Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 88.1%

Time: 19.4s

Alternatives: 10

Speedup: 1.8×

Specification

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 80.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := 1 - {\left(\frac{M_m \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+183}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M_m}{4}}{\frac{\ell}{h \cdot \left(D \cdot \frac{M_m}{d}\right)}} \cdot \frac{D}{d}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (pow (/ (* M_m D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 5e+183)
     (* w0 (sqrt t_0))
     (*
      w0
      (sqrt
       (- 1.0 (* (/ (/ M_m 4.0) (/ l (* h (* D (/ M_m d))))) (/ D d))))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = 1.0 - (pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+183) {
		tmp = w0 * sqrt(t_0);
	} else {
		tmp = w0 * sqrt((1.0 - (((M_m / 4.0) / (l / (h * (D * (M_m / d))))) * (D / d))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((((m_m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))
    if (t_0 <= 5d+183) then
        tmp = w0 * sqrt(t_0)
    else
        tmp = w0 * sqrt((1.0d0 - (((m_m / 4.0d0) / (l / (h * (d * (m_m / d_1))))) * (d / d_1))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = 1.0 - (Math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 5e+183) {
		tmp = w0 * Math.sqrt(t_0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((M_m / 4.0) / (l / (h * (D * (M_m / d))))) * (D / d))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = 1.0 - (math.pow(((M_m * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 5e+183:
		tmp = w0 * math.sqrt(t_0)
	else:
		tmp = w0 * math.sqrt((1.0 - (((M_m / 4.0) / (l / (h * (D * (M_m / d))))) * (D / d))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 5e+183)
		tmp = Float64(w0 * sqrt(t_0));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m / 4.0) / Float64(l / Float64(h * Float64(D * Float64(M_m / d))))) * Float64(D / d)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = 1.0 - ((((M_m * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 5e+183)
		tmp = w0 * sqrt(t_0);
	else
		tmp = w0 * sqrt((1.0 - (((M_m / 4.0) / (l / (h * (D * (M_m / d))))) * (D / d))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(1.0 - N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+183], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m / 4.0), $MachinePrecision] / N[(l / N[(h * N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 5.00000000000000009e183

   1. Initial program 99.9%

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

   if 5.00000000000000009e183 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 54.1%

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

      1. associate-/l* 58.4%

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

      2. clear-num 58.3%

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

      3. associate-/r/ 58.4%

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

      4. associate-/r/ 58.4%

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

      5. associate-/r* 58.4%

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

      6. metadata-eval 58.4%

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

   3. Applied egg-rr 58.4%

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

   5. Applied egg-rr 59.8%

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

      1. div-inv 59.8%

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

      2. *-commutative 59.8%

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

      3. clear-num 59.8%

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

      4. associate-/r* 59.8%

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

      5. associate-/l/ 71.6%

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

      6. *-commutative 71.6%

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

      7. associate-/r/ 68.5%

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

      8. *-commutative 68.5%

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

   7. Applied egg-rr 68.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{+183}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{4}}{\frac{\ell}{h \cdot \left(D \cdot \frac{M}{d}\right)}} \cdot \frac{D}{d}}\\ \end{array} \]

