Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.4% → 86.8%

Time: 18.4s

Alternatives: 12

Speedup: 1.0×

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: 80.4% 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: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D_m}{2 \cdot d_m}\\ \mathbf{if}\;t_0 \leq 0.0001:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M_m \cdot \left(D_m \cdot \frac{0.5}{d_m}\right)\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+153}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{d_m}{M_m} \cdot \frac{2}{D_m}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{\ell} \cdot \left(h \cdot {\left(M_m \cdot D_m\right)}^{2}\right)}{{d_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) (* 2.0 d_m))))
   (if (<= t_0 0.0001)
     (* w0 (sqrt (- 1.0 (* h (/ (pow (* M_m (* D_m (/ 0.5 d_m))) 2.0) l)))))
     (if (<= t_0 4e+153)
       (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ d_m M_m) (/ 2.0 D_m)) -2.0)))))
       (*
        w0
        (sqrt
         (-
          1.0
          (/ (* (/ 0.25 l) (* h (pow (* M_m D_m) 2.0))) (pow d_m 2.0)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = w0 * sqrt((1.0 - (h * (pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))));
	} else if (t_0 <= 4e+153) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((d_m / M_m) * (2.0 / D_m)), -2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (((0.25 / l) * (h * pow((M_m * D_m), 2.0))) / pow(d_m, 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * d_m) / (2.0d0 * d_m_1)
    if (t_0 <= 0.0001d0) then
        tmp = w0 * sqrt((1.0d0 - (h * (((m_m * (d_m * (0.5d0 / d_m_1))) ** 2.0d0) / l))))
    else if (t_0 <= 4d+153) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d_m_1 / m_m) * (2.0d0 / d_m)) ** (-2.0d0)))))
    else
        tmp = w0 * sqrt((1.0d0 - (((0.25d0 / l) * (h * ((m_m * d_m) ** 2.0d0))) / (d_m_1 ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / (2.0 * d_m);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))));
	} else if (t_0 <= 4e+153) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((d_m / M_m) * (2.0 / D_m)), -2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((0.25 / l) * (h * Math.pow((M_m * D_m), 2.0))) / Math.pow(d_m, 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	t_0 = (M_m * D_m) / (2.0 * d_m)
	tmp = 0
	if t_0 <= 0.0001:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))))
	elif t_0 <= 4e+153:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((d_m / M_m) * (2.0 / D_m)), -2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (((0.25 / l) * (h * math.pow((M_m * D_m), 2.0))) / math.pow(d_m, 2.0))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / Float64(2.0 * d_m))
	tmp = 0.0
	if (t_0 <= 0.0001)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(D_m * Float64(0.5 / d_m))) ^ 2.0) / l)))));
	elseif (t_0 <= 4e+153)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(d_m / M_m) * Float64(2.0 / D_m)) ^ -2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(0.25 / l) * Float64(h * (Float64(M_m * D_m) ^ 2.0))) / (d_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	t_0 = (M_m * D_m) / (2.0 * d_m);
	tmp = 0.0;
	if (t_0 <= 0.0001)
		tmp = w0 * sqrt((1.0 - (h * (((M_m * (D_m * (0.5 / d_m))) ^ 2.0) / l))));
	elseif (t_0 <= 4e+153)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((d_m / M_m) * (2.0 / D_m)) ^ -2.0))));
	else
		tmp = w0 * sqrt((1.0 - (((0.25 / l) * (h * ((M_m * D_m) ^ 2.0))) / (d_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0001], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+153], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(d$95$m / M$95$m), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(0.25 / l), $MachinePrecision] * N[(h * N[Power[N[(M$95$m * D$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[d$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 1.00000000000000005e-4

   1. Initial program 82.9%

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

   3. Simplified 82.0%

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

      1. associate-*r/ 88.9%

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

      2. clear-num 88.9%

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

      3. associate-*l/ 89.9%

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

      4. div-inv 89.9%

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

      5. associate-*l* 89.9%

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

      6. associate-/r* 89.9%

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

      7. metadata-eval 89.9%

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

   5. Applied egg-rr 89.9%

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

      1. clear-num 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-*l* 88.9%

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

      4. associate-*r/ 82.0%

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

      5. expm1-log1p-u 81.7%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      6. expm1-udef 81.7%

