Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.8% → 86.9%

Time: 17.8s

Alternatives: 12

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: 80.8% 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: 86.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M\_m \cdot D\_m}{2 \cdot d\_m} \leq 5 \cdot 10^{+65}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\frac{h}{\frac{-4}{M\_m}}}{\frac{d\_m}{M\_m \cdot D\_m}}}{\ell} \cdot \frac{D\_m}{d\_m}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\frac{M\_m \cdot \left(M\_m \cdot D\_m\right)}{d\_m \cdot -4}}{\ell} \cdot \frac{D\_m}{\frac{1}{h}}}{d\_m}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ (* M_m D_m) (* 2.0 d_m)) 5e+65)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (/ (/ h (/ -4.0 M_m)) (/ d_m (* M_m D_m))) l) (/ D_m d_m)))))
   (*
    w0
    (sqrt
     (+
      1.0
      (/
       (* (/ (/ (* M_m (* M_m D_m)) (* d_m -4.0)) l) (/ D_m (/ 1.0 h)))
       d_m))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+65) {
		tmp = w0 * sqrt((1.0 + ((((h / (-4.0 / M_m)) / (d_m / (M_m * D_m))) / l) * (D_m / d_m))));
	} else {
		tmp = w0 * sqrt((1.0 + (((((M_m * (M_m * D_m)) / (d_m * -4.0)) / l) * (D_m / (1.0 / h))) / d_m)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (((m_m * d_m) / (2.0d0 * d_m_1)) <= 5d+65) then
        tmp = w0 * sqrt((1.0d0 + ((((h / ((-4.0d0) / m_m)) / (d_m_1 / (m_m * d_m))) / l) * (d_m / d_m_1))))
    else
        tmp = w0 * sqrt((1.0d0 + (((((m_m * (m_m * d_m)) / (d_m_1 * (-4.0d0))) / l) * (d_m / (1.0d0 / h))) / d_m_1)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+65) {
		tmp = w0 * Math.sqrt((1.0 + ((((h / (-4.0 / M_m)) / (d_m / (M_m * D_m))) / l) * (D_m / d_m))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((((M_m * (M_m * D_m)) / (d_m * -4.0)) / l) * (D_m / (1.0 / h))) / d_m)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if ((M_m * D_m) / (2.0 * d_m)) <= 5e+65:
		tmp = w0 * math.sqrt((1.0 + ((((h / (-4.0 / M_m)) / (d_m / (M_m * D_m))) / l) * (D_m / d_m))))
	else:
		tmp = w0 * math.sqrt((1.0 + (((((M_m * (M_m * D_m)) / (d_m * -4.0)) / l) * (D_m / (1.0 / h))) / d_m)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) <= 5e+65)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(h / Float64(-4.0 / M_m)) / Float64(d_m / Float64(M_m * D_m))) / l) * Float64(D_m / d_m)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(Float64(M_m * Float64(M_m * D_m)) / Float64(d_m * -4.0)) / l) * Float64(D_m / Float64(1.0 / h))) / d_m))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (((M_m * D_m) / (2.0 * d_m)) <= 5e+65)
		tmp = w0 * sqrt((1.0 + ((((h / (-4.0 / M_m)) / (d_m / (M_m * D_m))) / l) * (D_m / d_m))));
	else
		tmp = w0 * sqrt((1.0 + (((((M_m * (M_m * D_m)) / (d_m * -4.0)) / l) * (D_m / (1.0 / h))) / d_m)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d$95$m), $MachinePrecision]), $MachinePrecision], 5e+65], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(h / N[(-4.0 / M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d$95$m / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(N[(M$95$m * N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * -4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) < 4.99999999999999973e65

   1. Initial program 85.7%

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

   3. Simplified 79.3%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D\right)}{\ell \cdot d}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right) \cdot D}{\ell \cdot d}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell} \cdot \frac{D}{d}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell}\right), \left(\frac{D}{d}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 89.6%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}}}{\ell} \cdot \frac{D}{d}}} \]
   7. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{h}{\frac{-4}{M} \cdot \frac{d}{D \cdot M}}\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{h}{\frac{-4}{M}}}{\frac{d}{D \cdot M}}\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{h}{\frac{-4}{M}}\right), \left(\frac{d}{D \cdot M}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \left(\frac{-4}{M}\right)\right), \left(\frac{d}{D \cdot M}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(-4, M\right)\right), \left(\frac{d}{D \cdot M}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(-4, M\right)\right), \mathsf{/.f64}\left(d, \left(D \cdot M\right)\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(-4, M\right)\right), \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 88.1%

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(-4, M\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 88.1%

      \[\leadsto w0 \cdot \sqrt{1 + \frac{\color{blue}{\frac{\frac{h}{\frac{-4}{M}}}{\frac{d}{M \cdot D}}}}{\ell} \cdot \frac{D}{d}} \]

   if 4.99999999999999973e65 < (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d))

   1. Initial program 66.2%

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

   3. Simplified 57.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right) \cdot \frac{h}{\ell}\right), d\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right) \cdot \frac{1}{\frac{\ell}{h}}\right), d\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\frac{\ell}{h}}\right), d\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D}{\frac{\ell}{h}}\right), d\right)\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D}{\ell \cdot \frac{1}{h}}\right), d\right)\right)\right)\right) \]

      6. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell} \cdot \frac{D}{\frac{1}{h}}\right), d\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell}\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right), \ell\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      9. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot \left(M \cdot D\right)}{-4 \cdot d}\right), \ell\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \ell\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \ell\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(D \cdot M\right)\right), \left(-4 \cdot d\right)\right), \ell\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(-4 \cdot d\right)\right), \ell\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(d \cdot -4\right)\right), \ell\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \ell\right), \left(\frac{D}{\frac{1}{h}}\right)\right), d\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \ell\right), \mathsf{/.f64}\left(D, \left(\frac{1}{h}\right)\right)\right), d\right)\right)\right)\right) \]