Alternative 2: 84.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-318}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(\frac{M_m}{d} \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M_m}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (/ h l) -4e-318)
   (*
    w0
    (sqrt (- 1.0 (* D (* (/ M_m d) (* 0.25 (* (/ h l) (* D (/ M_m d)))))))))
   w0))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -4e-318) {
		tmp = w0 * sqrt((1.0 - (D * ((M_m / d) * (0.25 * ((h / l) * (D * (M_m / d))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((h / l) <= (-4d-318)) then
        tmp = w0 * sqrt((1.0d0 - (d * ((m_m / d_1) * (0.25d0 * ((h / l) * (d * (m_m / d_1))))))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -4e-318) {
		tmp = w0 * Math.sqrt((1.0 - (D * ((M_m / d) * (0.25 * ((h / l) * (D * (M_m / d))))))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (h / l) <= -4e-318:
		tmp = w0 * math.sqrt((1.0 - (D * ((M_m / d) * (0.25 * ((h / l) * (D * (M_m / d))))))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -4e-318)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(D * Float64(Float64(M_m / d) * Float64(0.25 * Float64(Float64(h / l) * Float64(D * Float64(M_m / d)))))))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if ((h / l) <= -4e-318)
		tmp = w0 * sqrt((1.0 - (D * ((M_m / d) * (0.25 * ((h / l) * (D * (M_m / d))))))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -4e-318], N[(w0 * N[Sqrt[N[(1.0 - N[(D * N[(N[(M$95$m / d), $MachinePrecision] * N[(0.25 * N[(N[(h / l), $MachinePrecision] * N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -3.9999999e-318

   1. Initial program 82.5%

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

      1. associate-/l* 83.8%

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

      2. clear-num 83.8%

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

      3. associate-/r/ 83.8%

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

      4. associate-/r/ 83.7%

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

      5. associate-/r* 83.7%

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

      6. metadata-eval 83.7%

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

   3. Applied egg-rr 83.7%

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

   5. Applied egg-rr 81.9%

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

      1. associate-/r/ 83.8%

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

      2. associate-*l/ 85.0%

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

      3. *-commutative 85.0%

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

      4. clear-num 85.0%

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

      5. div-inv 85.0%

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

      6. clear-num 84.4%

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

      7. associate-/l/ 84.7%

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

      8. un-div-inv 84.7%

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

      9. associate-/l* 84.7%

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

      10. *-commutative 84.7%

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

      11. div-inv 84.7%

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

      12. *-commutative 84.7%

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

      13. associate-*l* 84.7%

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

      14. associate-/r/ 82.6%

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

      15. *-commutative 82.6%

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

   7. Applied egg-rr 80.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(D \cdot \frac{M}{d}\right) \cdot \left(\left(M \cdot \left(\frac{D}{d} \cdot \frac{h}{\ell}\right)\right) \cdot 0.25\right)}} \]
   8. Step-by-step derivation

      1. associate-*l* 78.2%

         \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{D \cdot \left(\frac{M}{d} \cdot \left(\left(M \cdot \left(\frac{D}{d} \cdot \frac{h}{\ell}\right)\right) \cdot 0.25\right)\right)}} \]

      2. *-commutative 78.2%

         \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \color{blue}{\left(0.25 \cdot \left(M \cdot \left(\frac{D}{d} \cdot \frac{h}{\ell}\right)\right)\right)}\right)} \]

      3. associate-*r* 80.1%

         \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{h}{\ell}\right)}\right)\right)} \]

      4. associate-*r/ 78.8%

         \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \left(\color{blue}{\frac{M \cdot D}{d}} \cdot \frac{h}{\ell}\right)\right)\right)} \]

      5. associate-*l/ 80.7%

         \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \left(\color{blue}{\left(\frac{M}{d} \cdot D\right)} \cdot \frac{h}{\ell}\right)\right)\right)} \]

      6. associate-/r/ 80.1%

         \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \left(\color{blue}{\frac{M}{\frac{d}{D}}} \cdot \frac{h}{\ell}\right)\right)\right)} \]

      7. associate-/r/ 79.0%

         \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \color{blue}{\frac{M}{\frac{\frac{d}{D}}{\frac{h}{\ell}}}}\right)\right)} \]

      8. associate-/l* 77.0%

         \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \color{blue}{\frac{M \cdot \frac{h}{\ell}}{\frac{d}{D}}}\right)\right)} \]

      9. associate-*r/ 76.3%

         \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \frac{\color{blue}{\frac{M \cdot h}{\ell}}}{\frac{d}{D}}\right)\right)} \]

