Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.5% → 87.0%

Time: 19.1s

Alternatives: 6

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: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-67}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(\frac{\frac{D}{d}}{2} \cdot M\_m\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot {\left(\frac{\frac{D}{2} \cdot \frac{M\_m}{d}}{\sqrt{\ell}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= l 5e-67)
   (* w0 (sqrt (+ 1.0 (* (* h (pow (* (/ (/ D d) 2.0) M_m) 2.0)) (/ -1.0 l)))))
   (* w0 (sqrt (- 1.0 (* h (pow (/ (* (/ D 2.0) (/ M_m d)) (sqrt l)) 2.0)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (l <= 5e-67) {
		tmp = w0 * sqrt((1.0 + ((h * pow((((D / d) / 2.0) * M_m), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * pow((((D / 2.0) * (M_m / d)) / sqrt(l)), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 5d-67) then
        tmp = w0 * sqrt((1.0d0 + ((h * ((((d / d_1) / 2.0d0) * m_m) ** 2.0d0)) * ((-1.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * ((((d / 2.0d0) * (m_m / d_1)) / sqrt(l)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (l <= 5e-67) {
		tmp = w0 * Math.sqrt((1.0 + ((h * Math.pow((((D / d) / 2.0) * M_m), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * Math.pow((((D / 2.0) * (M_m / d)) / Math.sqrt(l)), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if l <= 5e-67:
		tmp = w0 * math.sqrt((1.0 + ((h * math.pow((((D / d) / 2.0) * M_m), 2.0)) * (-1.0 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * math.pow((((D / 2.0) * (M_m / d)) / math.sqrt(l)), 2.0))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (l <= 5e-67)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * (Float64(Float64(Float64(D / d) / 2.0) * M_m) ^ 2.0)) * Float64(-1.0 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * (Float64(Float64(Float64(D / 2.0) * Float64(M_m / d)) / sqrt(l)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (l <= 5e-67)
		tmp = w0 * sqrt((1.0 + ((h * ((((D / d) / 2.0) * M_m) ^ 2.0)) * (-1.0 / l))));
	else
		tmp = w0 * sqrt((1.0 - (h * ((((D / 2.0) * (M_m / d)) / sqrt(l)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[l, 5e-67], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[Power[N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[Power[N[(N[(N[(D / 2.0), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999999e-67

   1. Initial program 80.4%

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

   3. Simplified 79.8%

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

      1. unpow2 79.8%

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

      2. unpow2 79.8%

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

      3. associate-/r* 79.8%

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

      4. associate-*r/ 80.4%

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

      5. *-commutative 80.4%

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

      6. associate-*r/ 88.4%

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

      7. clear-num 88.4%

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

      8. *-commutative 88.4%

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

      9. associate-*l/ 88.4%

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

      10. associate-/r* 88.4%

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

      11. clear-num 88.4%

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

      12. associate-*l/ 88.4%

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

      13. *-un-lft-identity 88.4%

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

      14. associate-/r/ 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. associate-/r/ 88.4%