      17. /-lowering-/.f64 64.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \ell\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(1, h\right)\right)\right), d\right)\right)\right)\right) \]

   6. Applied egg-rr 64.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 5 \cdot 10^{+65}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\frac{h}{\frac{-4}{M}}}{\frac{d}{M \cdot D}}}{\ell} \cdot \frac{D}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d \cdot -4}}{\ell} \cdot \frac{D}{\frac{1}{h}}}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-278}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\frac{h}{\frac{-4}{M\_m}}}{\frac{d\_m}{M\_m \cdot D\_m}}}{\ell} \cdot \frac{D\_m}{d\_m}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -4e-278)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (/ (/ h (/ -4.0 M_m)) (/ d_m (* M_m D_m))) l) (/ D_m d_m)))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -4e-278) {
		tmp = w0 * sqrt((1.0 + ((((h / (-4.0 / M_m)) / (d_m / (M_m * D_m))) / l) * (D_m / d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if ((h / l) <= (-4d-278)) then
        tmp = w0 * sqrt((1.0d0 + ((((h / ((-4.0d0) / m_m)) / (d_m_1 / (m_m * d_m))) / l) * (d_m / d_m_1))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -4e-278) {
		tmp = w0 * Math.sqrt((1.0 + ((((h / (-4.0 / M_m)) / (d_m / (M_m * D_m))) / l) * (D_m / d_m))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -4e-278:
		tmp = w0 * math.sqrt((1.0 + ((((h / (-4.0 / M_m)) / (d_m / (M_m * D_m))) / l) * (D_m / d_m))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -4e-278)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(h / Float64(-4.0 / M_m)) / Float64(d_m / Float64(M_m * D_m))) / l) * Float64(D_m / d_m)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -4e-278)
		tmp = w0 * sqrt((1.0 + ((((h / (-4.0 / M_m)) / (d_m / (M_m * D_m))) / l) * (D_m / d_m))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -4e-278], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(h / N[(-4.0 / M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d$95$m / N[(M$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -3.99999999999999975e-278

   1. Initial program 79.5%

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

   3. Simplified 73.7%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D\right)}{\ell \cdot d}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right) \cdot D}{\ell \cdot d}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell} \cdot \frac{D}{d}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell}\right), \left(\frac{D}{d}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 79.6%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}}}{\ell} \cdot \frac{D}{d}}} \]
   7. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{h}{\frac{-4}{M} \cdot \frac{d}{D \cdot M}}\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{h}{\frac{-4}{M}}}{\frac{d}{D \cdot M}}\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{h}{\frac{-4}{M}}\right), \left(\frac{d}{D \cdot M}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \left(\frac{-4}{M}\right)\right), \left(\frac{d}{D \cdot M}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(-4, M\right)\right), \left(\frac{d}{D \cdot M}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(-4, M\right)\right), \mathsf{/.f64}\left(d, \left(D \cdot M\right)\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(-4, M\right)\right), \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 77.3%

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(h, \mathsf{/.f64}\left(-4, M\right)\right), \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(M, D\right)\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 77.3%

      \[\leadsto w0 \cdot \sqrt{1 + \frac{\color{blue}{\frac{\frac{h}{\frac{-4}{M}}}{\frac{d}{M \cdot D}}}}{\ell} \cdot \frac{D}{d}} \]

   if -3.99999999999999975e-278 < (/.f64 h l)

   1. Initial program 85.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 95.5%

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

Alternative 3: 87.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;d\_m \leq 520000000:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M\_m \cdot D\_m\right) \cdot \left(\frac{h}{-4} \cdot \frac{M\_m}{d\_m}\right)}{d\_m} \cdot \frac{D\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D\_m}{d\_m} \cdot \frac{\left(D\_m \cdot -0.25\right) \cdot \frac{h \cdot \left(M\_m \cdot M\_m\right)}{d\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= d_m 520000000.0)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (* (* M_m D_m) (* (/ h -4.0) (/ M_m d_m))) d_m) (/ D_m l)))))
   (*
    w0
    (sqrt
     (+
      1.0
      (* (/ D_m d_m) (/ (* (* D_m -0.25) (/ (* h (* M_m M_m)) d_m)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 520000000.0) {
		tmp = w0 * sqrt((1.0 + ((((M_m * D_m) * ((h / -4.0) * (M_m / d_m))) / d_m) * (D_m / l))));
	} else {
		tmp = w0 * sqrt((1.0 + ((D_m / d_m) * (((D_m * -0.25) * ((h * (M_m * M_m)) / d_m)) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (d_m_1 <= 520000000.0d0) then
        tmp = w0 * sqrt((1.0d0 + ((((m_m * d_m) * ((h / (-4.0d0)) * (m_m / d_m_1))) / d_m_1) * (d_m / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((d_m / d_m_1) * (((d_m * (-0.25d0)) * ((h * (m_m * m_m)) / d_m_1)) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (d_m <= 520000000.0) {
		tmp = w0 * Math.sqrt((1.0 + ((((M_m * D_m) * ((h / -4.0) * (M_m / d_m))) / d_m) * (D_m / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D_m / d_m) * (((D_m * -0.25) * ((h * (M_m * M_m)) / d_m)) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if d_m <= 520000000.0:
		tmp = w0 * math.sqrt((1.0 + ((((M_m * D_m) * ((h / -4.0) * (M_m / d_m))) / d_m) * (D_m / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((D_m / d_m) * (((D_m * -0.25) * ((h * (M_m * M_m)) / d_m)) / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (d_m <= 520000000.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(M_m * D_m) * Float64(Float64(h / -4.0) * Float64(M_m / d_m))) / d_m) * Float64(D_m / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m / d_m) * Float64(Float64(Float64(D_m * -0.25) * Float64(Float64(h * Float64(M_m * M_m)) / d_m)) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (d_m <= 520000000.0)
		tmp = w0 * sqrt((1.0 + ((((M_m * D_m) * ((h / -4.0) * (M_m / d_m))) / d_m) * (D_m / l))));
	else
		tmp = w0 * sqrt((1.0 + ((D_m / d_m) * (((D_m * -0.25) * ((h * (M_m * M_m)) / d_m)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[d$95$m, 520000000.0], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(N[(h / -4.0), $MachinePrecision] * N[(M$95$m / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(N[(D$95$m * -0.25), $MachinePrecision] * N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.2e8