      10. *-commutative 76.3%

          \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \frac{\frac{\color{blue}{h \cdot M}}{\ell}}{\frac{d}{D}}\right)\right)} \]

      11. associate-/r* 75.6%

          \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \color{blue}{\frac{h \cdot M}{\ell \cdot \frac{d}{D}}}\right)\right)} \]

      12. times-frac 80.1%

          \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M}{\frac{d}{D}}\right)}\right)\right)} \]

      13. associate-/r/ 80.7%

          \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{M}{d} \cdot D\right)}\right)\right)\right)} \]

      14. *-commutative 80.7%

          \[\leadsto w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}\right)\right)\right)} \]

   9. Simplified 80.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)}} \]

   if -3.9999999e-318 < (/.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 86.3%

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

   4. Taylor expanded in M around 0 94.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4 \cdot 10^{-318}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \left(0.25 \cdot \left(\frac{h}{\ell} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \cdot D \leq 2.5 \cdot 10^{-290}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M_m \cdot D}{d \cdot 4} \cdot \frac{\left(M_m \cdot D\right) \cdot \frac{h}{d}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= (* M_m D) 2.5e-290)
   w0
   (*
    w0
    (sqrt (- 1.0 (* (/ (* M_m D) (* d 4.0)) (/ (* (* M_m D) (/ h d)) l)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((M_m * D) <= 2.5e-290) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 - (((M_m * D) / (d * 4.0)) * (((M_m * D) * (h / d)) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if ((m_m * d) <= 2.5d-290) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 - (((m_m * d) / (d_1 * 4.0d0)) * (((m_m * d) * (h / d_1)) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if ((M_m * D) <= 2.5e-290) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((M_m * D) / (d * 4.0)) * (((M_m * D) * (h / d)) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if (M_m * D) <= 2.5e-290:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 - (((M_m * D) / (d * 4.0)) * (((M_m * D) * (h / d)) / l))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (Float64(M_m * D) <= 2.5e-290)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m * D) / Float64(d * 4.0)) * Float64(Float64(Float64(M_m * D) * Float64(h / d)) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if ((M_m * D) <= 2.5e-290)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 - (((M_m * D) / (d * 4.0)) * (((M_m * D) * (h / d)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[N[(M$95$m * D), $MachinePrecision], 2.5e-290], w0, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m * D), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 2.5e-290

   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 83.9%

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

   4. Taylor expanded in M around 0 66.8%

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

   if 2.5e-290 < (*.f64 M D)

   1. Initial program 84.2%

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

      1. associate-/l* 85.9%

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

      2. clear-num 85.9%

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

      3. associate-/r/ 85.8%

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

      4. associate-/r/ 85.8%

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

      5. associate-/r* 85.8%

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

      6. metadata-eval 85.8%

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

   3. Applied egg-rr 85.8%

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

   5. Applied egg-rr 85.0%

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

      1. div-inv 84.3%

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

      2. *-commutative 84.3%

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

      3. clear-num 84.3%

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

      4. associate-/r* 84.3%

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

      5. associate-/l/ 87.7%

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

      6. *-commutative 87.7%

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

      7. associate-/r/ 86.1%

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

      8. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. associate-/l/ 86.1%

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

      2. clear-num 86.1%

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

      3. frac-times 89.4%

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

      4. *-commutative 89.4%

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

      5. *-un-lft-identity 89.4%

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

      6. clear-num 89.4%

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

      7. associate-*l/ 89.4%

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

      8. metadata-eval 89.4%

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

      9. clear-num 89.4%

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

      10. associate-/r* 86.8%

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

      11. associate-/r/ 85.9%

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

      12. clear-num 85.9%

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

      13. associate-*r/ 85.1%

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

      14. *-commutative 85.1%

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

   9. Applied egg-rr 85.1%

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

       1. *-commutative 85.1%

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

       2. associate-/r* 85.1%

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

       3. associate-/l* 85.1%

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

       4. associate-/r/ 85.1%

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

       5. associate-/l/ 85.1%

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

       6. *-commutative 85.1%

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

       7. associate-*l/ 89.3%

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

       8. associate-*r/ 86.2%

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

       9. associate-/l* 89.4%

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

       10. associate-/r/ 88.5%

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

       11. *-commutative 88.5%

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

   11. Simplified 88.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2.5 \cdot 10^{-290}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{d \cdot 4} \cdot \frac{\left(M \cdot D\right) \cdot \frac{h}{d}}{\ell}}\\ \end{array} \]

