Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 87.5%

Time: 17.7s

Alternatives: 6

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.4% 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: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2 \cdot 10^{-86}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D\_m}{d} \cdot \frac{M\_m \cdot 0.5}{\ell}\right) \cdot \left(h \cdot \left(\frac{D\_m}{d} \cdot \left(M\_m \cdot 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 2e-86)
   (* w0 (sqrt (- 1.0 (/ (* (pow (* M_m (* 0.5 (/ D_m d))) 2.0) h) l))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* (/ D_m d) (/ (* M_m 0.5) l)) (* h (* (/ D_m d) (* M_m 0.5)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2e-86) {
		tmp = w0 * sqrt((1.0 - ((pow((M_m * (0.5 * (D_m / d))), 2.0) * h) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (((D_m / d) * ((M_m * 0.5) / l)) * (h * ((D_m / d) * (M_m * 0.5))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: tmp
    if (m_m <= 2d-86) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m * (0.5d0 * (d_m / d))) ** 2.0d0) * h) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (((d_m / d) * ((m_m * 0.5d0) / l)) * (h * ((d_m / d) * (m_m * 0.5d0))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2e-86) {
		tmp = w0 * Math.sqrt((1.0 - ((Math.pow((M_m * (0.5 * (D_m / d))), 2.0) * h) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D_m / d) * ((M_m * 0.5) / l)) * (h * ((D_m / d) * (M_m * 0.5))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 2e-86:
		tmp = w0 * math.sqrt((1.0 - ((math.pow((M_m * (0.5 * (D_m / d))), 2.0) * h) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D_m / d) * ((M_m * 0.5) / l)) * (h * ((D_m / d) * (M_m * 0.5))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2e-86)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64((Float64(M_m * Float64(0.5 * Float64(D_m / d))) ^ 2.0) * h) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D_m / d) * Float64(Float64(M_m * 0.5) / l)) * Float64(h * Float64(Float64(D_m / d) * Float64(M_m * 0.5)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 2e-86)
		tmp = w0 * sqrt((1.0 - ((((M_m * (0.5 * (D_m / d))) ^ 2.0) * h) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((D_m / d) * ((M_m * 0.5) / l)) * (h * ((D_m / d) * (M_m * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2e-86], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[Power[N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * 0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(D$95$m / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.00000000000000017e-86

   1. Initial program 84.1%

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

   3. Simplified 84.1%

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

      1. associate-*r/ 88.0%

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

      2. add-sqr-sqrt 88.0%

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

      3. pow2 88.0%

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

      4. sqrt-pow1 88.0%

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

      5. metadata-eval 88.0%

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

      6. pow1 88.0%

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

      7. *-un-lft-identity 88.0%

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

      8. times-frac 88.0%

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

      9. metadata-eval 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 2.00000000000000017e-86 < M

   1. Initial program 79.5%

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

   3. Simplified 82.1%

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

      1. associate-*r/ 83.7%

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

      2. add-sqr-sqrt 83.7%

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

      3. pow2 83.7%

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

      4. sqrt-pow1 83.7%

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

      5. metadata-eval 83.7%

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

      6. pow1 83.7%

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

      7. *-un-lft-identity 83.7%

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

      8. times-frac 83.7%

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

      9. metadata-eval 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*r/ 82.1%

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

      2. clear-num 82.1%

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

      3. div-inv 82.1%

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

      4. unpow2 82.1%

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

      5. div-inv 82.1%

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

      6. times-frac 90.0%

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

      7. *-commutative 90.0%

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

      8. *-commutative 90.0%

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

      9. associate-*l* 90.0%

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

      10. *-commutative 90.0%

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

      11. *-commutative 90.0%

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

      12. associate-*l* 90.0%

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

   8. Applied egg-rr 90.0%

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

      1. associate-/l* 82.2%

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

      2. *-commutative 82.2%

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

      3. associate-/r/ 82.2%

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

      4. /-rgt-identity 82.2%

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

      5. *-commutative 82.2%

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

   10. Simplified 82.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-86}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M \cdot 0.5}{\ell}\right) \cdot \left(h \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

3. Simplified 83.5%

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

   1. associate-*r/ 86.7%

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

   2. add-sqr-sqrt 86.7%

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

   3. pow2 86.7%

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

   4. sqrt-pow1 86.7%

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

   5. metadata-eval 86.7%

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

   6. pow1 86.7%

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

   7. *-un-lft-identity 86.7%

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

   8. times-frac 86.7%

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

   9. metadata-eval 86.7%

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

6. Applied egg-rr 86.7%

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

   1. associate-*r/ 83.5%

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

   2. clear-num 83.5%

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

   3. div-inv 83.5%

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

   4. unpow2 83.5%

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

   5. div-inv 83.5%

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

   6. times-frac 88.9%

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

   7. *-commutative 88.9%

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

   8. *-commutative 88.9%

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

   9. associate-*l* 88.9%

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

   10. *-commutative 88.9%

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

   11. *-commutative 88.9%

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

   12. associate-*l* 88.9%

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

8. Applied egg-rr 88.9%

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

   1. associate-/l* 84.5%

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

   2. *-commutative 84.5%

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

   3. associate-/r/ 84.5%

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

   4. /-rgt-identity 84.5%

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

   5. *-commutative 84.5%

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

10. Simplified 84.5%

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

11. Taylor expanded in D around 0 81.4%

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

12. Final simplification 81.4%

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M \cdot 0.5}{\ell}\right) \cdot \left(0.5 \cdot \frac{D \cdot \left(M \cdot h\right)}{d}\right)} \]
13. Add Preprocessing