   1. Initial program 81.8%

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

   3. Simplified 76.9%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell}}{d}\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d \cdot \ell}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D\right)}{d \cdot \ell}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right) \cdot D}{d \cdot \ell}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d} \cdot \frac{D}{\ell}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right), \left(\frac{D}{\ell}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 78.4%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}}}{d} \cdot \frac{D}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{h}{-4} \cdot \frac{M}{\frac{d}{D \cdot M}}\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{h}{-4} \cdot \left(\frac{M}{d} \cdot \left(D \cdot M\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{h}{-4} \cdot \frac{M}{d}\right) \cdot \left(D \cdot M\right)\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{h}{-4} \cdot \frac{M}{d}\right), \left(D \cdot M\right)\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{h}{-4}\right), \left(\frac{M}{d}\right)\right), \left(D \cdot M\right)\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, -4\right), \left(\frac{M}{d}\right)\right), \left(D \cdot M\right)\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, -4\right), \mathsf{/.f64}\left(M, d\right)\right), \left(D \cdot M\right)\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, -4\right), \mathsf{/.f64}\left(M, d\right)\right), \left(M \cdot D\right)\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 79.0%

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, -4\right), \mathsf{/.f64}\left(M, d\right)\right), \mathsf{*.f64}\left(M, D\right)\right), d\right), \mathsf{/.f64}\left(D, \ell\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 79.0%

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

   if 5.2e8 < d

   1. Initial program 84.1%

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

   3. Simplified 72.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D\right)}{\ell \cdot d}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right) \cdot D}{\ell \cdot d}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell} \cdot \frac{D}{d}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell}\right), \left(\frac{D}{d}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 89.6%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}}}{\ell} \cdot \frac{D}{d}}} \]

   7. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \frac{D \cdot \left({M}^{2} \cdot h\right)}{d}\right)}, \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{d}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{4} \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{d}\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4} \cdot D\right), \left(\frac{{M}^{2} \cdot h}{d}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \left(\frac{{M}^{2} \cdot h}{d}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), d\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 81.8%

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

   9. Simplified 81.8%

      \[\leadsto w0 \cdot \sqrt{1 + \frac{\color{blue}{\left(-0.25 \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{d}}}{\ell} \cdot \frac{D}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 520000000:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot D\right) \cdot \left(\frac{h}{-4} \cdot \frac{M}{d}\right)}{d} \cdot \frac{D}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{d} \cdot \frac{\left(D \cdot -0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.5 \cdot 10^{-163}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D\_m}{d\_m} \cdot \frac{\left(D\_m \cdot -0.25\right) \cdot \frac{h \cdot \left(M\_m \cdot M\_m\right)}{d\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 4.5e-163)
   w0
   (*
    w0
    (sqrt
     (+
      1.0
      (* (/ D_m d_m) (/ (* (* D_m -0.25) (/ (* h (* M_m M_m)) d_m)) l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 4.5e-163) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 + ((D_m / d_m) * (((D_m * -0.25) * ((h * (M_m * M_m)) / d_m)) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (m_m <= 4.5d-163) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 + ((d_m / d_m_1) * (((d_m * (-0.25d0)) * ((h * (m_m * m_m)) / d_m_1)) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 4.5e-163) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D_m / d_m) * (((D_m * -0.25) * ((h * (M_m * M_m)) / d_m)) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 4.5e-163:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 + ((D_m / d_m) * (((D_m * -0.25) * ((h * (M_m * M_m)) / d_m)) / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 4.5e-163)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D_m / d_m) * Float64(Float64(Float64(D_m * -0.25) * Float64(Float64(h * Float64(M_m * M_m)) / d_m)) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 4.5e-163)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 + ((D_m / d_m) * (((D_m * -0.25) * ((h * (M_m * M_m)) / d_m)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 4.5e-163], w0, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(N[(D$95$m * -0.25), $MachinePrecision] * N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.4999999999999997e-163

   1. Initial program 83.6%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 77.3%

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

   if 4.4999999999999997e-163 < M

   1. Initial program 80.7%

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

   3. Simplified 71.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{\ell \cdot d}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D\right)}{\ell \cdot d}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right) \cdot D}{\ell \cdot d}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell} \cdot \frac{D}{d}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\ell}\right), \left(\frac{D}{d}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 81.8%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{h}{\frac{-4}{\frac{M}{\frac{d}{D \cdot M}}}}}{\ell} \cdot \frac{D}{d}}} \]

   7. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \frac{D \cdot \left({M}^{2} \cdot h\right)}{d}\right)}, \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{d}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{4} \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{d}\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4} \cdot D\right), \left(\frac{{M}^{2} \cdot h}{d}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \left(\frac{{M}^{2} \cdot h}{d}\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot h\right), d\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), d\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), d\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), d\right)\right), \ell\right), \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