Alternative 4: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot \left(w0 \cdot {M_m}^{2}\right)\right)}{\ell \cdot \frac{d}{D}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 4.8e-158)
   w0
   (+ w0 (* -0.125 (/ (* (/ D d) (* h (* w0 (pow M_m 2.0)))) (* l (/ d D)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.8e-158) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D / d) * (h * (w0 * pow(M_m, 2.0)))) / (l * (d / D))));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m_m <= 4.8d-158) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * (((d / d_1) * (h * (w0 * (m_m ** 2.0d0)))) / (l * (d_1 / d))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.8e-158) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * (((D / d) * (h * (w0 * Math.pow(M_m, 2.0)))) / (l * (d / D))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 4.8e-158:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * (((D / d) * (h * (w0 * math.pow(M_m, 2.0)))) / (l * (d / D))))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 4.8e-158)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(h * Float64(w0 * (M_m ^ 2.0)))) / Float64(l * Float64(d / D)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 4.8e-158)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * (((D / d) * (h * (w0 * (M_m ^ 2.0)))) / (l * (d / D))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 4.8e-158], w0, N[(w0 + N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(h * N[(w0 * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.80000000000000015e-158

   1. Initial program 82.2%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in M around 0 66.8%

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

   if 4.80000000000000015e-158 < M

   1. Initial program 86.0%

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

   3. Simplified 86.9%

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

      1. associate-*r/ 88.0%

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

      2. times-frac 87.1%

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

      3. associate-/l* 85.9%

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

      4. unpow2 85.9%

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

      5. times-frac 85.9%

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

      6. associate-*l/ 85.9%

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

      7. times-frac 86.9%

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

      8. associate-*l/ 86.9%

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

      9. frac-times 86.9%

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

      10. associate-/l/ 86.9%

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

      11. metadata-eval 86.9%

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

   5. Applied egg-rr 86.9%

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

   6. Taylor expanded in M around 0 52.1%

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

      1. times-frac 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. times-frac 67.7%

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

      5. unpow2 67.7%

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

   8. Simplified 67.7%

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

      1. unpow2 67.7%

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

      2. associate-*r/ 67.7%

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

   10. Applied egg-rr 67.7%

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

       1. associate-/l* 67.7%

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

       2. frac-times 69.0%

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

       3. *-commutative 69.0%

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

       4. associate-*l* 69.1%

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

   12. Applied egg-rr 69.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{\frac{D}{d} \cdot \left(h \cdot \left(w0 \cdot {M}^{2}\right)\right)}{\ell \cdot \frac{d}{D}}\\ \end{array} \]

Alternative 5: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 \cdot \sqrt{1 - \frac{\frac{M_m}{4}}{\frac{\ell}{h \cdot \left(D \cdot \frac{M_m}{d}\right)}} \cdot \frac{D}{d}} \end{array} \]

FPCore

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

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (((M_m / 4.0) / (l / (h * (D * (M_m / d))))) * (D / d))));
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - (((m_m / 4.0d0) / (l / (h * (d * (m_m / d_1))))) * (d / d_1))))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (((M_m / 4.0) / (l / (h * (D * (M_m / d))))) * (D / d))));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 * math.sqrt((1.0 - (((M_m / 4.0) / (l / (h * (D * (M_m / d))))) * (D / d))))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m / 4.0) / Float64(l / Float64(h * Float64(D * Float64(M_m / d))))) * Float64(D / d)))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 * sqrt((1.0 - (((M_m / 4.0) / (l / (h * (D * (M_m / d))))) * (D / d))));
end