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

      2. associate-*l/ 88.4%

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

      3. *-commutative 88.4%

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

      4. associate-*l/ 88.4%

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

      5. associate-/l/ 87.9%

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

      6. associate-/r/ 87.9%

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

   8. Simplified 87.9%

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

   if 4.9999999999999999e-67 < l

   1. Initial program 84.3%

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

   3. Simplified 85.6%

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

      1. clear-num 85.5%

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

      2. un-div-inv 86.8%

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

      3. clear-num 86.7%

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

      4. un-div-inv 86.7%

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

      5. associate-/r/ 86.7%

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

   6. Applied egg-rr 86.7%

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

      1. associate-/r/ 87.0%

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

      2. associate-*l/ 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-/l* 87.0%

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

      5. associate-*l/ 87.0%

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

      6. *-commutative 87.0%

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

      7. associate-*l/ 87.0%

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

      8. associate-/l/ 85.8%

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

      9. associate-/r/ 85.8%

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

      10. associate-*l/ 85.8%

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

      11. *-commutative 85.8%

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

      12. associate-*r/ 85.9%

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

      13. *-commutative 85.9%

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

      14. associate-/l/ 85.9%

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

      15. associate-/l* 87.0%

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

   8. Simplified 87.0%

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

      1. add-sqr-sqrt 87.0%

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

      2. sqrt-div 87.0%

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

      3. sqrt-pow1 84.5%

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

      4. metadata-eval 84.5%

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

      5. pow1 84.5%

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

      6. sqrt-div 84.5%

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

      7. sqrt-pow1 90.6%

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

      8. metadata-eval 90.6%

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

      9. pow1 90.6%

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

   10. Applied egg-rr 90.6%

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

       1. unpow2 90.6%

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

       2. associate-*r/ 88.3%

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

       3. times-frac 90.6%

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

   12. Simplified 90.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-67}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot {\left(\frac{\frac{D}{2} \cdot \frac{M}{d}}{\sqrt{\ell}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot {\left(\frac{D \cdot M\_m}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;t\_0 \leq 0.0004:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (* (/ h l) (pow (/ (* D M_m) (* d 2.0)) 2.0))))
   (if (<= t_0 0.0004) (* w0 (sqrt (- 1.0 t_0))) w0)))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (h / l) * pow(((D * M_m) / (d * 2.0)), 2.0);
	double tmp;
	if (t_0 <= 0.0004) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h / l) * (((d * m_m) / (d_1 * 2.0d0)) ** 2.0d0)
    if (t_0 <= 0.0004d0) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (h / l) * Math.pow(((D * M_m) / (d * 2.0)), 2.0);
	double tmp;
	if (t_0 <= 0.0004) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = (h / l) * math.pow(((D * M_m) / (d * 2.0)), 2.0)
	tmp = 0
	if t_0 <= 0.0004:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(Float64(h / l) * (Float64(Float64(D * M_m) / Float64(d * 2.0)) ^ 2.0))
	tmp = 0.0
	if (t_0 <= 0.0004)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = (h / l) * (((D * M_m) / (d * 2.0)) ^ 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0004)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0004], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 4.00000000000000019e-4

   1. Initial program 91.3%

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

   if 4.00000000000000019e-4 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in D around 0 69.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \leq 0.0004:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(\frac{\frac{D}{d}}{2} \cdot M\_m\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (/ h l) -5e-251)
   (* w0 (sqrt (+ 1.0 (* (* h (pow (* (/ (/ D d) 2.0) M_m) 2.0)) (/ -1.0 l)))))
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-251) {
		tmp = w0 * sqrt((1.0 + ((h * pow((((D / d) / 2.0) * M_m), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-5d-251)) then
        tmp = w0 * sqrt((1.0d0 + ((h * ((((d / d_1) / 2.0d0) * m_m) ** 2.0d0)) * ((-1.0d0) / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-251) {
		tmp = w0 * Math.sqrt((1.0 + ((h * Math.pow((((D / d) / 2.0) * M_m), 2.0)) * (-1.0 / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (h / l) <= -5e-251:
		tmp = w0 * math.sqrt((1.0 + ((h * math.pow((((D / d) / 2.0) * M_m), 2.0)) * (-1.0 / l))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-251)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * (Float64(Float64(Float64(D / d) / 2.0) * M_m) ^ 2.0)) * Float64(-1.0 / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -5e-251)
		tmp = w0 * sqrt((1.0 + ((h * ((((D / d) / 2.0) * M_m) ^ 2.0)) * (-1.0 / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-251], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[Power[N[(N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision] * M$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.0000000000000003e-251

   1. Initial program 80.2%

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

   3. Simplified 80.3%

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

      1. unpow2 80.3%

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

      2. unpow2 80.3%

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

      3. associate-/r* 80.3%

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

      4. associate-*r/ 80.2%

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

      5. *-commutative 80.2%

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

      6. associate-*r/ 86.8%

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

      7. clear-num 86.8%

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

      8. *-commutative 86.8%

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

      9. associate-*l/ 87.5%

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

      10. associate-/r* 87.5%

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

      11. clear-num 87.5%

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

      12. associate-*l/ 87.5%

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

      13. *-un-lft-identity 87.5%

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

      14. associate-/r/ 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. associate-/r/ 87.4%

