Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 88.5%

Time: 16.0s

Alternatives: 6

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: 81.1% 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: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+190}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(D\_m \cdot \left(\frac{M\_m \cdot \left(D\_m \cdot 0.5\right)}{d} \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right) \cdot \frac{M\_m \cdot \frac{D\_m}{2 \cdot d}}{\ell}\right)}\\ \end{array} \end{array} \]

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
 (if (<= (- 1.0 (* (pow (/ (* M_m D_m) (* 2.0 d)) 2.0) (/ h l))) 1e+190)
   (*
    w0
    (sqrt
     (-
      1.0
      (* (/ h l) (* D_m (* (/ (* M_m (* D_m 0.5)) d) (* M_m (/ 0.5 d))))))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* h (* (* M_m (* 0.5 (/ D_m d))) (/ (* M_m (/ D_m (* 2.0 d))) 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 ((1.0 - (pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 1e+190) {
		tmp = w0 * sqrt((1.0 - ((h / l) * (D_m * (((M_m * (D_m * 0.5)) / d) * (M_m * (0.5 / d)))))));
	} else {
		tmp = w0 * sqrt((1.0 - (h * ((M_m * (0.5 * (D_m / d))) * ((M_m * (D_m / (2.0 * d))) / 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 ((1.0d0 - ((((m_m * d_m) / (2.0d0 * d)) ** 2.0d0) * (h / l))) <= 1d+190) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (d_m * (((m_m * (d_m * 0.5d0)) / d) * (m_m * (0.5d0 / d)))))))
    else
        tmp = w0 * sqrt((1.0d0 - (h * ((m_m * (0.5d0 * (d_m / d))) * ((m_m * (d_m / (2.0d0 * d))) / 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 ((1.0 - (Math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 1e+190) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * (D_m * (((M_m * (D_m * 0.5)) / d) * (M_m * (0.5 / d)))))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * ((M_m * (0.5 * (D_m / d))) * ((M_m * (D_m / (2.0 * d))) / 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 (1.0 - (math.pow(((M_m * D_m) / (2.0 * d)), 2.0) * (h / l))) <= 1e+190:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * (D_m * (((M_m * (D_m * 0.5)) / d) * (M_m * (0.5 / d)))))))
	else:
		tmp = w0 * math.sqrt((1.0 - (h * ((M_m * (0.5 * (D_m / d))) * ((M_m * (D_m / (2.0 * d))) / 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 (Float64(1.0 - Float64((Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 1e+190)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * Float64(D_m * Float64(Float64(Float64(M_m * Float64(D_m * 0.5)) / d) * Float64(M_m * Float64(0.5 / d))))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(M_m * Float64(0.5 * Float64(D_m / d))) * Float64(Float64(M_m * Float64(D_m / Float64(2.0 * d))) / 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 ((1.0 - ((((M_m * D_m) / (2.0 * d)) ^ 2.0) * (h / l))) <= 1e+190)
		tmp = w0 * sqrt((1.0 - ((h / l) * (D_m * (((M_m * (D_m * 0.5)) / d) * (M_m * (0.5 / d)))))));
	else
		tmp = w0 * sqrt((1.0 - (h * ((M_m * (0.5 * (D_m / d))) * ((M_m * (D_m / (2.0 * d))) / 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[N[(1.0 - N[(N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+190], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(D$95$m * N[(N[(N[(M$95$m * N[(D$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l))) < 1.0000000000000001e190

   1. Initial program 100.0%

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

   3. Simplified 98.9%

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

      1. unpow2 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r* 97.8%

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

      4. *-commutative 97.8%

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

      5. associate-*l/ 97.8%

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

      6. associate-*r/ 96.6%

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

      7. times-frac 97.8%

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

      8. associate-/l* 96.6%

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

      9. div-inv 96.6%

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

      10. associate-/r* 96.6%

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

      11. metadata-eval 96.6%

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

      12. div-inv 96.6%

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

      13. metadata-eval 96.6%

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

      14. associate-/l* 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. associate-*r* 97.8%

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

      2. associate-/l* 97.8%

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

      3. associate-*l* 97.8%

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

   8. Applied egg-rr 97.8%

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

   if 1.0000000000000001e190 < (-.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64)) (/.f64 h l)))