Wolfram

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

Derivation

1. Initial program 83.6%

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

   1. associate-/l* 84.8%

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

   2. clear-num 84.8%

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

   3. associate-/r/ 84.8%

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

   4. associate-/r/ 84.8%

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

   5. associate-/r* 84.8%

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

   6. metadata-eval 84.8%

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

3. Applied egg-rr 84.8%

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

5. Applied egg-rr 84.9%

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

   1. div-inv 84.5%

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

   2. *-commutative 84.5%

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

   3. clear-num 84.5%

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

   4. associate-/r* 84.5%

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

   5. associate-/l/ 88.8%

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

   6. *-commutative 88.8%

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

   7. associate-/r/ 86.1%

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

   8. *-commutative 86.1%

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

7. Applied egg-rr 86.1%

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

8. Final simplification 86.1%

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

Alternative 6: 72.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ w0 + -0.125 \cdot \frac{\frac{D}{d}}{\frac{\ell}{h \cdot \left(w0 \cdot {M_m}^{2}\right)} \cdot \frac{d}{D}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (+ w0 (* -0.125 (/ (/ D d) (* (/ l (* h (* w0 (pow M_m 2.0)))) (/ d D))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 + (-0.125 * ((D / d) / ((l / (h * (w0 * pow(M_m, 2.0)))) * (d / D))));
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 + ((-0.125d0) * ((d / d_1) / ((l / (h * (w0 * (m_m ** 2.0d0)))) * (d_1 / d))))
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 + (-0.125 * ((D / d) / ((l / (h * (w0 * Math.pow(M_m, 2.0)))) * (d / D))));
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	return w0 + (-0.125 * ((D / d) / ((l / (h * (w0 * math.pow(M_m, 2.0)))) * (d / D))))

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	return Float64(w0 + Float64(-0.125 * Float64(Float64(D / d) / Float64(Float64(l / Float64(h * Float64(w0 * (M_m ^ 2.0)))) * Float64(d / D)))))
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0 + (-0.125 * ((D / d) / ((l / (h * (w0 * (M_m ^ 2.0)))) * (d / D))));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 + N[(-0.125 * N[(N[(D / d), $MachinePrecision] / N[(N[(l / N[(h * N[(w0 * N[Power[M$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 84.8%

