44.8%

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

   6. Simplified 44.8%

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

   7. Taylor expanded in l around 0 44.8%

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

      1. *-commutative 44.8%

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

      2. times-frac 45.2%

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

      3. unpow2 45.2%

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

      4. unpow2 45.2%

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

      5. times-frac 62.4%

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

   9. Simplified 62.4%

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

   if 3.2e5 < d

   1. Initial program 77.5%

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

      1. times-frac 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in M around 0 76.6%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 320000:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{\frac{\ell}{h} \cdot \left(\frac{d}{M} \cdot \frac{d}{M}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 10: 71.4% accurate, 10.3× speedup

Math

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

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 (<= M 1.25)
   w0
   (* -0.125 (* (* (/ D d) (/ D d)) (/ w0 (/ l (* h (* M M))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if M < 1.25

   1. Initial program 81.0%

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

      1. times-frac 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in M around 0 72.9%

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

   if 1.25 < M

   1. Initial program 74.9%

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

      1. times-frac 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in M around 0 38.0%

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

      1. *-commutative 38.0%

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

      2. associate-/l* 42.8%

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

      3. unpow2 42.8%

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

      4. unpow2 42.8%

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

      5. *-commutative 42.8%

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

      6. unpow2 42.8%

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

   6. Simplified 42.8%

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

   7. Taylor expanded in l around 0 42.8%

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

      1. *-commutative 42.8%

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

      2. times-frac 38.2%

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

      3. unpow2 38.2%

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

      4. unpow2 38.2%

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

      5. times-frac 54.4%

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

   9. Simplified 54.4%

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

   10. Taylor expanded in D around inf 24.2%

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

       1. associate-*r/ 24.2%

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

       2. *-commutative 24.2%

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

       3. associate-*r/ 24.2%

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

       4. times-frac 24.1%

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

       5. associate-*r* 22.4%

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

       6. *-commutative 22.4%

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

       7. unpow2 22.4%

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

       8. associate-*r* 22.8%

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

       9. *-commutative 22.8%

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

       10. associate-*r/ 23.2%

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

       11. *-commutative 23.2%

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

       12. unpow2 23.2%

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

       13. unpow2 23.2%

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

       14. times-frac 26.9%

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

       15. unpow2 26.9%

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

       16. associate-*l* 27.0%

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

   12. Simplified 24.1%

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

       1. frac-times 27.8%

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

   14. Applied egg-rr 27.8%

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

4. Final simplification 62.0%

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

Alternative 11: 67.6% 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 79.5%

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

   1. times-frac 79.9%

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

3. Simplified 79.9%

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

4. Taylor expanded in M around 0 66.1%

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

5. Final simplification 66.1%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023230 
(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))))))