   9. Simplified 69.2%

      \[\leadsto w0 \cdot \sqrt{1 + \frac{\color{blue}{\left(-0.25 \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{d}}}{\ell} \cdot \frac{D}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.5 \cdot 10^{-163}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{d} \cdot \frac{\left(D \cdot -0.25\right) \cdot \frac{h \cdot \left(M \cdot M\right)}{d}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1.8 \cdot 10^{-256}:\\ \;\;\;\;w0 + \frac{D\_m}{d\_m} \cdot \frac{D\_m \cdot \left(\left(M\_m \cdot \left(M\_m \cdot h\right)\right) \cdot \left(w0 \cdot -0.125\right)\right)}{d\_m \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -1.8e-256)
   (+
    w0
    (* (/ D_m d_m) (/ (* D_m (* (* M_m (* M_m h)) (* w0 -0.125))) (* d_m l))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1.8e-256) {
		tmp = w0 + ((D_m / d_m) * ((D_m * ((M_m * (M_m * h)) * (w0 * -0.125))) / (d_m * l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if ((h / l) <= (-1.8d-256)) then
        tmp = w0 + ((d_m / d_m_1) * ((d_m * ((m_m * (m_m * h)) * (w0 * (-0.125d0)))) / (d_m_1 * l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1.8e-256) {
		tmp = w0 + ((D_m / d_m) * ((D_m * ((M_m * (M_m * h)) * (w0 * -0.125))) / (d_m * l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -1.8e-256:
		tmp = w0 + ((D_m / d_m) * ((D_m * ((M_m * (M_m * h)) * (w0 * -0.125))) / (d_m * l)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -1.8e-256)
		tmp = Float64(w0 + Float64(Float64(D_m / d_m) * Float64(Float64(D_m * Float64(Float64(M_m * Float64(M_m * h)) * Float64(w0 * -0.125))) / Float64(d_m * l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -1.8e-256)
		tmp = w0 + ((D_m / d_m) * ((D_m * ((M_m * (M_m * h)) * (w0 * -0.125))) / (d_m * l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -1.8e-256], N[(w0 + N[(N[(D$95$m / d$95$m), $MachinePrecision] * N[(N[(D$95$m * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] * N[(w0 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.8000000000000001e-256

   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 73.4%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

   7. Simplified 46.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\left(\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D\right) \cdot D}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\left(\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D\right) \cdot D}{\left(d \cdot \ell\right) \cdot \color{blue}{d}}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D}{d \cdot \ell} \cdot \color{blue}{\frac{D}{d}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(\frac{\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D}{d \cdot \ell}\right), \color{blue}{\left(\frac{D}{d}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D\right), \left(d \cdot \ell\right)\right), \left(\frac{\color{blue}{D}}{d}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\frac{-1}{8} \cdot w0\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{-1}{8} \cdot w0\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot h\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot \left(M \cdot h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \left(w0 \cdot \frac{-1}{8}\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), \left(d \cdot \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(d, \ell\right)\right), \left(\frac{D}{d}\right)\right)\right) \]

      17. /-lowering-/.f64 57.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(d, \ell\right)\right), \mathsf{/.f64}\left(D, \color{blue}{d}\right)\right)\right) \]

   9. Applied egg-rr 57.9%

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

   if -1.8000000000000001e-256 < (/.f64 h l)

   1. Initial program 85.4%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 94.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1.8 \cdot 10^{-256}:\\ \;\;\;\;w0 + \frac{D}{d} \cdot \frac{D \cdot \left(\left(M \cdot \left(M \cdot h\right)\right) \cdot \left(w0 \cdot -0.125\right)\right)}{d \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1.8 \cdot 10^{-256}:\\ \;\;\;\;w0 + \frac{D\_m \cdot \left(\left(M\_m \cdot \left(M\_m \cdot h\right)\right) \cdot \left(w0 \cdot -0.125\right)\right)}{d\_m} \cdot \frac{D\_m}{d\_m \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -1.8e-256)
   (+
    w0
    (* (/ (* D_m (* (* M_m (* M_m h)) (* w0 -0.125))) d_m) (/ D_m (* d_m l))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1.8e-256) {
		tmp = w0 + (((D_m * ((M_m * (M_m * h)) * (w0 * -0.125))) / d_m) * (D_m / (d_m * l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if ((h / l) <= (-1.8d-256)) then
        tmp = w0 + (((d_m * ((m_m * (m_m * h)) * (w0 * (-0.125d0)))) / d_m_1) * (d_m / (d_m_1 * l)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -1.8e-256) {
		tmp = w0 + (((D_m * ((M_m * (M_m * h)) * (w0 * -0.125))) / d_m) * (D_m / (d_m * l)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -1.8e-256:
		tmp = w0 + (((D_m * ((M_m * (M_m * h)) * (w0 * -0.125))) / d_m) * (D_m / (d_m * l)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -1.8e-256)
		tmp = Float64(w0 + Float64(Float64(Float64(D_m * Float64(Float64(M_m * Float64(M_m * h)) * Float64(w0 * -0.125))) / d_m) * Float64(D_m / Float64(d_m * l))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -1.8e-256)
		tmp = w0 + (((D_m * ((M_m * (M_m * h)) * (w0 * -0.125))) / d_m) * (D_m / (d_m * l)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -1.8e-256], N[(w0 + N[(N[(N[(D$95$m * N[(N[(M$95$m * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] * N[(w0 * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision] * N[(D$95$m / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -1.8000000000000001e-256

   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 73.4%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

   7. Simplified 46.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\left(\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D\right) \cdot D}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D}{d} \cdot \color{blue}{\frac{D}{d \cdot \ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(\frac{\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D}{d}\right), \color{blue}{\left(\frac{D}{d \cdot \ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D\right), d\right), \left(\frac{\color{blue}{D}}{d \cdot \ell}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\frac{-1}{8} \cdot w0\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot h\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot \left(M \cdot h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \left(w0 \cdot \frac{-1}{8}\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right) \]

      16. *-lowering-*.f64 57.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right) \]

   9. Applied egg-rr 57.9%

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

   if -1.8000000000000001e-256 < (/.f64 h l)

