Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.0% → 89.7%

Time: 18.3s

Alternatives: 4

Speedup: 1.7×

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: 81.0% 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: 89.7% accurate, 1.7× speedup

Math

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

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 2.0)))))
   (* w0 (sqrt (+ 1.0 (* (/ t_0 l) (/ t_0 (/ -1.0 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 * 2.0));
	return w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
}

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
    t_0 = d_m * (m_m / (d * 2.0d0))
    code = w0 * sqrt((1.0d0 + ((t_0 / l) * (t_0 / ((-1.0d0) / h)))))
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 * 2.0));
	return w0 * Math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
}

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 * 2.0))
	return w0 * math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))))

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 / Float64(d * 2.0)))
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 / l) * Float64(t_0 / Float64(-1.0 / h))))))
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)
	t_0 = D_m * (M_m / (d * 2.0));
	tmp = w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
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 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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 81.4%

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

   1. associate-*r/ 86.2%

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

   2. clear-num 86.2%

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

   3. add-sqr-sqrt 86.2%

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

   4. pow2 86.2%

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

   5. unpow2 86.2%

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

   6. sqrt-prod 51.6%

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

   7. add-sqr-sqrt 86.2%

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

   8. *-un-lft-identity 86.2%

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

   9. times-frac 86.2%

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

   10. metadata-eval 86.2%

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

6. Applied egg-rr 86.2%

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

   1. associate-/r/ 86.3%

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

   2. *-commutative 86.3%

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

   3. associate-*r/ 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-/l* 86.0%

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

   6. associate-*l/ 85.5%

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

   7. *-commutative 85.5%

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

   8. associate-*l* 85.5%

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

8. Simplified 85.5%

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

   1. *-commutative 85.5%

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

   2. *-commutative 85.5%

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

   3. associate-*r* 85.5%

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

   4. *-commutative 85.5%

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

   5. associate-*r* 80.6%

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

   6. div-inv 80.6%

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

   7. clear-num 80.6%

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

   8. un-div-inv 80.6%

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

   9. *-commutative 80.6%

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

   10. associate-*r* 80.6%

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

   11. clear-num 80.6%

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

   12. un-div-inv 81.0%

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

10. Applied egg-rr 81.0%

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

    1. associate-/r/ 81.0%

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

12. Simplified 81.0%

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

    1. unpow2 81.0%

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

    2. div-inv 81.0%

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

    3. times-frac 87.0%

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

    4. *-commutative 87.0%

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

    5. clear-num 87.0%

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

    6. un-div-inv 87.0%

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

    7. div-inv 87.0%

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

    8. metadata-eval 87.0%

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

    9. *-commutative 87.0%

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

    10. clear-num 87.0%

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

    11. un-div-inv 87.0%

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

    12. div-inv 87.0%

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

    13. metadata-eval 87.0%

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

14. Applied egg-rr 87.0%

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

15. Final simplification 87.0%

    \[\leadsto w0 \cdot \sqrt{1 + \frac{D \cdot \frac{M}{d \cdot 2}}{\ell} \cdot \frac{D \cdot \frac{M}{d \cdot 2}}{\frac{-1}{h}}} \]
16. Add Preprocessing

Alternative 2: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d \cdot 2}\\ \mathbf{if}\;D\_m \leq 5.6 \cdot 10^{-94}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right)}\\ \end{array} \end{array} \]

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 2.0)))))
   (if (<= D_m 5.6e-94) w0 (* w0 (sqrt (- 1.0 (* t_0 (* t_0 (/ h 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) {
	double t_0 = D_m * (M_m / (d * 2.0));
	double tmp;
	if (D_m <= 5.6e-94) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	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 * 2.0d0))
    if (d_m <= 5.6d-94) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 * (h / l)))))
    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 * 2.0));
	double tmp;
	if (D_m <= 5.6e-94) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	}
	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 * 2.0))
	tmp = 0
	if D_m <= 5.6e-94:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 * (h / l)))))
	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 / Float64(d * 2.0)))
	tmp = 0.0
	if (D_m <= 5.6e-94)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(h / l))))));
	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 * 2.0));
	tmp = 0.0;
	if (D_m <= 5.6e-94)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 * (h / l)))));
	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 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D$95$m, 5.6e-94], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 5.5999999999999995e-94

   1. Initial program 83.7%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in M around 0 77.2%

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

   if 5.5999999999999995e-94 < D

   1. Initial program 74.1%

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

   3. Simplified 73.9%

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

      1. associate-*r/ 75.6%

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

      2. clear-num 75.6%

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

      3. add-sqr-sqrt 75.6%

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

      4. pow2 75.6%

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

      5. unpow2 75.6%

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

      6. sqrt-prod 50.3%

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

      7. add-sqr-sqrt 75.6%

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

      8. *-un-lft-identity 75.6%

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

      9. times-frac 75.6%

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

      10. metadata-eval 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. associate-/r/ 75.6%

