Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.4% → 89.6%

Time: 16.7s

Alternatives: 5

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: 89.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.1%

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

   1. *-commutative 79.1%

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

   2. frac-times 79.5%

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

   3. *-commutative 79.5%

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

   4. associate-*l/ 85.0%

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

   5. associate-*l/ 84.7%

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

   6. *-commutative 84.7%

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

   7. div-inv 84.6%

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

   8. associate-/r* 84.6%

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

   9. metadata-eval 84.6%

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

6. Applied egg-rr 84.6%

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

   1. associate-*l/ 79.1%

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

   2. unpow2 79.1%

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

   3. associate-*r* 80.7%

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

   4. associate-*r/ 80.7%

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

   5. associate-*r/ 80.7%

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

8. Applied egg-rr 80.7%

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

   1. associate-*l/ 87.3%

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

   2. associate-/l* 87.3%

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

10. Applied egg-rr 87.3%

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

Alternative 2: 87.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot \left(M \cdot 0.5\right)}{d}\\ t_1 := w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D\_m \cdot \left(M \cdot h\right)}{d}}{\ell} \cdot \left(D\_m \cdot \frac{M \cdot 0.5}{d}\right)}\\ \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{+264}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-95}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (/ (* D_m (* M 0.5)) d))
        (t_1
         (*
          w0
          (sqrt
           (-
            1.0
            (*
             (/ (* 0.5 (/ (* D_m (* M h)) d)) l)
             (* D_m (/ (* M 0.5) d))))))))
   (if (<= (/ h l) -1e+264)
     t_1
     (if (<= (/ h l) -1e-95)
       (* w0 (sqrt (- 1.0 (* (* (/ h l) t_0) t_0))))
       t_1))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = (D_m * (M * 0.5)) / d;
	double t_1 = w0 * sqrt((1.0 - (((0.5 * ((D_m * (M * h)) / d)) / l) * (D_m * ((M * 0.5) / d)))));
	double tmp;
	if ((h / l) <= -1e+264) {
		tmp = t_1;
	} else if ((h / l) <= -1e-95) {
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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) :: t_1
    real(8) :: tmp
    t_0 = (d_m * (m * 0.5d0)) / d
    t_1 = w0 * sqrt((1.0d0 - (((0.5d0 * ((d_m * (m * h)) / d)) / l) * (d_m * ((m * 0.5d0) / d)))))
    if ((h / l) <= (-1d+264)) then
        tmp = t_1
    else if ((h / l) <= (-1d-95)) then
        tmp = w0 * sqrt((1.0d0 - (((h / l) * t_0) * t_0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = (D_m * (M * 0.5)) / d;
	double t_1 = w0 * Math.sqrt((1.0 - (((0.5 * ((D_m * (M * h)) / d)) / l) * (D_m * ((M * 0.5) / d)))));
	double tmp;
	if ((h / l) <= -1e+264) {
		tmp = t_1;
	} else if ((h / l) <= -1e-95) {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * t_0) * t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	t_0 = (D_m * (M * 0.5)) / d
	t_1 = w0 * math.sqrt((1.0 - (((0.5 * ((D_m * (M * h)) / d)) / l) * (D_m * ((M * 0.5) / d)))))
	tmp = 0
	if (h / l) <= -1e+264:
		tmp = t_1
	elif (h / l) <= -1e-95:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * t_0) * t_0)))
	else:
		tmp = t_1
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(Float64(D_m * Float64(M * 0.5)) / d)
	t_1 = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(Float64(D_m * Float64(M * h)) / d)) / l) * Float64(D_m * Float64(Float64(M * 0.5) / d))))))
	tmp = 0.0
	if (Float64(h / l) <= -1e+264)
		tmp = t_1;
	elseif (Float64(h / l) <= -1e-95)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * t_0) * t_0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	t_0 = (D_m * (M * 0.5)) / d;
	t_1 = w0 * sqrt((1.0 - (((0.5 * ((D_m * (M * h)) / d)) / l) * (D_m * ((M * 0.5) / d)))));
	tmp = 0.0;
	if ((h / l) <= -1e+264)
		tmp = t_1;
	elseif ((h / l) <= -1e-95)
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * t_0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D$95$m * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(0.5 * N[(N[(D$95$m * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -1e+264], t$95$1, If[LessEqual[N[(h / l), $MachinePrecision], -1e-95], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.00000000000000004e264 or -9.99999999999999989e-96 < (/.f64 h l)