   1. Initial program 85.4%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 94.9%

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

Alternative 7: 74.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;w0 + \frac{D\_m}{d\_m \cdot \ell} \cdot \left(\left(\left(M\_m \cdot D\_m\right) \cdot \left(M\_m \cdot h\right)\right) \cdot \frac{w0 \cdot -0.125}{d\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= (/ h l) -5e-18)
   (+
    w0
    (* (/ D_m (* d_m l)) (* (* (* M_m D_m) (* M_m h)) (/ (* w0 -0.125) d_m))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-18) {
		tmp = w0 + ((D_m / (d_m * l)) * (((M_m * D_m) * (M_m * h)) * ((w0 * -0.125) / d_m)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if ((h / l) <= (-5d-18)) then
        tmp = w0 + ((d_m / (d_m_1 * l)) * (((m_m * d_m) * (m_m * h)) * ((w0 * (-0.125d0)) / d_m_1)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if ((h / l) <= -5e-18) {
		tmp = w0 + ((D_m / (d_m * l)) * (((M_m * D_m) * (M_m * h)) * ((w0 * -0.125) / d_m)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if (h / l) <= -5e-18:
		tmp = w0 + ((D_m / (d_m * l)) * (((M_m * D_m) * (M_m * h)) * ((w0 * -0.125) / d_m)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (Float64(h / l) <= -5e-18)
		tmp = Float64(w0 + Float64(Float64(D_m / Float64(d_m * l)) * Float64(Float64(Float64(M_m * D_m) * Float64(M_m * h)) * Float64(Float64(w0 * -0.125) / d_m))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if ((h / l) <= -5e-18)
		tmp = w0 + ((D_m / (d_m * l)) * (((M_m * D_m) * (M_m * h)) * ((w0 * -0.125) / d_m)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-18], N[(w0 + N[(N[(D$95$m / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * D$95$m), $MachinePrecision] * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * -0.125), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.00000000000000036e-18

   1. Initial program 77.1%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

   7. Simplified 41.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\left(\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D\right) \cdot D}{\color{blue}{d} \cdot \left(d \cdot \ell\right)}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D}{d} \cdot \color{blue}{\frac{D}{d \cdot \ell}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(\frac{\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D}{d}\right), \color{blue}{\left(\frac{D}{d \cdot \ell}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right) \cdot D\right), d\right), \left(\frac{\color{blue}{D}}{d \cdot \ell}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\frac{-1}{8} \cdot \left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\frac{-1}{8} \cdot w0\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot h\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot \left(M \cdot h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \left(\frac{-1}{8} \cdot w0\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \left(w0 \cdot \frac{-1}{8}\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right) \]

      16. *-lowering-*.f64 52.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(w0, \frac{-1}{8}\right)\right)\right), d\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right) \]

   9. Applied egg-rr 52.9%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(\frac{\left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \cdot \left(w0 \cdot \frac{-1}{8}\right)}{d}\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(\left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \cdot \frac{w0 \cdot \frac{-1}{8}}{d}\right), \mathsf{/.f64}\left(\color{blue}{D}, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right), \left(\frac{w0 \cdot \frac{-1}{8}}{d}\right)\right), \mathsf{/.f64}\left(\color{blue}{D}, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(D \cdot M\right) \cdot \left(M \cdot h\right)\right), \left(\frac{w0 \cdot \frac{-1}{8}}{d}\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(D \cdot M\right), \left(M \cdot h\right)\right), \left(\frac{w0 \cdot \frac{-1}{8}}{d}\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(M \cdot h\right)\right), \left(\frac{w0 \cdot \frac{-1}{8}}{d}\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(M \cdot h\right)\right), \left(\frac{w0 \cdot \frac{-1}{8}}{d}\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \left(h \cdot M\right)\right), \left(\frac{w0 \cdot \frac{-1}{8}}{d}\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(h, M\right)\right), \left(\frac{w0 \cdot \frac{-1}{8}}{d}\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(h, M\right)\right), \mathsf{/.f64}\left(\left(w0 \cdot \frac{-1}{8}\right), d\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

       11. *-lowering-*.f64 60.8%

           \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, D\right), \mathsf{*.f64}\left(h, M\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \frac{-1}{8}\right), d\right)\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right) \]

   11. Applied egg-rr 60.8%

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

   if -5.00000000000000036e-18 < (/.f64 h l)

   1. Initial program 85.3%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 88.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-18}:\\ \;\;\;\;w0 + \frac{D}{d \cdot \ell} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot h\right)\right) \cdot \frac{w0 \cdot -0.125}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 5 \cdot 10^{+89}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(D\_m \cdot -0.125\right) \cdot \left(\frac{D\_m}{d\_m \cdot \ell} \cdot \frac{w0 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)}{d\_m}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 5e+89)
   w0
   (* (* D_m -0.125) (* (/ D_m (* d_m l)) (/ (* w0 (* h (* M_m M_m))) d_m)))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 5e+89) {
		tmp = w0;
	} else {
		tmp = (D_m * -0.125) * ((D_m / (d_m * l)) * ((w0 * (h * (M_m * M_m))) / d_m));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (m_m <= 5d+89) then
        tmp = w0
    else
        tmp = (d_m * (-0.125d0)) * ((d_m / (d_m_1 * l)) * ((w0 * (h * (m_m * m_m))) / d_m_1))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 5e+89) {
		tmp = w0;
	} else {
		tmp = (D_m * -0.125) * ((D_m / (d_m * l)) * ((w0 * (h * (M_m * M_m))) / d_m));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 5e+89:
		tmp = w0
	else:
		tmp = (D_m * -0.125) * ((D_m / (d_m * l)) * ((w0 * (h * (M_m * M_m))) / d_m))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 5e+89)
		tmp = w0;
	else
		tmp = Float64(Float64(D_m * -0.125) * Float64(Float64(D_m / Float64(d_m * l)) * Float64(Float64(w0 * Float64(h * Float64(M_m * M_m))) / d_m)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 5e+89)
		tmp = w0;
	else
		tmp = (D_m * -0.125) * ((D_m / (d_m * l)) * ((w0 * (h * (M_m * M_m))) / d_m));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 5e+89], w0, N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(N[(D$95$m / N[(d$95$m * l), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.99999999999999983e89