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

      2. *-commutative 75.6%

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

      3. associate-*r/ 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-/l* 75.8%

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

      6. associate-*l/ 77.0%

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

      7. *-commutative 77.0%

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

      8. associate-*l* 77.0%

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

   8. Simplified 77.0%

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

      1. *-commutative 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*r* 77.0%

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

      4. *-commutative 77.0%

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

      5. associate-*r* 75.3%

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

      6. div-inv 75.2%

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

      7. clear-num 75.2%

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

      8. un-div-inv 75.2%

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

      9. *-commutative 75.2%

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

      10. associate-*r* 75.2%

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

      11. clear-num 75.2%

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

      12. un-div-inv 75.2%

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

   10. Applied egg-rr 75.2%

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

       1. associate-/r/ 75.3%

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

   12. Simplified 75.3%

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

       1. div-inv 75.3%

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

       2. unpow2 75.3%

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

       3. clear-num 75.3%

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

       4. associate-*l* 76.7%

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

       5. *-commutative 76.7%

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

       6. clear-num 76.7%

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

       7. un-div-inv 76.8%

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

       8. div-inv 76.8%

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

       9. metadata-eval 76.8%

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

       10. *-commutative 76.8%

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

       11. clear-num 76.8%

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

       12. un-div-inv 76.7%

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

       13. div-inv 76.7%

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

       14. metadata-eval 76.7%

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

   14. Applied egg-rr 76.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5.6 \cdot 10^{-94}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \left(\left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \frac{h}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d \cdot 2}\\ \mathbf{if}\;D\_m \leq 9.2 \cdot 10^{-97}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t\_0 \cdot \frac{t\_0}{\frac{\ell}{h}}}\\ \end{array} \end{array} \]

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 2.0)))))
   (if (<= D_m 9.2e-97) w0 (* w0 (sqrt (- 1.0 (* t_0 (/ t_0 (/ l 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 * 2.0));
	double tmp;
	if (D_m <= 9.2e-97) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 / (l / 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 * 2.0d0))
    if (d_m <= 9.2d-97) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (t_0 * (t_0 / (l / 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 * 2.0));
	double tmp;
	if (D_m <= 9.2e-97) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (t_0 / (l / 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 * 2.0))
	tmp = 0
	if D_m <= 9.2e-97:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (t_0 / (l / 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 / Float64(d * 2.0)))
	tmp = 0.0
	if (D_m <= 9.2e-97)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(t_0 / Float64(l / 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 * 2.0));
	tmp = 0.0;
	if (D_m <= 9.2e-97)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (t_0 * (t_0 / (l / 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 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D$95$m, 9.2e-97], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(t$95$0 / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 9.19999999999999976e-97

   1. Initial program 83.7%

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

   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. Taylor expanded in M around 0 77.1%

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

   if 9.19999999999999976e-97 < 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 74.2%

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

      1. associate-*r/ 75.9%

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

      2. clear-num 75.9%

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

      3. add-sqr-sqrt 75.9%

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

      4. pow2 75.9%

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

      5. unpow2 75.9%

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

      6. sqrt-prod 51.0%

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

      7. add-sqr-sqrt 75.9%

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

      8. *-un-lft-identity 75.9%

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

      9. times-frac 75.9%

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

      10. metadata-eval 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. associate-/r/ 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r/ 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-/l* 76.1%

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

      6. associate-*l/ 77.3%

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

      7. *-commutative 77.3%

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

      8. associate-*l* 77.3%

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

   8. Simplified 77.3%

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

      1. *-commutative 77.3%

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

      2. *-commutative 77.3%

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

      3. associate-*r* 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-*r* 75.6%

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

      6. div-inv 75.6%

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

      7. clear-num 75.6%

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

      8. un-div-inv 75.6%

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

      9. *-commutative 75.6%

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

      10. associate-*r* 75.6%

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

      11. clear-num 75.6%

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

      12. un-div-inv 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. associate-/r/ 75.6%

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

   12. Simplified 75.6%

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

       1. unpow2 75.6%

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

       2. *-un-lft-identity 75.6%

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

       3. times-frac 77.1%

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

       4. *-commutative 77.1%

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

       5. clear-num 77.1%

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

       6. un-div-inv 77.1%

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

       7. div-inv 77.1%

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

       8. metadata-eval 77.1%

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

       9. *-commutative 77.1%

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

       10. clear-num 77.1%

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

       11. un-div-inv 77.1%

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

       12. div-inv 77.1%

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

       13. metadata-eval 77.1%

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

   14. Applied egg-rr 77.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 9.2 \cdot 10^{-97}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(D \cdot \frac{M}{d \cdot 2}\right) \cdot \frac{D \cdot \frac{M}{d \cdot 2}}{\frac{\ell}{h}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% 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 81.1%

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

3. Simplified 81.4%

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

5. Taylor expanded in M around 0 69.7%

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

6. Final simplification 69.7%

   \[\leadsto w0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024045 
(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))))))