   1. Initial program 81.1%

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

   3. Simplified 81.0%

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

      1. *-commutative 81.0%

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

      2. frac-times 81.1%

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

      3. *-commutative 81.1%

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

      4. associate-*l/ 89.7%

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

      5. associate-*l/ 89.7%

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

      6. *-commutative 89.7%

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

      7. div-inv 89.7%

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

      8. associate-/r* 89.7%

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

      9. metadata-eval 89.7%

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

   6. Applied egg-rr 89.7%

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

      1. associate-*l/ 81.0%

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

      2. unpow2 81.0%

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

      3. associate-*r* 83.4%

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

      4. associate-*r/ 83.4%

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

      5. associate-*r/ 83.4%

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

   8. Applied egg-rr 83.4%

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

      1. associate-*l/ 93.5%

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

      2. associate-/l* 93.5%

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

   10. Applied egg-rr 93.5%

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

   11. Taylor expanded in h around 0 90.1%

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

   if -1.00000000000000004e264 < (/.f64 h l) < -9.99999999999999989e-96

   1. Initial program 75.9%

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

   3. Simplified 74.8%

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

      1. *-commutative 74.8%

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

      2. frac-times 75.9%

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

      3. *-commutative 75.9%

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

      4. associate-*l/ 74.8%

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

      5. associate-*l/ 73.6%

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

      6. *-commutative 73.6%

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

      7. div-inv 73.6%

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

      8. associate-/r* 73.6%

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

      9. metadata-eval 73.6%

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

   6. Applied egg-rr 73.6%

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

      1. associate-*l/ 74.8%

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

      2. unpow2 74.8%

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

      3. associate-*r* 74.7%

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

      4. associate-*r/ 74.8%

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

      5. associate-*r/ 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. associate-*r/ 74.8%

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

   10. Applied egg-rr 74.8%

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

       1. associate-*r/ 77.1%

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

   12. Applied egg-rr 77.1%

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

Alternative 3: 76.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M \cdot 0.5}{d}\\ \mathbf{if}\;M \leq 4.1 \cdot 10^{-225}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot t\_0\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ (* M 0.5) d))))
   (if (<= M 4.1e-225) w0 (* w0 (sqrt (- 1.0 (* (* (/ h l) t_0) t_0)))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * ((M * 0.5) / d);
	double tmp;
	if (M <= 4.1e-225) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * t_0)));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 * 0.5d0) / d)
    if (m <= 4.1d-225) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (((h / l) * t_0) * t_0)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * ((M * 0.5) / d);
	double tmp;
	if (M <= 4.1e-225) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((h / l) * t_0) * t_0)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	t_0 = D_m * ((M * 0.5) / d)
	tmp = 0
	if M <= 4.1e-225:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (((h / l) * t_0) * t_0)))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(Float64(M * 0.5) / d))
	tmp = 0.0
	if (M <= 4.1e-225)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h / l) * t_0) * t_0))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	t_0 = D_m * ((M * 0.5) / d);
	tmp = 0.0;
	if (M <= 4.1e-225)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (((h / l) * t_0) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 4.1e-225], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 4.10000000000000022e-225

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in D around 0 72.4%

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

   if 4.10000000000000022e-225 < M

   1. Initial program 81.7%

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

   3. Simplified 80.7%

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

      1. *-commutative 80.7%

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

      2. frac-times 81.7%

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

      3. *-commutative 81.7%

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

      4. associate-*l/ 85.7%

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

      5. associate-*l/ 84.7%

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

      6. *-commutative 84.7%

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

      7. div-inv 84.7%

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

      8. associate-/r* 84.7%

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

      9. metadata-eval 84.7%

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

   6. Applied egg-rr 84.7%

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

      1. associate-*l/ 80.7%

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

      2. unpow2 80.7%

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

      3. associate-*r* 81.7%

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

      4. associate-*r/ 81.7%

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

      5. associate-*r/ 81.7%

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

   8. Applied egg-rr 81.7%

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

Alternative 4: 84.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [w0, M, D_m, h, l, d] = \mathsf{sort}([w0, M, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M \cdot 0.5}{d}\\ \mathbf{if}\;d \leq 10^{-45}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot D\_m}{\ell \cdot \frac{d}{M \cdot 0.5}} \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.5 \cdot \frac{D\_m \cdot \left(M \cdot h\right)}{d}}{\ell} \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M D_m h l d)
 :precision binary64
 (let* ((t_0 (* D_m (/ (* M 0.5) d))))
   (if (<= d 1e-45)
     (* w0 (sqrt (- 1.0 (* (/ (* h D_m) (* l (/ d (* M 0.5)))) t_0))))
     (* w0 (sqrt (- 1.0 (* (/ (* 0.5 (/ (* D_m (* M h)) d)) l) t_0)))))))