   1. Initial program 83.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 76.1%

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

   if 4.99999999999999983e89 < M

   1. Initial program 75.7%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

   7. Simplified 35.5%

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

   8. Taylor expanded in M around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}\right), \color{blue}{\ell}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \left(d \cdot d\right)\right)\right), \ell\right) \]

      16. *-lowering-*.f64 21.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \mathsf{*.f64}\left(d, d\right)\right)\right), \ell\right) \]

   10. Simplified 21.8%

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

       1. associate-/l* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot D\right) \cdot D\right) \cdot \frac{\color{blue}{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}}{\ell} \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot D\right) \cdot \color{blue}{\left(D \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot D\right), \color{blue}{\left(D \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(D \cdot \frac{-1}{8}\right), \left(\color{blue}{D} \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\color{blue}{D} \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)}\right)\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \color{blue}{\ell}}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(M \cdot M\right) \cdot \left(h \cdot w0\right)\right), \color{blue}{\left(\left(d \cdot d\right) \cdot \ell\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(h \cdot w0\right) \cdot \left(M \cdot M\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \left(M \cdot M\right)\right)\right), \left(\left(d \cdot \color{blue}{d}\right) \cdot \ell\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\left(d \cdot d\right) \cdot \ell\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\ell \cdot \color{blue}{\left(d \cdot d\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(d \cdot d\right)}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 20.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 20.3%

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

       1. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\frac{D \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\frac{D \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(\ell \cdot d\right) \cdot \color{blue}{d}}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\frac{D \cdot \left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right)}{\left(d \cdot \ell\right) \cdot d}\right)\right) \]

       4. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\frac{D}{d \cdot \ell} \cdot \color{blue}{\frac{h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)}{d}}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(\frac{D}{d \cdot \ell}\right), \color{blue}{\left(\frac{h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)}{d}\right)}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \left(d \cdot \ell\right)\right), \left(\frac{\color{blue}{h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)}}{d}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right), \left(\frac{h \cdot \color{blue}{\left(w0 \cdot \left(M \cdot M\right)\right)}}{d}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right), \mathsf{/.f64}\left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right), \color{blue}{d}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right), \mathsf{/.f64}\left(\left(\left(w0 \cdot \left(M \cdot M\right)\right) \cdot h\right), d\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right), \mathsf{/.f64}\left(\left(w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)\right), d\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \left(\left(M \cdot M\right) \cdot h\right)\right), d\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right), d\right)\right)\right) \]

       13. *-lowering-*.f64 25.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right), d\right)\right)\right) \]

   14. Applied egg-rr 25.1%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{+89}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(D \cdot -0.125\right) \cdot \left(\frac{D}{d \cdot \ell} \cdot \frac{w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.1 \cdot 10^{+93}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(D\_m \cdot -0.125\right) \cdot \left(h \cdot \left(D\_m \cdot \left(\frac{M\_m \cdot M\_m}{d\_m \cdot d\_m} \cdot \frac{w0}{\ell}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 2.1e+93)
   w0
   (* (* D_m -0.125) (* h (* D_m (* (/ (* M_m M_m) (* d_m d_m)) (/ w0 l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.1e+93) {
		tmp = w0;
	} else {
		tmp = (D_m * -0.125) * (h * (D_m * (((M_m * M_m) / (d_m * d_m)) * (w0 / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (m_m <= 2.1d+93) then
        tmp = w0
    else
        tmp = (d_m * (-0.125d0)) * (h * (d_m * (((m_m * m_m) / (d_m_1 * d_m_1)) * (w0 / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.1e+93) {
		tmp = w0;
	} else {
		tmp = (D_m * -0.125) * (h * (D_m * (((M_m * M_m) / (d_m * d_m)) * (w0 / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 2.1e+93:
		tmp = w0
	else:
		tmp = (D_m * -0.125) * (h * (D_m * (((M_m * M_m) / (d_m * d_m)) * (w0 / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 2.1e+93)
		tmp = w0;
	else
		tmp = Float64(Float64(D_m * -0.125) * Float64(h * Float64(D_m * Float64(Float64(Float64(M_m * M_m) / Float64(d_m * d_m)) * Float64(w0 / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 2.1e+93)
		tmp = w0;
	else
		tmp = (D_m * -0.125) * (h * (D_m * (((M_m * M_m) / (d_m * d_m)) * (w0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 2.1e+93], w0, N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(h * N[(D$95$m * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(w0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.0999999999999998e93

   1. Initial program 83.0%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 75.5%

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

   if 2.0999999999999998e93 < M

   1. Initial program 79.1%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

   7. Simplified 38.2%

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

   8. Taylor expanded in M around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}\right), \color{blue}{\ell}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \left(d \cdot d\right)\right)\right), \ell\right) \]

      16. *-lowering-*.f64 23.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \mathsf{*.f64}\left(d, d\right)\right)\right), \ell\right) \]

