Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 90.1%

Time: 14.0s

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: 81.2% 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: 90.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 81.7%

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

   1. unpow2 81.7%

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

   2. associate-*r/ 80.6%

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

   3. clear-num 80.6%

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

   4. un-div-inv 80.6%

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

   5. *-un-lft-identity 80.6%

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

   6. times-frac 80.6%

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

   7. metadata-eval 80.6%

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

   8. times-frac 81.7%

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

6. Applied egg-rr 81.7%

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

   1. frac-times 87.6%

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

   2. associate-*r/ 87.6%

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

   3. frac-times 86.2%

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

   4. *-commutative 86.2%

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

8. Applied egg-rr 86.2%

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

   1. *-un-lft-identity 86.2%

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

   2. associate-*r/ 88.1%

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

   3. associate-/l* 88.9%

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

   4. associate-*r/ 87.0%

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

   5. associate-/l* 87.0%

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

   6. *-commutative 87.0%

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

   7. associate-/l* 87.0%

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

10. Applied egg-rr 87.0%

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

    1. *-lft-identity 87.0%

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

    2. associate-*r/ 86.3%

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

    3. *-commutative 86.3%

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

    4. associate-*l* 86.3%

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

    5. associate-*l/ 88.1%

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

    6. *-commutative 88.1%

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

    7. *-commutative 88.1%

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

    8. metadata-eval 88.1%

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

    9. times-frac 88.1%

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

    10. *-rgt-identity 88.1%

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

    11. associate-*r/ 86.3%

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

    12. associate-*r/ 88.1%

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

    13. times-frac 86.7%

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

    14. associate-*r/ 86.7%

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

    15. times-frac 88.1%

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

12. Simplified 88.1%

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

Alternative 2: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d} \cdot \frac{M\_m \cdot D\_m}{d \cdot 2}\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -1e-309)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* (/ (* M_m (* D_m 0.5)) d) (/ (* M_m D_m) (* d 2.0))) (/ h l)))))
   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) {
	double tmp;
	if ((h / l) <= -1e-309) {
		tmp = w0 * sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d) * ((M_m * D_m) / (d * 2.0))) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((h / l) <= (-1d-309)) then
        tmp = w0 * sqrt((1.0d0 - ((((m_m * (d_m * 0.5d0)) / d) * ((m_m * d_m) / (d * 2.0d0))) * (h / l))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -1e-309) {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d) * ((M_m * D_m) / (d * 2.0))) * (h / l))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (h / l) <= -1e-309:
		tmp = w0 * math.sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d) * ((M_m * D_m) / (d * 2.0))) * (h / l))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -1e-309)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d) * Float64(Float64(M_m * D_m) / Float64(d * 2.0))) * Float64(h / l)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -1e-309)
		tmp = w0 * sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d) * ((M_m * D_m) / (d * 2.0))) * (h / l))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -1e-309], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.000000000000002e-309

   1. Initial program 74.8%

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

   3. Simplified 75.6%

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

      1. associate-*r/ 74.8%

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

      2. unpow2 74.8%

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

      3. associate-/r* 74.8%

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

      4. frac-times 59.6%

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

      5. associate-/l* 59.6%

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

      6. div-inv 59.6%

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

      7. metadata-eval 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. times-frac 74.8%

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

      2. *-commutative 74.8%

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

      3. *-commutative 74.8%

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

   8. Applied egg-rr 74.8%

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

   if -1.000000000000002e-309 < (/.f64 h l)