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

   1. associate-*r/ 86.8%

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

   2. times-frac 86.1%

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

   3. associate-/l* 84.0%

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

   4. unpow2 84.0%

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

   5. times-frac 83.6%

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

   6. associate-*l/ 83.6%

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

   7. times-frac 85.1%

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

   8. associate-*l/ 85.1%

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

   9. frac-times 85.1%

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

   10. associate-/l/ 85.1%

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

   11. metadata-eval 85.1%

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

5. Applied egg-rr 85.1%

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

6. Taylor expanded in M around 0 48.9%

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

   1. times-frac 49.8%

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

   2. unpow2 49.8%

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

   3. unpow2 49.8%

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

   4. times-frac 62.1%

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

   5. unpow2 62.1%

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

8. Simplified 62.1%

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

   1. unpow2 62.1%

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

   2. associate-*r/ 61.7%

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

10. Applied egg-rr 61.7%

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

    1. *-commutative 61.7%

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

    2. clear-num 61.6%

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

    3. associate-/l* 62.1%

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

    4. frac-times 65.5%

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

    5. *-un-lft-identity 65.5%

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

    6. *-commutative 65.5%

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

    7. associate-*l* 69.9%

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

12. Applied egg-rr 69.9%

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

13. Final simplification 69.9%

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

Alternative 7: 72.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 2.25 \cdot 10^{-166}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot w0\right) \cdot \frac{D \cdot \left(\frac{M_m}{d} \cdot \frac{M_m \cdot D}{d}\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 2.25e-166)
   w0
   (+ w0 (* -0.125 (* (* h w0) (/ (* D (* (/ M_m d) (/ (* M_m D) d))) l))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.25e-166) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * w0) * ((D * ((M_m / d) * ((M_m * D) / d))) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m_m <= 2.25d-166) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((h * w0) * ((d * ((m_m / d_1) * ((m_m * d) / d_1))) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.25e-166) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * w0) * ((D * ((M_m / d) * ((M_m * D) / d))) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 2.25e-166:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((h * w0) * ((D * ((M_m / d) * ((M_m * D) / d))) / l)))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 2.25e-166)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * w0) * Float64(Float64(D * Float64(Float64(M_m / d) * Float64(Float64(M_m * D) / d))) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.25e-166)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((h * w0) * ((D * ((M_m / d) * ((M_m * D) / d))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 2.25e-166], w0, N[(w0 + N[(-0.125 * N[(N[(h * w0), $MachinePrecision] * N[(N[(D * N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(M$95$m * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.2499999999999999e-166

   1. Initial program 82.0%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in M around 0 67.0%

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

   if 2.2499999999999999e-166 < M

   1. Initial program 86.2%

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

   3. Simplified 87.2%

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

      1. associate-*r/ 88.3%

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

      2. times-frac 87.3%

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

      3. associate-/l* 86.2%

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

      4. unpow2 86.2%

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

      5. times-frac 86.2%

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

      6. associate-*l/ 86.2%

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

      7. times-frac 87.1%

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

      8. associate-*l/ 87.1%

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

      9. frac-times 87.1%

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

      10. associate-/l/ 87.1%

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

      11. metadata-eval 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in M around 0 52.0%

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

      1. times-frac 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. times-frac 67.3%

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

      5. unpow2 67.3%

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

   8. Simplified 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-/l* 67.3%

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

      3. associate-*l/ 68.3%

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

      4. unpow2 68.3%

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

      5. unpow2 68.3%

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

      6. swap-sqr 72.7%

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

      7. clear-num 72.7%

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

      8. div-inv 72.7%

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

      9. clear-num 72.7%

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

      10. div-inv 72.7%

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

      11. pow1 72.7%

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

      12. pow1 72.7%

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

      13. pow-sqr 72.7%

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

      14. associate-/r/ 71.7%

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

      15. *-commutative 71.7%

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

      16. metadata-eval 71.7%

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

   10. Applied egg-rr 71.7%

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

       1. associate-/r/ 74.6%

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

   12. Simplified 74.6%

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

       1. unpow2 74.6%

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

       2. *-commutative 74.6%

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

       3. associate-*r* 71.7%

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

       4. associate-*r/ 71.6%

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

       5. *-commutative 71.6%

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

   14. Applied egg-rr 71.6%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{-166}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot w0\right) \cdot \frac{D \cdot \left(\frac{M}{d} \cdot \frac{M \cdot D}{d}\right)}{\ell}\right)\\ \end{array} \]

Alternative 8: 72.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D}{d}\\ \mathbf{if}\;M_m \leq 1.6 \cdot 10^{-165}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot w0\right) \cdot \frac{t_0 \cdot t_0}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M_m D) d)))
   (if (<= M_m 1.6e-165) w0 (+ w0 (* -0.125 (* (* h w0) (/ (* t_0 t_0) l)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (M_m * D) / d;
	double tmp;
	if (M_m <= 1.6e-165) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * w0) * ((t_0 * t_0) / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * d) / d_1
    if (m_m <= 1.6d-165) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((h * w0) * ((t_0 * t_0) / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (M_m * D) / d;
	double tmp;
	if (M_m <= 1.6e-165) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((h * w0) * ((t_0 * t_0) / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = (M_m * D) / d
	tmp = 0
	if M_m <= 1.6e-165:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((h * w0) * ((t_0 * t_0) / l)))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(Float64(M_m * D) / d)
	tmp = 0.0
	if (M_m <= 1.6e-165)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * w0) * Float64(Float64(t_0 * t_0) / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = (M_m * D) / d;
	tmp = 0.0;
	if (M_m <= 1.6e-165)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((h * w0) * ((t_0 * t_0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M$95$m * D), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[M$95$m, 1.6e-165], w0, N[(w0 + N[(-0.125 * N[(N[(h * w0), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 1.60000000000000006e-165