C

D_m = fabs(D);
assert(w0 < M && M < D_m && D_m < h && h < l && l < d);
double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * ((M * 0.5) / d);
	double tmp;
	if (d <= 1e-45) {
		tmp = w0 * sqrt((1.0 - (((h * D_m) / (l * (d / (M * 0.5)))) * t_0)));
	} else {
		tmp = w0 * sqrt((1.0 - (((0.5 * ((D_m * (M * h)) / d)) / l) * t_0)));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: 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 * 0.5d0) / d)
    if (d <= 1d-45) then
        tmp = w0 * sqrt((1.0d0 - (((h * d_m) / (l * (d / (m * 0.5d0)))) * t_0)))
    else
        tmp = w0 * sqrt((1.0d0 - (((0.5d0 * ((d_m * (m * h)) / d)) / l) * t_0)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert w0 < M && M < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M, double D_m, double h, double l, double d) {
	double t_0 = D_m * ((M * 0.5) / d);
	double tmp;
	if (d <= 1e-45) {
		tmp = w0 * Math.sqrt((1.0 - (((h * D_m) / (l * (d / (M * 0.5)))) * t_0)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((0.5 * ((D_m * (M * h)) / d)) / l) * t_0)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[w0, M, D_m, h, l, d] = sort([w0, M, D_m, h, l, d])
def code(w0, M, D_m, h, l, d):
	t_0 = D_m * ((M * 0.5) / d)
	tmp = 0
	if d <= 1e-45:
		tmp = w0 * math.sqrt((1.0 - (((h * D_m) / (l * (d / (M * 0.5)))) * t_0)))
	else:
		tmp = w0 * math.sqrt((1.0 - (((0.5 * ((D_m * (M * h)) / d)) / l) * t_0)))
	return tmp

Julia

D_m = abs(D)
w0, M, D_m, h, l, d = sort([w0, M, D_m, h, l, d])
function code(w0, M, D_m, h, l, d)
	t_0 = Float64(D_m * Float64(Float64(M * 0.5) / d))
	tmp = 0.0
	if (d <= 1e-45)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(h * D_m) / Float64(l * Float64(d / Float64(M * 0.5)))) * t_0))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(0.5 * Float64(Float64(D_m * Float64(M * h)) / d)) / l) * t_0))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
w0, M, D_m, h, l, d = num2cell(sort([w0, M, D_m, h, l, d])){:}
function tmp_2 = code(w0, M, D_m, h, l, d)
	t_0 = D_m * ((M * 0.5) / d);
	tmp = 0.0;
	if (d <= 1e-45)
		tmp = w0 * sqrt((1.0 - (((h * D_m) / (l * (d / (M * 0.5)))) * t_0)));
	else
		tmp = w0 * sqrt((1.0 - (((0.5 * ((D_m * (M * h)) / d)) / l) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 1e-45], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(h * D$95$m), $MachinePrecision] / N[(l * N[(d / N[(M * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(0.5 * N[(N[(D$95$m * N[(M * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 9.99999999999999984e-46

   1. Initial program 76.4%

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

   3. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. frac-times 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-*l/ 82.6%

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

      5. associate-*l/ 82.6%

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

      6. *-commutative 82.6%

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

      7. div-inv 82.5%

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

      8. associate-/r* 82.5%

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

      9. metadata-eval 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-*l/ 76.3%

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

      2. unpow2 76.3%

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

      3. associate-*r* 78.1%

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

      4. associate-*r/ 78.1%

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

      5. associate-*r/ 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. associate-*r/ 77.0%

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

   10. Applied egg-rr 77.0%

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

       1. associate-/l* 78.1%

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

       2. frac-times 79.7%

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

   12. Applied egg-rr 79.7%

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

   if 9.99999999999999984e-46 < d

   1. Initial program 86.9%

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

   3. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. frac-times 86.9%

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

      3. *-commutative 86.9%

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

      4. associate-*l/ 90.9%

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

      5. associate-*l/ 89.7%

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

      6. *-commutative 89.7%

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

      7. div-inv 89.7%

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

      8. associate-/r* 89.7%

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

      9. metadata-eval 89.7%

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

   6. Applied egg-rr 89.7%

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

      1. associate-*l/ 85.7%

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

      2. unpow2 85.7%

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

      3. associate-*r* 87.1%

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

      4. associate-*r/ 87.1%

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

      5. associate-*r/ 87.1%

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

   8. Applied egg-rr 87.1%

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

      1. associate-*l/ 92.3%

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

      2. associate-/l* 92.3%

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

   10. Applied egg-rr 92.3%

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

   11. Taylor expanded in h around 0 88.3%

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

Alternative 5: 68.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.1%

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

5. Taylor expanded in D around 0 68.2%

   \[\leadsto \color{blue}{w0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024034 -o generate:simplify
(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))))))