         \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]

   7. Applied egg-rr 82.6%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 82.6%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      2. expm1-log1p 82.9%

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

      3. associate-*r/ 89.9%

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

      4. associate-*l/ 90.4%

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

      5. *-commutative 90.4%

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

      6. associate-*l* 90.3%

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

   9. Simplified 90.3%

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

   if 1.00000000000000005e-4 < (/.f64 (*.f64 M D) (*.f64 2 d)) < 4e153

   1. Initial program 85.3%

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

   3. Simplified 74.7%

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

      1. associate-*l/ 85.3%

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

      2. clear-num 85.4%

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

      3. *-un-lft-identity 85.4%

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

      4. times-frac 85.4%

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

      5. metadata-eval 85.4%

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

   5. Applied egg-rr 85.4%

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

      1. associate-*r/ 78.4%

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

      2. inv-pow 78.4%

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

      3. pow-pow 78.5%

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

      4. associate-*r/ 78.5%

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

      5. metadata-eval 78.5%

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

   7. Applied egg-rr 78.5%

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

      1. associate-*r/ 85.4%

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

      2. *-commutative 85.4%

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

      3. *-commutative 85.4%

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

      4. times-frac 80.2%

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

   9. Simplified 80.2%

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

   if 4e153 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 36.8%

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

   3. Simplified 36.8%

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

   4. Taylor expanded in M around 0 40.4%

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

      1. associate-*r/ 40.4%

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

      2. *-commutative 40.4%

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

      3. times-frac 40.8%

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

      4. *-commutative 40.8%

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

      5. *-commutative 40.8%

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

      6. associate-*l* 40.8%

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

      7. unpow2 40.8%

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

      8. unpow2 40.8%

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

      9. swap-sqr 49.9%

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

      10. unpow2 49.9%

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

      11. *-commutative 49.9%

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

   6. Simplified 49.9%

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

      1. associate-*r/ 54.2%

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

      2. *-commutative 54.2%

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

   8. Applied egg-rr 54.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 0.0001:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 4 \cdot 10^{+153}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{0.25}{\ell} \cdot \left(h \cdot {\left(M \cdot D\right)}^{2}\right)}{{d}^{2}}}\\ \end{array} \]

Alternative 2: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2}} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D_m \cdot \left(M_m \cdot \frac{0.5}{d_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M_m \cdot \left(D_m \cdot \frac{0.5}{d_m}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<=
      (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)))))
      5e+295)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D_m (* M_m (/ 0.5 d_m))) 2.0)))))
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* M_m (* D_m (/ 0.5 d_m))) 2.0) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((w0 * sqrt((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0))))) <= 5e+295) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D_m * (M_m * (0.5 / d_m))), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((w0 * sqrt((1.0d0 - ((h / l) * (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0))))) <= 5d+295) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d_m * (m_m * (0.5d0 / d_m_1))) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m_m * (d_m * (0.5d0 / d_m_1))) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0))))) <= 5e+295) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * (M_m * (0.5 / d_m))), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0))))) <= 5e+295:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D_m * (M_m * (0.5 / d_m))), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0))))) <= 5e+295)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(M_m * Float64(0.5 / d_m))) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(D_m * Float64(0.5 / d_m))) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((w0 * sqrt((1.0 - ((h / l) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0))))) <= 5e+295)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D_m * (M_m * (0.5 / d_m))) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - (h * (((M_m * (D_m * (0.5 / d_m))) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+295], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 4.99999999999999991e295

   1. Initial program 92.8%

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

      1. associate-*l/ 89.9%

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

      2. div-inv 89.9%

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

      3. associate-/r* 89.9%

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

      4. metadata-eval 89.9%

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

   3. Applied egg-rr 89.9%

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

   if 4.99999999999999991e295 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))))

