Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 89.1%

Time: 14.7s

Alternatives: 3

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: 81.3% 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: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ t_1 := \frac{M\_m}{\frac{d}{D \cdot 0.5}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;w0 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(t\_1 \cdot \frac{t\_1}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l))))
        (t_1 (/ M_m (/ d (* D 0.5)))))
   (if (<= t_0 5e+292)
     (* w0 (sqrt t_0))
     (* w0 (sqrt (- 1.0 (* h (* t_1 (/ t_1 l)))))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l));
	double t_1 = M_m / (d / (D * 0.5));
	double tmp;
	if (t_0 <= 5e+292) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (h * (t_1 * (t_1 / l)))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    t_1 = m_m / (d_1 / (d * 0.5d0))
    if (t_0 <= 5d+292) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (h * (t_1 * (t_1 / l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l));
	double t_1 = M_m / (d / (D * 0.5));
	double tmp;
	if (t_0 <= 5e+292) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (t_1 * (t_1 / l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l))
	t_1 = M_m / (d / (D * 0.5))
	tmp = 0
	if t_0 <= 5e+292:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (h * (t_1 * (t_1 / l)))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	t_1 = Float64(M_m / Float64(d / Float64(D * 0.5)))
	tmp = 0.0
	if (t_0 <= 5e+292)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_1 * Float64(t_1 / l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = 1.0 - ((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l));
	t_1 = M_m / (d / (D * 0.5));
	tmp = 0.0;
	if (t_0 <= 5e+292)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (h * (t_1 * (t_1 / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m / N[(d / N[(D * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+292], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 4.9999999999999996e292

   1. Initial program 99.9%

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

   if 4.9999999999999996e292 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 39.5%

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

   3. Simplified 41.7%

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

      1. associate-/l* 39.5%

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

      2. clear-num 39.5%

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

      3. un-div-inv 39.9%

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

      4. associate-/l* 42.1%

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

      5. div-inv 42.1%

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

      6. clear-num 42.1%

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

      7. associate-/r* 42.1%

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

   6. Applied egg-rr 42.1%

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

      1. associate-/r/ 59.9%

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

      2. associate-*r/ 59.4%

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

      3. div-inv 59.4%

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

      4. metadata-eval 59.4%

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

   8. Applied egg-rr 59.4%

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

      1. unpow2 59.4%

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

      2. *-un-lft-identity 59.4%

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

      3. times-frac 67.3%

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

      4. associate-/l* 64.2%

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

      5. associate-/l* 67.2%

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

   10. Applied egg-rr 67.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\frac{M}{\frac{d}{D \cdot 0.5}} \cdot \frac{\frac{M}{\frac{d}{D \cdot 0.5}}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \left(D \cdot 0.5\right) \cdot \frac{M\_m}{d}\\ w0 \cdot \sqrt{1 - h \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)} \end{array} \end{array} \]

FPCore

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

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (D * 0.5) * (M_m / d);
	return w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    t_0 = (d * 0.5d0) * (m_m / d_1)
    code = w0 * sqrt((1.0d0 - (h * (t_0 * (t_0 / l)))))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (D * 0.5) * (M_m / d);
	return w0 * Math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = (D * 0.5) * (M_m / d)
	return w0 * math.sqrt((1.0 - (h * (t_0 * (t_0 / l)))))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(Float64(D * 0.5) * Float64(M_m / d))
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(t_0 * Float64(t_0 / l))))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	t_0 = (D * 0.5) * (M_m / d);
	tmp = w0 * sqrt((1.0 - (h * (t_0 * (t_0 / l)))));
end

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Simplified 79.0%

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

   1. associate-/l* 79.4%

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

   2. clear-num 79.4%

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

   3. un-div-inv 79.5%

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

   4. associate-/l* 79.2%

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

   5. div-inv 79.1%

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

   6. clear-num 79.2%

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

   7. associate-/r* 79.2%

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

6. Applied egg-rr 79.2%

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

   1. associate-/r/ 83.3%

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

   2. associate-*r/ 84.3%

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

   3. div-inv 84.3%

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

   4. metadata-eval 84.3%

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

8. Applied egg-rr 84.3%

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

   1. unpow2 84.3%

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

   2. associate-/l* 82.6%

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

   3. associate-/l* 83.3%

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

10. Applied egg-rr 83.3%

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

    1. associate-*r/ 85.8%

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

    2. *-commutative 85.8%

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

    3. associate-/r/ 85.1%

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

    4. associate-/r/ 87.2%

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

12. Applied egg-rr 87.2%

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

13. Final simplification 87.2%

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

Alternative 3: 67.6% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Simplified 79.0%

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

5. Taylor expanded in M around 0 63.4%

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

6. Final simplification 63.4%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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))))))