   10. Simplified 23.0%

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

       1. associate-/l* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot D\right) \cdot D\right) \cdot \frac{\color{blue}{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}}{\ell} \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot D\right) \cdot \color{blue}{\left(D \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot D\right), \color{blue}{\left(D \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(D \cdot \frac{-1}{8}\right), \left(\color{blue}{D} \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\color{blue}{D} \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)}\right)\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \color{blue}{\ell}}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(M \cdot M\right) \cdot \left(h \cdot w0\right)\right), \color{blue}{\left(\left(d \cdot d\right) \cdot \ell\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(h \cdot w0\right) \cdot \left(M \cdot M\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \left(M \cdot M\right)\right)\right), \left(\left(d \cdot \color{blue}{d}\right) \cdot \ell\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\left(d \cdot d\right) \cdot \ell\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\ell \cdot \color{blue}{\left(d \cdot d\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(d \cdot d\right)}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 21.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 21.3%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\frac{h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot \color{blue}{D}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\left(h \cdot \frac{w0 \cdot \left(M \cdot M\right)}{\ell \cdot \left(d \cdot d\right)}\right) \cdot D\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(h \cdot \color{blue}{\left(\frac{w0 \cdot \left(M \cdot M\right)}{\ell \cdot \left(d \cdot d\right)} \cdot D\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \color{blue}{\left(\frac{w0 \cdot \left(M \cdot M\right)}{\ell \cdot \left(d \cdot d\right)} \cdot D\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\left(\frac{w0 \cdot \left(M \cdot M\right)}{\ell \cdot \left(d \cdot d\right)}\right), \color{blue}{D}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot w0}{\ell \cdot \left(d \cdot d\right)}\right), D\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\left(\frac{\left(M \cdot M\right) \cdot w0}{\left(d \cdot d\right) \cdot \ell}\right), D\right)\right)\right) \]

       8. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\left(\frac{M \cdot M}{d \cdot d} \cdot \frac{w0}{\ell}\right), D\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot M}{d \cdot d}\right), \left(\frac{w0}{\ell}\right)\right), D\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot M\right), \left(d \cdot d\right)\right), \left(\frac{w0}{\ell}\right)\right), D\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(d \cdot d\right)\right), \left(\frac{w0}{\ell}\right)\right), D\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(d, d\right)\right), \left(\frac{w0}{\ell}\right)\right), D\right)\right)\right) \]

       13. /-lowering-/.f64 23.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{/.f64}\left(w0, \ell\right)\right), D\right)\right)\right) \]

   14. Applied egg-rr 23.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+93}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(D \cdot -0.125\right) \cdot \left(h \cdot \left(D \cdot \left(\frac{M \cdot M}{d \cdot d} \cdot \frac{w0}{\ell}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.4 \cdot 10^{+95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(D\_m \cdot -0.125\right) \cdot \left(D\_m \cdot \frac{\frac{h}{\ell} \cdot \left(w0 \cdot \left(M\_m \cdot M\_m\right)\right)}{d\_m \cdot d\_m}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 3.4e+95)
   w0
   (* (* D_m -0.125) (* D_m (/ (* (/ h l) (* w0 (* M_m M_m))) (* d_m d_m))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.4e+95) {
		tmp = w0;
	} else {
		tmp = (D_m * -0.125) * (D_m * (((h / l) * (w0 * (M_m * M_m))) / (d_m * d_m)));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (m_m <= 3.4d+95) then
        tmp = w0
    else
        tmp = (d_m * (-0.125d0)) * (d_m * (((h / l) * (w0 * (m_m * m_m))) / (d_m_1 * d_m_1)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 3.4e+95) {
		tmp = w0;
	} else {
		tmp = (D_m * -0.125) * (D_m * (((h / l) * (w0 * (M_m * M_m))) / (d_m * d_m)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 3.4e+95:
		tmp = w0
	else:
		tmp = (D_m * -0.125) * (D_m * (((h / l) * (w0 * (M_m * M_m))) / (d_m * d_m)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 3.4e+95)
		tmp = w0;
	else
		tmp = Float64(Float64(D_m * -0.125) * Float64(D_m * Float64(Float64(Float64(h / l) * Float64(w0 * Float64(M_m * M_m))) / Float64(d_m * d_m))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 3.4e+95)
		tmp = w0;
	else
		tmp = (D_m * -0.125) * (D_m * (((h / l) * (w0 * (M_m * M_m))) / (d_m * d_m)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 3.4e+95], w0, N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(D$95$m * N[(N[(N[(h / l), $MachinePrecision] * N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.40000000000000022e95

   1. Initial program 83.1%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 75.6%

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

   if 3.40000000000000022e95 < M

   1. Initial program 78.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

   7. Simplified 36.5%

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

   8. Taylor expanded in M around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}\right), \color{blue}{\ell}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \left(d \cdot d\right)\right)\right), \ell\right) \]

      16. *-lowering-*.f64 23.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \mathsf{*.f64}\left(d, d\right)\right)\right), \ell\right) \]

   10. Simplified 23.6%

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

       1. associate-/l* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot D\right) \cdot D\right) \cdot \frac{\color{blue}{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}}{\ell} \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot D\right) \cdot \color{blue}{\left(D \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot D\right), \color{blue}{\left(D \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(D \cdot \frac{-1}{8}\right), \left(\color{blue}{D} \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\color{blue}{D} \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)}\right)\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \color{blue}{\ell}}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(M \cdot M\right) \cdot \left(h \cdot w0\right)\right), \color{blue}{\left(\left(d \cdot d\right) \cdot \ell\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(h \cdot w0\right) \cdot \left(M \cdot M\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \left(M \cdot M\right)\right)\right), \left(\left(d \cdot \color{blue}{d}\right) \cdot \ell\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\left(d \cdot d\right) \cdot \ell\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\ell \cdot \color{blue}{\left(d \cdot d\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(d \cdot d\right)}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 21.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 21.8%

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

       1. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{h}{\ell} \cdot \color{blue}{\frac{w0 \cdot \left(M \cdot M\right)}{d \cdot d}}\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\frac{h}{\ell} \cdot \left(w0 \cdot \left(M \cdot M\right)\right)}{\color{blue}{d \cdot d}}\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{h}{\ell} \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right), \color{blue}{\left(d \cdot d\right)}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{h}{\ell}\right), \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{d} \cdot d\right)\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(d \cdot d\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(w0, \left(M \cdot M\right)\right)\right), \left(d \cdot d\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(d \cdot d\right)\right)\right)\right) \]

       8. *-lowering-*.f64 24.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right)\right) \]