   1. Initial program 28.5%

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

   3. Simplified 30.3%

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

      1. associate-*r/ 56.8%

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

      2. clear-num 56.8%

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

      3. associate-*l/ 55.2%

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

      4. div-inv 55.2%

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

      5. associate-*l* 55.1%

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

      6. associate-/r* 55.1%

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

      7. metadata-eval 55.1%

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

   5. Applied egg-rr 55.1%

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

      1. clear-num 55.1%

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

      2. *-commutative 55.1%

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

      3. associate-*l* 56.8%

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

      4. associate-*r/ 30.3%

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

      5. expm1-log1p-u 30.3%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      6. expm1-udef 30.3%

         \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]

   7. Applied egg-rr 28.5%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 28.5%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      2. expm1-log1p 28.5%

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

      3. associate-*r/ 55.2%

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

      4. associate-*l/ 55.2%

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

      5. *-commutative 55.2%

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

      6. associate-*l* 55.1%

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

   9. Simplified 55.1%

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 3: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M_m \cdot D_m}{2 \cdot d_m}\right)}^{2} \leq 10^{+258}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{1}{2 \cdot \frac{d_m}{M_m \cdot D_m}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M_m \cdot \left(D_m \cdot \frac{0.5}{d_m}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (- 1.0 (* (/ h l) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))) 1e+258)
   (*
    w0
    (sqrt (- 1.0 (* (/ h l) (pow (/ 1.0 (* 2.0 (/ d_m (* M_m D_m)))) 2.0)))))
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* M_m (* D_m (/ 0.5 d_m))) 2.0) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((1.0 - ((h / l) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0))) <= 1e+258) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((1.0 / (2.0 * (d_m / (M_m * D_m)))), 2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if ((1.0d0 - ((h / l) * (((m_m * d_m) / (2.0d0 * d_m_1)) ** 2.0d0))) <= 1d+258) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((1.0d0 / (2.0d0 * (d_m_1 / (m_m * d_m)))) ** 2.0d0))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m_m * (d_m * (0.5d0 / d_m_1))) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((1.0 - ((h / l) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0))) <= 1e+258) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((1.0 / (2.0 * (d_m / (M_m * D_m)))), 2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (1.0 - ((h / l) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0))) <= 1e+258:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((1.0 / (2.0 * (d_m / (M_m * D_m)))), 2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0))) <= 1e+258)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(1.0 / Float64(2.0 * Float64(d_m / Float64(M_m * D_m)))) ^ 2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(D_m * Float64(0.5 / d_m))) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((1.0 - ((h / l) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0))) <= 1e+258)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((1.0 / (2.0 * (d_m / (M_m * D_m)))) ^ 2.0))));
	else
		tmp = w0 * sqrt((1.0 - (h * (((M_m * (D_m * (0.5 / d_m))) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+258], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(1.0 / N[(2.0 * N[(d$95$m / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $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))) < 1.00000000000000006e258

   1. Initial program 99.9%

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

   3. Simplified 96.5%

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

      1. associate-*l/ 99.9%

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

      2. clear-num 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. times-frac 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 1.00000000000000006e258 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 37.7%

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

   3. Simplified 38.8%

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

      1. associate-*r/ 56.9%

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

      2. clear-num 57.0%

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

      3. associate-*l/ 57.0%

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

      4. div-inv 57.0%

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

      5. associate-*l* 56.9%

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

      6. associate-/r* 56.9%

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

      7. metadata-eval 56.9%

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

   5. Applied egg-rr 56.9%

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

      1. clear-num 56.9%

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

      2. *-commutative 56.9%

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

      3. associate-*l* 56.9%

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

      4. associate-*r/ 38.8%

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

      5. expm1-log1p-u 38.5%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      6. expm1-udef 38.5%

         \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]

   7. Applied egg-rr 37.4%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 37.4%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      2. expm1-log1p 37.7%