   1. Initial program 47.2%

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

   3. Simplified 50.7%

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

      1. *-commutative 50.7%

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

      2. associate-*l/ 47.2%

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

      3. associate-*r/ 50.7%

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

      4. times-frac 47.2%

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

      5. unpow2 47.2%

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

      6. associate-/r* 47.2%

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

      7. frac-times 43.3%

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

      8. div-inv 43.3%

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

      9. *-commutative 43.3%

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

      10. metadata-eval 43.3%

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

      11. associate-*r* 43.3%

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

      12. *-commutative 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. associate-*r/ 55.9%

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

      2. times-frac 62.3%

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

      3. associate-*r* 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. *-commutative 62.3%

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

   8. Applied egg-rr 62.3%

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

      1. pow1 62.3%

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

      2. *-commutative 62.3%

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

      3. associate-/l* 62.3%

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

      4. associate-*r* 60.0%

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

      5. associate-/l* 64.6%

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

   10. Applied egg-rr 64.6%

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

       1. unpow1 64.6%

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

       2. associate-*l* 63.5%

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

       3. associate-*r/ 63.5%

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

       4. *-commutative 63.5%

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

       5. associate-*r/ 63.5%

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

   12. Simplified 63.5%

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

       1. associate-/l* 63.5%

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

       2. associate-*r* 64.6%

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

       3. associate-*r/ 64.6%

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

       4. associate-*r/ 60.0%

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

   14. Applied egg-rr 60.0%

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

       1. associate-/l* 67.2%

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

       2. associate-/l* 67.2%

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

       3. associate-/l* 73.5%

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

       4. *-commutative 73.5%

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

   16. Simplified 73.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+190}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot \left(D \cdot \left(\frac{M \cdot \left(D \cdot 0.5\right)}{d} \cdot \left(M \cdot \frac{0.5}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right) \cdot \frac{M \cdot \frac{D}{2 \cdot d}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-294}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\left(M\_m \cdot \left(0.5 \cdot \frac{D\_m}{d}\right)\right) \cdot \frac{M\_m \cdot \frac{D\_m}{2 \cdot d}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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-294)
   (*
    w0
    (sqrt
     (-
      1.0
      (* h (* (* M_m (* 0.5 (/ D_m d))) (/ (* M_m (/ D_m (* 2.0 d))) 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-294) {
		tmp = w0 * sqrt((1.0 - (h * ((M_m * (0.5 * (D_m / d))) * ((M_m * (D_m / (2.0 * d))) / 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-294)) then
        tmp = w0 * sqrt((1.0d0 - (h * ((m_m * (0.5d0 * (d_m / d))) * ((m_m * (d_m / (2.0d0 * d))) / 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-294) {
		tmp = w0 * Math.sqrt((1.0 - (h * ((M_m * (0.5 * (D_m / d))) * ((M_m * (D_m / (2.0 * d))) / 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-294:
		tmp = w0 * math.sqrt((1.0 - (h * ((M_m * (0.5 * (D_m / d))) * ((M_m * (D_m / (2.0 * d))) / 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-294)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(M_m * Float64(0.5 * Float64(D_m / d))) * Float64(Float64(M_m * Float64(D_m / Float64(2.0 * d))) / 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-294)
		tmp = w0 * sqrt((1.0 - (h * ((M_m * (0.5 * (D_m / d))) * ((M_m * (D_m / (2.0 * d))) / 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-294], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(M$95$m * N[(0.5 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.00000000000000002e-294

   1. Initial program 77.9%

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

   3. Simplified 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-*l/ 77.9%

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

      3. associate-*r/ 80.0%

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

      4. times-frac 77.9%

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

      5. unpow2 77.9%

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

      6. associate-/r* 77.9%

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

      7. frac-times 70.0%

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

      8. div-inv 70.0%

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

      9. *-commutative 70.0%

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

      10. metadata-eval 70.0%

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

      11. associate-*r* 70.0%

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

      12. *-commutative 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. associate-*r/ 71.8%

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

      2. times-frac 81.1%

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

      3. associate-*r* 81.1%

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

      4. *-commutative 81.1%

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

      5. *-commutative 81.1%

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

      6. *-commutative 81.1%

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

   8. Applied egg-rr 81.1%

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

      1. pow1 81.1%

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

      2. *-commutative 81.1%

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

      3. associate-/l* 81.1%

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

      4. associate-*r* 79.8%

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

      5. associate-/l* 82.5%

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

   10. Applied egg-rr 82.5%

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

       1. unpow1 82.5%

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

       2. associate-*l* 81.9%

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

       3. associate-*r/ 81.9%

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

       4. *-commutative 81.9%

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

       5. associate-*r/ 81.9%

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

   12. Simplified 81.9%

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

       1. associate-/l* 81.8%

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

       2. associate-*r* 82.5%

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

       3. associate-*r/ 82.5%

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

       4. associate-*r/ 79.8%

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

   14. Applied egg-rr 79.8%

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

       1. associate-/l* 82.6%

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

       2. associate-/l* 82.6%

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

       3. associate-/l* 86.3%

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

       4. *-commutative 86.3%

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

   16. Simplified 86.3%

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

   if -1.00000000000000002e-294 < (/.f64 h l)