Alternative 3: 87.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

3. Simplified 83.5%

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

   1. associate-*r/ 86.7%

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

   2. add-sqr-sqrt 86.7%

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

   3. pow2 86.7%

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

   4. sqrt-pow1 86.7%

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

   5. metadata-eval 86.7%

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

   6. pow1 86.7%

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

   7. *-un-lft-identity 86.7%

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

   8. times-frac 86.7%

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

   9. metadata-eval 86.7%

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

6. Applied egg-rr 86.7%

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

   1. associate-*r/ 83.5%

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

   2. clear-num 83.5%

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

   3. div-inv 83.5%

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

   4. unpow2 83.5%

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

   5. div-inv 83.5%

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

   6. times-frac 88.9%

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

   7. *-commutative 88.9%

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

   8. *-commutative 88.9%

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

   9. associate-*l* 88.9%

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

   10. *-commutative 88.9%

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

   11. *-commutative 88.9%

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

   12. associate-*l* 88.9%

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

8. Applied egg-rr 88.9%

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

   1. associate-/l* 84.5%

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

   2. *-commutative 84.5%

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

   3. associate-/r/ 84.5%

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

   4. /-rgt-identity 84.5%

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

   5. *-commutative 84.5%

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

10. Simplified 84.5%

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

11. Final simplification 84.5%

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{M \cdot 0.5}{\ell}\right) \cdot \left(h \cdot \left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)\right)} \]
12. Add Preprocessing

Alternative 4: 81.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    code = w0 + ((-0.125d0) * (w0 * (h * (((d_m * (m_m / d)) ** 2.0d0) / l))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

3. Simplified 83.5%

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

5. Taylor expanded in M around 0 48.3%

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

   1. associate-/l* 49.0%

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

   2. times-frac 50.1%

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

7. Simplified 50.1%

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

   1. associate-*r* 49.7%

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

   2. clear-num 49.7%

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

   3. un-div-inv 49.7%

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

   4. add-sqr-sqrt 49.7%

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

   5. pow2 49.7%

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

   6. sqrt-div 49.7%

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

   7. sqrt-pow1 56.4%

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

   8. metadata-eval 56.4%

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

   9. pow1 56.4%

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

   10. sqrt-pow1 61.2%

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

   11. metadata-eval 61.2%

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

   12. pow1 61.2%

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

9. Applied egg-rr 61.2%

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

    1. pow2 61.2%

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

11. Applied egg-rr 61.2%

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

    1. associate-/r/ 62.4%

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

    2. pow2 62.4%

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

    3. unpow-prod-down 73.7%

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

    4. associate-*r* 80.4%

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

13. Applied egg-rr 80.4%

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

14. Final simplification 80.4%

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

Alternative 5: 78.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d}\\ \mathbf{if}\;D\_m \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{t\_0 \cdot t\_0}{\ell} \cdot \left(w0 \cdot h\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m d))))
   (if (<= D_m 2.6e+90) w0 (+ w0 (* -0.125 (* (/ (* t_0 t_0) l) (* w0 h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (D_m <= 2.6e+90) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((t_0 * t_0) / l) * (w0 * h)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m * (m_m / d)
    if (d_m <= 2.6d+90) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((t_0 * t_0) / l) * (w0 * h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m * (M_m / d);
	double tmp;
	if (D_m <= 2.6e+90) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((t_0 * t_0) / l) * (w0 * h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m * (M_m / d)
	tmp = 0
	if D_m <= 2.6e+90:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((t_0 * t_0) / l) * (w0 * h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(M_m / d))
	tmp = 0.0
	if (D_m <= 2.6e+90)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(t_0 * t_0) / l) * Float64(w0 * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m * (M_m / d);
	tmp = 0.0;
	if (D_m <= 2.6e+90)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((t_0 * t_0) / l) * (w0 * h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D$95$m, 2.6e+90], w0, N[(w0 + N[(-0.125 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / l), $MachinePrecision] * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 2.5999999999999998e90

   1. Initial program 84.6%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in M around 0 72.8%

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

   if 2.5999999999999998e90 < D

   1. Initial program 74.4%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in M around 0 45.2%

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

      1. associate-/l* 47.5%

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

      2. times-frac 45.1%

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

   7. Simplified 45.1%

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

      1. unpow2 45.1%

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

      2. unpow2 45.1%

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

      3. times-frac 47.7%

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

   9. Applied egg-rr 47.7%

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

   10. Taylor expanded in D around 0 45.2%

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

       1. associate-*r* 45.2%

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

       2. times-frac 45.1%

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

       3. associate-*r/ 44.9%

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

       4. unpow2 44.9%

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

       5. unpow2 44.9%

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

       6. unpow2 44.9%

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

       7. times-frac 47.6%

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

       8. swap-sqr 68.2%

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

       9. unpow2 68.2%

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

       10. associate-*r/ 66.1%

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

       11. associate-*l/ 68.2%

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

   12. Simplified 68.2%

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

       1. unpow2 68.2%

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

   14. Applied egg-rr 68.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.6 \cdot 10^{+90}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\ell} \cdot \left(w0 \cdot h\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

3. Simplified 83.5%

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

5. Taylor expanded in M around 0 66.1%

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

6. Final simplification 66.1%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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))))))