   14. Applied egg-rr 24.5%

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

Alternative 11: 70.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ d_m = \left|d\right| \\ [w0, M_m, D_m, h, l, d_m] = \mathsf{sort}([w0, M_m, D_m, h, l, d_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.55 \cdot 10^{+95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\left(D\_m \cdot -0.125\right) \cdot \left(D\_m \cdot \frac{h \cdot \left(w0 \cdot \left(M\_m \cdot M\_m\right)\right)}{\ell \cdot \left(d\_m \cdot d\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d_m)
 :precision binary64
 (if (<= M_m 2.55e+95)
   w0
   (* (* D_m -0.125) (* D_m (/ (* h (* w0 (* M_m M_m))) (* l (* d_m d_m)))))))

C

M_m = fabs(M);
D_m = fabs(D);
d_m = fabs(d);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m);
double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.55e+95) {
		tmp = w0;
	} else {
		tmp = (D_m * -0.125) * (D_m * ((h * (w0 * (M_m * M_m))) / (l * (d_m * d_m))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    real(8) :: tmp
    if (m_m <= 2.55d+95) then
        tmp = w0
    else
        tmp = (d_m * (-0.125d0)) * (d_m * ((h * (w0 * (m_m * m_m))) / (l * (d_m_1 * d_m_1))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
d_m = Math.abs(d);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d_m;
public static double code(double w0, double M_m, double D_m, double h, double l, double d_m) {
	double tmp;
	if (M_m <= 2.55e+95) {
		tmp = w0;
	} else {
		tmp = (D_m * -0.125) * (D_m * ((h * (w0 * (M_m * M_m))) / (l * (d_m * d_m))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
d_m = math.fabs(d)
[w0, M_m, D_m, h, l, d_m] = sort([w0, M_m, D_m, h, l, d_m])
def code(w0, M_m, D_m, h, l, d_m):
	tmp = 0
	if M_m <= 2.55e+95:
		tmp = w0
	else:
		tmp = (D_m * -0.125) * (D_m * ((h * (w0 * (M_m * M_m))) / (l * (d_m * d_m))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d_m = abs(d)
w0, M_m, D_m, h, l, d_m = sort([w0, M_m, D_m, h, l, d_m])
function code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0
	if (M_m <= 2.55e+95)
		tmp = w0;
	else
		tmp = Float64(Float64(D_m * -0.125) * Float64(D_m * Float64(Float64(h * Float64(w0 * Float64(M_m * M_m))) / Float64(l * Float64(d_m * d_m)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d_m = abs(d);
w0, M_m, D_m, h, l, d_m = num2cell(sort([w0, M_m, D_m, h, l, d_m])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d_m)
	tmp = 0.0;
	if (M_m <= 2.55e+95)
		tmp = w0;
	else
		tmp = (D_m * -0.125) * (D_m * ((h * (w0 * (M_m * M_m))) / (l * (d_m * d_m))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d$95$m_] := If[LessEqual[M$95$m, 2.55e+95], w0, N[(N[(D$95$m * -0.125), $MachinePrecision] * N[(D$95$m * N[(N[(h * N[(w0 * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.55000000000000001e95

   1. Initial program 83.0%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 75.5%

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

   if 2.55000000000000001e95 < M

   1. Initial program 79.1%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2} \cdot \left(h \cdot w0\right)\right)\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(h \cdot w0\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(w0 \cdot h\right) \cdot {M}^{2}\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left(h \cdot {M}^{2}\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(w0 \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \left({M}^{2} \cdot h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

   7. Simplified 38.2%

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

   8. Taylor expanded in M around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}}{\color{blue}{\ell}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2}}\right), \color{blue}{\ell}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {D}^{2}\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2}\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot D\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)\right), \ell\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(h \cdot w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \left({d}^{2}\right)\right)\right), \ell\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \left(d \cdot d\right)\right)\right), \ell\right) \]

      16. *-lowering-*.f64 23.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{*.f64}\left(h, w0\right)\right), \mathsf{*.f64}\left(d, d\right)\right)\right), \ell\right) \]

   10. Simplified 23.0%

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

       1. associate-/l* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot \left(D \cdot D\right)\right) \cdot \color{blue}{\frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}} \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(\frac{-1}{8} \cdot D\right) \cdot D\right) \cdot \frac{\color{blue}{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}}{\ell} \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{8} \cdot D\right) \cdot \color{blue}{\left(D \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot D\right), \color{blue}{\left(D \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(D \cdot \frac{-1}{8}\right), \left(\color{blue}{D} \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \left(\color{blue}{D} \cdot \frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}}{\ell}\right)}\right)\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \left(\frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\left(d \cdot d\right) \cdot \color{blue}{\ell}}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(M \cdot M\right) \cdot \left(h \cdot w0\right)\right), \color{blue}{\left(\left(d \cdot d\right) \cdot \ell\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(h \cdot w0\right) \cdot \left(M \cdot M\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(h \cdot \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(w0 \cdot \left(M \cdot M\right)\right)\right), \left(\color{blue}{\left(d \cdot d\right)} \cdot \ell\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \left(M \cdot M\right)\right)\right), \left(\left(d \cdot \color{blue}{d}\right) \cdot \ell\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\left(d \cdot d\right) \cdot \ell\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \left(\ell \cdot \color{blue}{\left(d \cdot d\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(d \cdot d\right)}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 21.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, \frac{-1}{8}\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(M, M\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(d, \color{blue}{d}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 21.3%

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

Alternative 12: 67.5% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

M_m = abs(m)
D_m = abs(d)
d_m = abs(d)
NOTE: w0, M_m, D_m, h, l, and d_m should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d_m_1)
    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_m_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 75.6%

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

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{w0} \]
6. Step-by-step derivation

7. Simplified 71.9%

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

Reproduce

herbie shell --seed 2024161 
(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))))))