   1. Initial program 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in D around 0 96.9%

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

Alternative 3: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.62 \cdot 10^{+94}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D\_m \cdot h}{\ell} \cdot \left(\left(M\_m \cdot \frac{D\_m \cdot -0.5}{d}\right) \cdot \frac{M\_m \cdot 0.5}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= d 1.62e+94)
   (*
    w0
    (sqrt
     (+
      1.0
      (* (/ (* D_m h) l) (* (* M_m (/ (* D_m -0.5) d)) (/ (* M_m 0.5) d))))))
   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 (d <= 1.62e+94) {
		tmp = w0 * sqrt((1.0 + (((D_m * h) / l) * ((M_m * ((D_m * -0.5) / d)) * ((M_m * 0.5) / d)))));
	} 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 (d <= 1.62d+94) then
        tmp = w0 * sqrt((1.0d0 + (((d_m * h) / l) * ((m_m * ((d_m * (-0.5d0)) / d)) * ((m_m * 0.5d0) / d)))))
    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 (d <= 1.62e+94) {
		tmp = w0 * Math.sqrt((1.0 + (((D_m * h) / l) * ((M_m * ((D_m * -0.5) / d)) * ((M_m * 0.5) / d)))));
	} 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 d <= 1.62e+94:
		tmp = w0 * math.sqrt((1.0 + (((D_m * h) / l) * ((M_m * ((D_m * -0.5) / d)) * ((M_m * 0.5) / d)))))
	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 (d <= 1.62e+94)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D_m * h) / l) * Float64(Float64(M_m * Float64(Float64(D_m * -0.5) / d)) * Float64(Float64(M_m * 0.5) / d))))));
	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 (d <= 1.62e+94)
		tmp = w0 * sqrt((1.0 + (((D_m * h) / l) * ((M_m * ((D_m * -0.5) / d)) * ((M_m * 0.5) / d)))));
	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[d, 1.62e+94], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D$95$m * h), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m * N[(N[(D$95$m * -0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.61999999999999997e94

   1. Initial program 83.3%

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

   3. Simplified 83.4%

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

      1. unpow2 83.4%

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

      2. *-commutative 83.4%

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

      3. associate-*r* 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-*l/ 81.5%

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

      6. associate-*r/ 81.4%

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

      7. times-frac 81.5%

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

      8. associate-/l* 81.4%

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

      9. div-inv 81.4%

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

      10. associate-/r* 81.4%

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

      11. metadata-eval 81.4%

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

      12. div-inv 81.4%

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

      13. metadata-eval 81.4%

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

      14. associate-/l* 81.4%

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

   6. Applied egg-rr 81.4%

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

      1. *-un-lft-identity 81.4%

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

      2. associate-*l* 80.8%

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

      3. associate-*l* 80.4%

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

   8. Applied egg-rr 80.4%

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

      1. *-lft-identity 80.4%

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

      2. sub-neg 80.4%

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

      3. *-commutative 80.4%

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

      4. distribute-rgt-neg-in 80.4%

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

      5. associate-*r/ 79.6%

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

      6. associate-*r* 80.0%

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

      7. distribute-lft-neg-in 80.0%

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

   10. Simplified 80.0%

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

   if 1.61999999999999997e94 < d

   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 82.8%

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

   5. Taylor expanded in D around 0 91.0%

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

Alternative 4: 71.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.4 \cdot 10^{+40}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(w0 \cdot \frac{h \cdot {d}^{-2}}{\ell}\right)\right)\\ \end{array} \end{array} \]

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 4.4e+40)
   w0
   (* -0.125 (* (* (* M_m D_m) (* M_m D_m)) (* w0 (/ (* h (pow d -2.0)) l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.4e+40) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * ((h * pow(d, -2.0)) / 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 (m_m <= 4.4d+40) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((m_m * d_m) * (m_m * d_m)) * (w0 * ((h * (d ** (-2.0d0))) / 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 (M_m <= 4.4e+40) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * ((h * Math.pow(d, -2.0)) / 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 M_m <= 4.4e+40:
		tmp = w0
	else:
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * ((h * math.pow(d, -2.0)) / 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 (M_m <= 4.4e+40)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(w0 * Float64(Float64(h * (d ^ -2.0)) / 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 (M_m <= 4.4e+40)
		tmp = w0;
	else
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * (w0 * ((h * (d ^ -2.0)) / 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[M$95$m, 4.4e+40], w0, N[(-0.125 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0 * N[(N[(h * N[Power[d, -2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.3999999999999998e40