   1. Initial program 82.0%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in M around 0 67.0%

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

   if 1.60000000000000006e-165 < M

   1. Initial program 86.2%

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

   3. Simplified 87.2%

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

      1. associate-*r/ 88.3%

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

      2. times-frac 87.3%

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

      3. associate-/l* 86.2%

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

      4. unpow2 86.2%

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

      5. times-frac 86.2%

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

      6. associate-*l/ 86.2%

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

      7. times-frac 87.1%

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

      8. associate-*l/ 87.1%

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

      9. frac-times 87.1%

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

      10. associate-/l/ 87.1%

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

      11. metadata-eval 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in M around 0 52.0%

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

      1. times-frac 52.2%

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

      2. unpow2 52.2%

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

      3. unpow2 52.2%

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

      4. times-frac 67.3%

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

      5. unpow2 67.3%

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

   8. Simplified 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-/l* 67.3%

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

      3. associate-*l/ 68.3%

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

      4. unpow2 68.3%

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

      5. unpow2 68.3%

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

      6. swap-sqr 72.7%

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

      7. clear-num 72.7%

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

      8. div-inv 72.7%

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

      9. clear-num 72.7%

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

      10. div-inv 72.7%

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

      11. pow1 72.7%

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

      12. pow1 72.7%

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

      13. pow-sqr 72.7%

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

      14. associate-/r/ 71.7%

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

      15. *-commutative 71.7%

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

      16. metadata-eval 71.7%

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

   10. Applied egg-rr 71.7%

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

       1. associate-/r/ 74.6%

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

   12. Simplified 74.6%

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

       1. unpow2 74.6%

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

       2. associate-*r/ 74.6%

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

       3. associate-*r/ 75.6%

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

       4. *-commutative 75.6%

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

       5. *-commutative 75.6%

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

   14. Applied egg-rr 75.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.6 \cdot 10^{-165}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(h \cdot w0\right) \cdot \frac{\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}}{\ell}\right)\\ \end{array} \]

Alternative 9: 72.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [w0, M_m, D, h, l, d] = \mathsf{sort}([w0, M_m, D, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{M_m \cdot D}{d}\\ \mathbf{if}\;M_m \leq 4 \cdot 10^{-222}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(t_0 \cdot \frac{t_0}{\ell}\right) \cdot \left(h \cdot w0\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M_m D) d)))
   (if (<= M_m 4e-222) w0 (+ w0 (* -0.125 (* (* t_0 (/ t_0 l)) (* h w0)))))))

C

M_m = fabs(M);
assert(w0 < M_m && M_m < D && D < h && h < l && l < d);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (M_m * D) / d;
	double tmp;
	if (M_m <= 4e-222) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((t_0 * (t_0 / l)) * (h * w0)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m_m * d) / d_1
    if (m_m <= 4d-222) then
        tmp = w0
    else
        tmp = w0 + ((-0.125d0) * ((t_0 * (t_0 / l)) * (h * w0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert w0 < M_m && M_m < D && D < h && h < l && l < d;
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = (M_m * D) / d;
	double tmp;
	if (M_m <= 4e-222) {
		tmp = w0;
	} else {
		tmp = w0 + (-0.125 * ((t_0 * (t_0 / l)) * (h * w0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[w0, M_m, D, h, l, d] = sort([w0, M_m, D, h, l, d])
def code(w0, M_m, D, h, l, d):
	t_0 = (M_m * D) / d
	tmp = 0
	if M_m <= 4e-222:
		tmp = w0
	else:
		tmp = w0 + (-0.125 * ((t_0 * (t_0 / l)) * (h * w0)))
	return tmp