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

      3. associate-*r/ 57.0%

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

      4. associate-*l/ 57.0%

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

      5. *-commutative 57.0%

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

      6. associate-*l* 57.0%

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

   9. Simplified 57.0%

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+258}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{1}{2 \cdot \frac{d}{M \cdot D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d_m \leq 1.45 \cdot 10^{-124}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{d_m}{M_m} \cdot \frac{2}{D_m}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M_m \cdot \left(D_m \cdot \frac{0.5}{d_m}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= d_m 1.45e-124)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ d_m M_m) (/ 2.0 D_m)) -2.0)))))
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* M_m (* D_m (/ 0.5 d_m))) 2.0) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1.45e-124) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((d_m / M_m) * (2.0 / D_m)), -2.0))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (d_m_1 <= 1.45d-124) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((d_m_1 / m_m) * (2.0d0 / d_m)) ** (-2.0d0)))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * (((m_m * (d_m * (0.5d0 / d_m_1))) ** 2.0d0) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 1.45e-124) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((d_m / M_m) * (2.0 / D_m)), -2.0))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 1.45e-124:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((d_m / M_m) * (2.0 / D_m)), -2.0))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((M_m * (D_m * (0.5 / d_m))), 2.0) / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 1.45e-124)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(d_m / M_m) * Float64(2.0 / D_m)) ^ -2.0)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(M_m * Float64(D_m * Float64(0.5 / d_m))) ^ 2.0) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 1.45e-124)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((d_m / M_m) * (2.0 / D_m)) ^ -2.0))));
	else
		tmp = w0 * sqrt((1.0 - (h * (((M_m * (D_m * (0.5 / d_m))) ^ 2.0) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 1.45e-124], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(d$95$m / M$95$m), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.4500000000000001e-124

   1. Initial program 76.1%

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

   3. Simplified 73.5%

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

      1. associate-*l/ 76.1%

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

      2. clear-num 76.1%

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

      3. *-un-lft-identity 76.1%

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

      4. times-frac 76.1%

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

      5. metadata-eval 76.1%

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

   5. Applied egg-rr 76.1%

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

      1. associate-*r/ 77.2%

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

      2. inv-pow 77.2%

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

      3. pow-pow 77.2%

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

      4. associate-*r/ 77.2%

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

      5. metadata-eval 77.2%

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

   7. Applied egg-rr 77.2%

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

      1. associate-*r/ 76.0%

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

      2. *-commutative 76.0%

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

      3. *-commutative 76.0%

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

      4. times-frac 75.2%

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

   9. Simplified 75.2%

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

   if 1.4500000000000001e-124 < d

   1. Initial program 83.1%

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

   3. Simplified 82.2%

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

      1. associate-*r/ 92.7%

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

      2. clear-num 92.7%

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

      3. associate-*l/ 94.5%

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

      4. div-inv 94.5%

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

      5. associate-*l* 93.6%

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

      6. associate-/r* 93.6%

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

      7. metadata-eval 93.6%

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

   5. Applied egg-rr 93.6%

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

      1. clear-num 93.6%

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

      2. *-commutative 93.6%

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

      3. associate-*l* 92.7%

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

      4. associate-*r/ 82.2%

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

      5. expm1-log1p-u 81.8%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      6. expm1-udef 81.8%

         \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]

   7. Applied egg-rr 82.7%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 82.7%

         \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

      2. expm1-log1p 83.1%

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

      3. associate-*r/ 94.5%

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

      4. associate-*l/ 94.5%

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

      5. *-commutative 94.5%

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

      6. associate-*l* 93.6%

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

   9. Simplified 93.6%

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.45 \cdot 10^{-124}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* h (/ (pow (* M_m (* D_m (/ 0.5 d_m))) 2.0) l))))))

C

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

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0 * sqrt((1.0d0 - (h * (((m_m * (d_m * (0.5d0 / d_m_1))) ** 2.0d0) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(M$95$m * N[(D$95$m * N[(0.5 / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

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

3. Simplified 77.1%

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

   1. associate-*r/ 82.9%

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

   2. clear-num 82.9%

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

   3. associate-*l/ 84.4%

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

   4. div-inv 84.4%

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

   5. associate-*l* 84.4%

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

   6. associate-/r* 84.4%

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

   7. metadata-eval 84.4%

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

5. Applied egg-rr 84.4%

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

   1. clear-num 84.3%

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

   2. *-commutative 84.3%

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

   3. associate-*l* 82.9%

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

   4. associate-*r/ 77.1%

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

   5. expm1-log1p-u 76.7%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

   6. expm1-udef 76.7%

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot \frac{0.5}{d}\right) \cdot D\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]

7. Applied egg-rr 78.5%

   \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)} - 1\right)} \]
8. Step-by-step derivation