   1. Initial program 88.9%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in M around 0 95.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot \left(D \cdot 0.5\right)}{d} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;D\_m \leq 10^{+111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M\_m \cdot \frac{D\_m \cdot 0.5}{d}\right)}{\ell \cdot \frac{d \cdot 2}{M\_m \cdot D\_m}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d} \cdot \frac{M\_m \cdot D\_m}{d \cdot 2}\right) \cdot \frac{h}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= D_m 1e+111)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ (* h (* M_m (/ (* D_m 0.5) d))) (* l (/ (* d 2.0) (* M_m D_m)))))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (* (/ (* M_m (* D_m 0.5)) d) (/ (* M_m D_m) (* d 2.0))) (/ 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 tmp;
	if (D_m <= 1e+111) {
		tmp = w0 * sqrt((1.0 - ((h * (M_m * ((D_m * 0.5) / d))) / (l * ((d * 2.0) / (M_m * D_m))))));
	} else {
		tmp = w0 * sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d) * ((M_m * D_m) / (d * 2.0))) * (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) :: tmp
    if (d_m <= 1d+111) then
        tmp = w0 * sqrt((1.0d0 - ((h * (m_m * ((d_m * 0.5d0) / d))) / (l * ((d * 2.0d0) / (m_m * d_m))))))
    else
        tmp = w0 * sqrt((1.0d0 - ((((m_m * (d_m * 0.5d0)) / d) * ((m_m * d_m) / (d * 2.0d0))) * (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 tmp;
	if (D_m <= 1e+111) {
		tmp = w0 * Math.sqrt((1.0 - ((h * (M_m * ((D_m * 0.5) / d))) / (l * ((d * 2.0) / (M_m * D_m))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d) * ((M_m * D_m) / (d * 2.0))) * (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):
	tmp = 0
	if D_m <= 1e+111:
		tmp = w0 * math.sqrt((1.0 - ((h * (M_m * ((D_m * 0.5) / d))) / (l * ((d * 2.0) / (M_m * D_m))))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d) * ((M_m * D_m) / (d * 2.0))) * (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)
	tmp = 0.0
	if (D_m <= 1e+111)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * Float64(M_m * Float64(Float64(D_m * 0.5) / d))) / Float64(l * Float64(Float64(d * 2.0) / Float64(M_m * D_m)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d) * Float64(Float64(M_m * D_m) / Float64(d * 2.0))) * 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)
	tmp = 0.0;
	if (D_m <= 1e+111)
		tmp = w0 * sqrt((1.0 - ((h * (M_m * ((D_m * 0.5) / d))) / (l * ((d * 2.0) / (M_m * D_m))))));
	else
		tmp = w0 * sqrt((1.0 - ((((M_m * (D_m * 0.5)) / d) * ((M_m * D_m) / (d * 2.0))) * (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_] := If[LessEqual[D$95$m, 1e+111], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[(M$95$m * N[(N[(D$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[(d * 2.0), $MachinePrecision] / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 9.99999999999999957e110

   1. Initial program 82.4%

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

   3. Simplified 82.4%

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

      1. unpow2 82.4%

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

      2. associate-*r/ 82.0%

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

      3. clear-num 82.0%

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

      4. un-div-inv 82.0%

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

      5. *-un-lft-identity 82.0%

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

      6. times-frac 82.0%

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

      7. metadata-eval 82.0%

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

      8. times-frac 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. frac-times 88.7%

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

      2. associate-*r/ 88.7%

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

      3. frac-times 88.0%

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

      4. *-commutative 88.0%

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

   8. Applied egg-rr 88.0%

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

   if 9.99999999999999957e110 < D

   1. Initial program 77.3%

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

   3. Simplified 77.2%

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

      1. associate-*r/ 77.3%

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

      2. unpow2 77.3%

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

      3. associate-/r* 77.3%

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

      4. frac-times 50.7%

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

      5. associate-/l* 50.7%

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

      6. div-inv 50.7%

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

      7. metadata-eval 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. times-frac 77.3%

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

      2. *-commutative 77.3%

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

      3. *-commutative 77.3%

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

   8. Applied egg-rr 77.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 10^{+111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot \left(M \cdot \frac{D \cdot 0.5}{d}\right)}{\ell \cdot \frac{d \cdot 2}{M \cdot D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot \left(D \cdot 0.5\right)}{d} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{h}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.35 \cdot 10^{-212}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{D\_m}{d} \cdot \left(\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right) \cdot \left(M\_m \cdot 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 4.35e-212)
   w0
   (*
    w0
    (sqrt
     (-
      1.0
      (* (/ h l) (* (/ D_m d) (* (* M_m (* 0.5 (/ D_m d))) (* M_m 0.5)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.35e-212) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D_m / d) * ((M_m * (0.5 * (D_m / d))) * (M_m * 0.5))))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 4.35d-212) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d_m / d) * ((m_m * (0.5d0 * (d_m / d))) * (m_m * 0.5d0))))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.35e-212) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * ((D_m / d) * ((M_m * (0.5 * (D_m / d))) * (M_m * 0.5))))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 4.35e-212:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * ((D_m / d) * ((M_m * (0.5 * (D_m / d))) * (M_m * 0.5))))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 4.35e-212)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(Float64(D_m / d) * Float64(Float64(M_m * Float64(0.5 * Float64(D_m / d))) * Float64(M_m * 0.5)))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 4.35e-212)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D_m / d) * ((M_m * (0.5 * (D_m / d))) * (M_m * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 4.35e-212], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.3499999999999999e-212

   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 79.3%

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

   5. Taylor expanded in M around 0 69.9%

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

   if 4.3499999999999999e-212 < M

   1. Initial program 84.6%

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

   3. Simplified 84.7%

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

      1. unpow2 84.7%

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

      2. associate-*r/ 83.8%

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

      3. times-frac 84.7%

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

      4. associate-*r* 83.9%

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

      5. *-un-lft-identity 83.9%

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

      6. times-frac 83.9%

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

      7. metadata-eval 83.9%

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

      8. div-inv 83.9%

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

      9. metadata-eval 83.9%

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

   6. Applied egg-rr 83.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.35 \cdot 10^{-212}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(\frac{D}{d} \cdot \left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \left(M \cdot 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% 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 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in M around 0 68.1%

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

Reproduce

herbie shell --seed 2024179 
(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))))))