Julia

M_m = abs(M)
w0, M_m, D, h, l, d = sort([w0, M_m, D, h, l, d])
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(Float64(M_m * D) / d)
	tmp = 0.0
	if (M_m <= 4e-222)
		tmp = w0;
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(t_0 * Float64(t_0 / l)) * Float64(h * w0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
w0, M_m, D, h, l, d = num2cell(sort([w0, M_m, D, h, l, d])){:}
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = (M_m * D) / d;
	tmp = 0.0;
	if (M_m <= 4e-222)
		tmp = w0;
	else
		tmp = w0 + (-0.125 * ((t_0 * (t_0 / l)) * (h * w0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: w0, M_m, D, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M$95$m * D), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[M$95$m, 4e-222], w0, N[(w0 + N[(-0.125 * N[(N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 4.00000000000000019e-222

   1. Initial program 80.5%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in M around 0 64.9%

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

   if 4.00000000000000019e-222 < M

   1. Initial program 87.7%

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

   3. Simplified 88.6%

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

      1. associate-*r/ 89.5%

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

      2. times-frac 88.7%

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

      3. associate-/l* 87.7%

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

      4. unpow2 87.7%

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

      5. times-frac 87.7%

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

      6. associate-*l/ 87.7%

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

      7. times-frac 88.5%

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

      8. associate-*l/ 88.5%

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

      9. frac-times 88.5%

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

      10. associate-/l/ 88.5%

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

      11. metadata-eval 88.5%

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

   5. Applied egg-rr 88.5%

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

   6. Taylor expanded in M around 0 51.8%

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

      1. times-frac 52.8%

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

      2. unpow2 52.8%

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

      3. unpow2 52.8%

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

      4. times-frac 69.1%

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

      5. unpow2 69.1%

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

   8. Simplified 69.1%

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

      1. *-commutative 69.1%

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

      2. associate-/l* 68.1%

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

      3. associate-*l/ 69.0%

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

      4. unpow2 69.0%

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

      5. unpow2 69.0%

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

      6. swap-sqr 73.9%

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

      7. clear-num 73.9%

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

      8. div-inv 73.9%

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

      9. clear-num 73.9%

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

      10. div-inv 73.9%

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

      11. pow1 73.9%

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

      12. pow1 73.9%

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

      13. pow-sqr 73.9%

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

      14. associate-/r/ 73.0%

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

      15. *-commutative 73.0%

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

      16. metadata-eval 73.0%

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

   10. Applied egg-rr 73.0%

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

       1. associate-/r/ 76.5%

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

   12. Simplified 76.5%

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

       1. unpow2 76.5%

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

       2. *-un-lft-identity 76.5%

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

       3. times-frac 78.3%

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

       4. associate-*r/ 77.4%

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

       5. *-commutative 77.4%

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

       6. associate-*r/ 79.2%

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

       7. *-commutative 79.2%

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

   14. Applied egg-rr 79.2%

       \[\leadsto w0 + -0.125 \cdot \left(\color{blue}{\left(\frac{\frac{M \cdot D}{d}}{1} \cdot \frac{\frac{M \cdot D}{d}}{\ell}\right)} \cdot \left(h \cdot w0\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-222}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(\frac{M \cdot D}{d} \cdot \frac{\frac{M \cdot D}{d}}{\ell}\right) \cdot \left(h \cdot w0\right)\right)\\ \end{array} \]

Alternative 10: 68.6% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

3. Simplified 84.8%

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

4. Taylor expanded in M around 0 63.8%

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

5. Final simplification 63.8%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023336 
(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))))))