   1. expm1-def 78.5%

      \[\leadsto w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)\right)} \]

   2. expm1-log1p 79.0%

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

   3. associate-*r/ 84.4%

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

   4. associate-*l/ 85.1%

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

   5. *-commutative 85.1%

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

   6. associate-*l* 85.1%

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

9. Simplified 85.1%

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

10. Final simplification 85.1%

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

Alternative 6: 79.7% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (* w0 (fma -0.125 (/ h (* l (pow (* M_m (/ D_m d_m)) -2.0))) 1.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * fma(-0.125, (h / (l * pow((M_m * (D_m / d_m)), -2.0))), 1.0);
}

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[(-0.125 * N[(h / N[(l * N[Power[N[(M$95$m * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

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

3. Simplified 77.1%

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

   1. associate-*r/ 82.9%

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

   2. clear-num 82.9%

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

   3. associate-*l/ 84.4%

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

   4. div-inv 84.4%

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

   5. associate-*l* 84.4%

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

   6. associate-/r* 84.4%

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

   7. metadata-eval 84.4%

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

5. Applied egg-rr 84.4%

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

6. Taylor expanded in l around inf 55.3%

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

   1. +-commutative 55.3%

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

   2. fma-def 55.3%

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

8. Simplified 77.7%

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

   1. div-inv 77.7%

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

   2. associate-*l/ 77.3%

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

   3. *-commutative 77.3%

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

   4. pow-flip 77.4%

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

   5. *-commutative 77.4%

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

   6. associate-*l/ 77.7%

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

   7. *-commutative 77.7%

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

   8. metadata-eval 77.7%

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

10. Applied egg-rr 77.7%

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

11. Final simplification 77.7%

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

Alternative 7: 79.9% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (* w0 (fma -0.125 (/ h (/ l (pow (/ M_m (/ d_m D_m)) 2.0))) 1.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * fma(-0.125, (h / (l / pow((M_m / (d_m / D_m)), 2.0))), 1.0);
}

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[(-0.125 * N[(h / N[(l / N[Power[N[(M$95$m / N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

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

3. Simplified 77.1%

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

   1. associate-*r/ 82.9%

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

   2. clear-num 82.9%

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

   3. associate-*l/ 84.4%

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

   4. div-inv 84.4%

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

   5. associate-*l* 84.4%

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

   6. associate-/r* 84.4%

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

   7. metadata-eval 84.4%

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

5. Applied egg-rr 84.4%

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

6. Taylor expanded in l around inf 55.3%

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

   1. +-commutative 55.3%

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

   2. fma-def 55.3%

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

8. Simplified 77.7%

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

   1. associate-*l/ 77.3%

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

   2. *-commutative 77.3%

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

   3. associate-/l* 77.7%

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

10. Applied egg-rr 77.7%

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

11. Final simplification 77.7%

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

Alternative 8: 80.0% accurate, 1.0× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (* w0 (fma -0.125 (/ h (/ l (pow (/ (* M_m D_m) d_m) 2.0))) 1.0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0 * fma(-0.125, (h / (l / pow(((M_m * D_m) / d_m), 2.0))), 1.0);
}

Julia

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := N[(w0 * N[(-0.125 * N[(h / N[(l / N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

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

3. Simplified 77.1%

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

   1. associate-*r/ 82.9%

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

   2. clear-num 82.9%

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

   3. associate-*l/ 84.4%

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

   4. div-inv 84.4%

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

   5. associate-*l* 84.4%

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

   6. associate-/r* 84.4%

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

   7. metadata-eval 84.4%

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

5. Applied egg-rr 84.4%

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

6. Taylor expanded in l around inf 55.3%

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

   1. +-commutative 55.3%

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

   2. fma-def 55.3%

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

8. Simplified 77.7%

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

   1. associate-*l/ 77.3%

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

   2. *-commutative 77.3%

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

10. Applied egg-rr 77.3%

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

11. Final simplification 77.3%

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

Alternative 9: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \cdot D_m \leq 2 \cdot 10^{-42}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{w0 \cdot h}{\ell} \cdot \left(\frac{M_m \cdot D_m}{d_m} \cdot \left(D_m \cdot \frac{M_m}{d_m}\right)\right), w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (* M_m D_m) 2e-42)
   w0
   (fma
    -0.125
    (* (/ (* w0 h) l) (* (/ (* M_m D_m) d_m) (* D_m (/ M_m d_m))))
    w0)))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((M_m * D_m) <= 2e-42) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, (((w0 * h) / l) * (((M_m * D_m) / d_m) * (D_m * (M_m / d_m)))), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-42)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(Float64(Float64(w0 * h) / l) * Float64(Float64(Float64(M_m * D_m) / d_m) * Float64(D_m * Float64(M_m / d_m)))), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-42], w0, N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000008e-42