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

      2. associate-*l/ 87.4%

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

      3. *-commutative 87.4%

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

      4. associate-*l/ 87.4%

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

      5. associate-/l/ 86.0%

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

      6. associate-/r/ 86.0%

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

   8. Simplified 86.0%

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

   if -5.0000000000000003e-251 < (/.f64 h l)

   1. Initial program 83.2%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in D around 0 91.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-251}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(h \cdot {\left(\frac{\frac{D}{d}}{2} \cdot M\right)}^{2}\right) \cdot \frac{-1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

3. Simplified 81.7%

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

   1. clear-num 81.6%

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

   2. un-div-inv 82.2%

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

   3. clear-num 82.2%

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

   4. un-div-inv 82.2%

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

   5. associate-/r/ 82.2%

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

6. Applied egg-rr 82.2%

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

   1. associate-/r/ 86.9%

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

   2. associate-*l/ 88.0%

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

   3. *-commutative 88.0%

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

   4. associate-/l* 86.9%

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

   5. associate-*l/ 86.9%

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

   6. *-commutative 86.9%

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

   7. associate-*l/ 86.9%

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

   8. associate-/l/ 86.1%

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

   9. associate-/r/ 86.1%

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

   10. associate-*l/ 86.1%

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

   11. *-commutative 86.1%

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

   12. associate-*r/ 86.5%

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

   13. *-commutative 86.5%

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

   14. associate-/l/ 86.5%

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

   15. associate-/l* 86.9%

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

8. Simplified 86.9%

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

9. Final simplification 86.9%

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

Alternative 5: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.6 \cdot 10^{+65}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 1.6e+65) w0 (log (exp w0))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.6e+65) {
		tmp = w0;
	} else {
		tmp = log(exp(w0));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m_m <= 1.6d+65) then
        tmp = w0
    else
        tmp = log(exp(w0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 1.6e+65) {
		tmp = w0;
	} else {
		tmp = Math.log(Math.exp(w0));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 1.6e+65:
		tmp = w0
	else:
		tmp = math.log(math.exp(w0))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 1.6e+65)
		tmp = w0;
	else
		tmp = log(exp(w0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 1.6e+65)
		tmp = w0;
	else
		tmp = log(exp(w0));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 1.6e+65], w0, N[Log[N[Exp[w0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.60000000000000003e65

   1. Initial program 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in D around 0 77.4%

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

   if 1.60000000000000003e65 < M

   1. Initial program 71.1%

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

   3. Simplified 71.1%

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

      1. add-sqr-sqrt 46.6%

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

      2. sqrt-unprod 36.5%

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

      3. *-commutative 36.5%

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

      4. *-commutative 36.5%

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

      5. swap-sqr 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. associate-*l/ 36.2%

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

      2. *-commutative 36.2%

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

      3. associate-*l/ 36.2%

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

      4. associate-/l/ 36.3%

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

      5. associate-/r/ 36.3%

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

   8. Simplified 36.3%

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

   9. Taylor expanded in h around 0 32.7%

      \[\leadsto \sqrt{\color{blue}{{w0}^{2}}} \]
   10. Step-by-step derivation

       1. sqrt-pow1 43.0%

          \[\leadsto \color{blue}{{w0}^{\left(\frac{2}{2}\right)}} \]

       2. metadata-eval 43.0%

          \[\leadsto {w0}^{\color{blue}{1}} \]

       3. pow1 43.0%

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

       4. add-log-exp 31.5%

          \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]

   11. Applied egg-rr 31.5%

       \[\leadsto \color{blue}{\log \left(e^{w0}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.6 \cdot 10^{+65}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{w0}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

3. Simplified 81.7%

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

5. Taylor expanded in D around 0 71.6%

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

6. Final simplification 71.6%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))))