   1. Initial program 85.9%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in D around 0 75.4%

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

   if 4.3999999999999998e40 < M

   1. Initial program 69.2%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in D around 0 30.2%

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

      1. +-commutative 30.2%

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

      2. fma-define 30.2%

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

      3. associate-*r* 32.5%

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

      4. unpow2 32.5%

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

      5. unpow2 32.5%

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

      6. swap-sqr 46.0%

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

      7. unpow2 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in D around inf 17.9%

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

      1. associate-*r* 18.1%

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

      2. *-commutative 18.1%

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

      3. unpow2 18.1%

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

      4. unpow2 18.1%

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

      5. swap-sqr 21.7%

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

      6. unpow2 21.7%

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

      7. *-commutative 21.7%

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

      8. *-commutative 21.7%

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

      9. associate-*r/ 21.7%

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

      10. *-commutative 21.7%

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

      11. times-frac 19.7%

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

   10. Simplified 19.7%

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

       1. unpow2 19.7%

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

       2. *-commutative 19.7%

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

       3. *-commutative 19.7%

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

   12. Applied egg-rr 19.7%

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

       1. associate-*l/ 21.9%

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

       2. div-inv 21.9%

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

       3. pow-flip 21.9%

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

       4. metadata-eval 21.9%

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

   14. Applied egg-rr 21.9%

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

       1. associate-/l* 22.0%

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

   16. Simplified 22.0%

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

Alternative 5: 71.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.4 \cdot 10^{+40}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot D\_m\right)\right) \cdot \left(\frac{w0}{\ell} \cdot \frac{h}{d \cdot d}\right)\right)\\ \end{array} \end{array} \]

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 4.4e+40)
   w0
   (* -0.125 (* (* (* M_m D_m) (* M_m D_m)) (* (/ w0 l) (/ h (* d d)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.4e+40) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * ((w0 / l) * (h / (d * d))));
	}
	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 <= 4.4d+40) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((m_m * d_m) * (m_m * d_m)) * ((w0 / l) * (h / (d * d))))
    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 <= 4.4e+40) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * ((w0 / l) * (h / (d * d))));
	}
	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 <= 4.4e+40:
		tmp = w0
	else:
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * ((w0 / l) * (h / (d * d))))
	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 <= 4.4e+40)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * D_m)) * Float64(Float64(w0 / l) * Float64(h / Float64(d * d)))));
	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 <= 4.4e+40)
		tmp = w0;
	else
		tmp = -0.125 * (((M_m * D_m) * (M_m * D_m)) * ((w0 / l) * (h / (d * d))));
	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, 4.4e+40], w0, N[(-0.125 * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 / l), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.3999999999999998e40

   1. Initial program 85.9%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in D around 0 75.4%

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

   if 4.3999999999999998e40 < M

   1. Initial program 69.2%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in D around 0 30.2%

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

      1. +-commutative 30.2%

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

      2. fma-define 30.2%

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

      3. associate-*r* 32.5%

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

      4. unpow2 32.5%

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

      5. unpow2 32.5%

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

      6. swap-sqr 46.0%

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

      7. unpow2 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in D around inf 17.9%

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

      1. associate-*r* 18.1%

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

      2. *-commutative 18.1%

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

      3. unpow2 18.1%

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

      4. unpow2 18.1%

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

      5. swap-sqr 21.7%

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

      6. unpow2 21.7%

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

      7. *-commutative 21.7%

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

      8. *-commutative 21.7%

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

      9. associate-*r/ 21.7%

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

      10. *-commutative 21.7%

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

      11. times-frac 19.7%

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

   10. Simplified 19.7%

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

       1. unpow2 19.7%

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

       2. *-commutative 19.7%

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

       3. *-commutative 19.7%

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

   12. Applied egg-rr 19.7%

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

       1. unpow2 19.7%

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

   14. Applied egg-rr 19.7%

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

Alternative 6: 68.5% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Simplified 83.3%

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

5. Taylor expanded in D around 0 70.4%

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

Reproduce

herbie shell --seed 2024135 
(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))))))