   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 79.6%

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

   4. Taylor expanded in M around 0 74.9%

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

   if 2.00000000000000008e-42 < (*.f64 M D)

   1. Initial program 70.0%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in M around 0 39.2%

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

      1. +-commutative 39.2%

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

      2. fma-def 39.2%

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

      3. associate-*r* 39.2%

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

      4. times-frac 40.9%

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

      5. *-commutative 40.9%

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

      6. unpow2 40.9%

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

      7. unpow2 40.9%

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

      8. swap-sqr 52.4%

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

      9. unpow2 52.4%

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

      10. *-commutative 52.4%

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

   6. Simplified 52.4%

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

      1. add-sqr-sqrt 52.4%

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

      2. sqrt-div 52.4%

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

      3. unpow2 52.4%

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

      4. sqrt-prod 52.4%

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

      5. add-sqr-sqrt 52.4%

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

      6. *-commutative 52.4%

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

      7. unpow2 52.4%

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

      8. sqrt-prod 29.7%

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

      9. add-sqr-sqrt 35.3%

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

      10. sqrt-div 35.3%

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

      11. unpow2 35.3%

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

      12. sqrt-prod 39.0%

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

      13. add-sqr-sqrt 39.0%

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

      14. *-commutative 39.0%

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

      15. unpow2 39.0%

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

      16. sqrt-prod 33.6%

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

      17. add-sqr-sqrt 56.5%

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

   8. Applied egg-rr 56.5%

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

   9. Taylor expanded in M around 0 56.5%

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

       1. associate-/l* 56.4%

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

   11. Simplified 56.4%

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

       1. associate-/r/ 56.5%

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

       2. associate-*l/ 56.5%

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

       3. *-commutative 56.5%

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

       4. associate-/l* 56.5%

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

       5. associate-/r/ 56.3%

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

   13. Applied egg-rr 56.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-42}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{w0 \cdot h}{\ell} \cdot \left(\frac{M \cdot D}{d} \cdot \left(D \cdot \frac{M}{d}\right)\right), w0\right)\\ \end{array} \]

Alternative 10: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D_m}{d_m}\\ \mathbf{if}\;M_m \cdot D_m \leq 2 \cdot 10^{-42}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(t_0 \cdot t_0\right) \cdot \frac{w0 \cdot h}{\ell}, w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ (* M_m D_m) d_m)))
   (if (<= (* M_m D_m) 2e-42)
     w0
     (fma -0.125 (* (* t_0 t_0) (/ (* w0 h) l)) w0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = (M_m * D_m) / d_m;
	double tmp;
	if ((M_m * D_m) <= 2e-42) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, ((t_0 * t_0) * ((w0 * h) / l)), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(Float64(M_m * D_m) / d_m)
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e-42)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(Float64(t_0 * t_0) * Float64(Float64(w0 * h) / l)), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(N[(M$95$m * D$95$m), $MachinePrecision] / d$95$m), $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e-42], w0, N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.00000000000000008e-42

   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 79.6%

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

   4. Taylor expanded in M around 0 74.9%

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

   if 2.00000000000000008e-42 < (*.f64 M D)

   1. Initial program 70.0%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in M around 0 39.2%

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

      1. +-commutative 39.2%

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

      2. fma-def 39.2%

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

      3. associate-*r* 39.2%

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

      4. times-frac 40.9%

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

      5. *-commutative 40.9%

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

      6. unpow2 40.9%

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

      7. unpow2 40.9%

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

      8. swap-sqr 52.4%

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

      9. unpow2 52.4%

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

      10. *-commutative 52.4%

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

   6. Simplified 52.4%

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

      1. add-sqr-sqrt 52.4%

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

      2. sqrt-div 52.4%

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

      3. unpow2 52.4%

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

      4. sqrt-prod 52.4%

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

      5. add-sqr-sqrt 52.4%

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

      6. *-commutative 52.4%

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

      7. unpow2 52.4%

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

      8. sqrt-prod 29.7%

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

      9. add-sqr-sqrt 35.3%

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

      10. sqrt-div 35.3%

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

      11. unpow2 35.3%

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

      12. sqrt-prod 39.0%

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

      13. add-sqr-sqrt 39.0%

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

      14. *-commutative 39.0%

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

      15. unpow2 39.0%

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

      16. sqrt-prod 33.6%

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

      17. add-sqr-sqrt 56.5%

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

   8. Applied egg-rr 56.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{-42}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{w0 \cdot h}{\ell}, w0\right)\\ \end{array} \]

Alternative 11: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} t_0 := \frac{D_m}{\frac{d_m}{M_m}}\\ \mathbf{if}\;M_m \leq 6.2 \cdot 10^{-171}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{w0 \cdot h}{\ell} \cdot \left(t_0 \cdot t_0\right), w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (let* ((t_0 (/ D_m (/ d_m M_m))))
   (if (<= M_m 6.2e-171) w0 (fma -0.125 (* (/ (* w0 h) l) (* t_0 t_0)) w0))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double t_0 = D_m / (d_m / M_m);
	double tmp;
	if (M_m <= 6.2e-171) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, (((w0 * h) / l) * (t_0 * t_0)), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	t_0 = Float64(D_m / Float64(d_m / M_m))
	tmp = 0.0
	if (M_m <= 6.2e-171)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(Float64(Float64(w0 * h) / l) * Float64(t_0 * t_0)), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := Block[{t$95$0 = N[(D$95$m / N[(d$95$m / M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 6.2e-171], w0, N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 6.2000000000000001e-171

   1. Initial program 79.0%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in M around 0 71.0%

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

   if 6.2000000000000001e-171 < M

   1. Initial program 79.1%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in M around 0 50.2%

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

      1. +-commutative 50.2%

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

      2. fma-def 50.2%

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

      3. associate-*r* 50.2%

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

      4. times-frac 54.4%

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

      5. *-commutative 54.4%

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

      6. unpow2 54.4%

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

      7. unpow2 54.4%

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

      8. swap-sqr 63.7%

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

      9. unpow2 63.7%

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

      10. *-commutative 63.7%

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

   6. Simplified 63.7%

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

      1. add-sqr-sqrt 63.7%

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

      2. sqrt-div 63.7%

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

      3. unpow2 63.7%

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

      4. sqrt-prod 38.6%

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

      5. add-sqr-sqrt 56.7%

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

      6. *-commutative 56.7%

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

      7. unpow2 56.7%

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

      8. sqrt-prod 30.5%

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

      9. add-sqr-sqrt 49.9%

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

      10. sqrt-div 49.9%

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

      11. unpow2 49.9%

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

      12. sqrt-prod 30.6%

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

      13. add-sqr-sqrt 54.6%

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

      14. *-commutative 54.6%

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

      15. unpow2 54.6%

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

      16. sqrt-prod 39.8%

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

      17. add-sqr-sqrt 69.6%

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

   8. Applied egg-rr 69.6%

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

   9. Taylor expanded in M around 0 69.6%

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

       1. associate-/l* 69.6%

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

   11. Simplified 69.6%

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

   12. Taylor expanded in M around 0 69.6%

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

       1. associate-/l* 69.6%

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

   14. Simplified 69.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6.2 \cdot 10^{-171}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{w0 \cdot h}{\ell} \cdot \left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right), w0\right)\\ \end{array} \]

Alternative 12: 67.6% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m) :precision binary64 w0)

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	return w0;
}

Fortran

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_m_1
    code = w0
end function

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	return w0

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := w0

Derivation

1. Initial program 79.0%

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

3. Simplified 77.1%

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

4. Taylor expanded in M around 0 65.2%

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

5. Final simplification 65.2%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023339 
(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))))))