Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.2% → 86.8%

Time: 17.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{D_m \cdot M_m}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-7}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D_m \cdot \frac{M_m}{d \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (* (pow (/ (* D_m M_m) (* d 2.0)) 2.0) (/ h l)) 1e-7)
   (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* D_m (/ M_m (* d 2.0))) 2.0)))))
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((pow(((D_m * M_m) / (d * 2.0)), 2.0) * (h / l)) <= 1e-7) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow((D_m * (M_m / (d * 2.0))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((((d_m * m_m) / (d * 2.0d0)) ** 2.0d0) * (h / l)) <= 1d-7) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * ((d_m * (m_m / (d * 2.0d0))) ** 2.0d0))))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((D_m * M_m) / (d * 2.0)), 2.0) * (h / l)) <= 1e-7) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow((D_m * (M_m / (d * 2.0))), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if (math.pow(((D_m * M_m) / (d * 2.0)), 2.0) * (h / l)) <= 1e-7:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow((D_m * (M_m / (d * 2.0))), 2.0))))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64((Float64(Float64(D_m * M_m) / Float64(d * 2.0)) ^ 2.0) * Float64(h / l)) <= 1e-7)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(D_m * Float64(M_m / Float64(d * 2.0))) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (((((D_m * M_m) / (d * 2.0)) ^ 2.0) * (h / l)) <= 1e-7)
		tmp = w0 * sqrt((1.0 - ((h / l) * ((D_m * (M_m / (d * 2.0))) ^ 2.0))));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(N[Power[N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], 1e-7], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 9.9999999999999995e-8

   1. Initial program 87.1%

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

   3. Simplified 88.0%

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

   if 9.9999999999999995e-8 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in M around 0 70.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{-7}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-314}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, w0 \cdot \frac{h \cdot {\left(M_m \cdot \frac{D_m}{d}\right)}^{2}}{\ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e-314)
   (fma -0.125 (* w0 (/ (* h (pow (* M_m (/ D_m d)) 2.0)) l)) w0)
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-314) {
		tmp = fma(-0.125, (w0 * ((h * pow((M_m * (D_m / d)), 2.0)) / l)), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-314)
		tmp = fma(-0.125, Float64(w0 * Float64(Float64(h * (Float64(M_m * Float64(D_m / d)) ^ 2.0)) / l)), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-314], N[(-0.125 * N[(w0 * N[(N[(h * N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.99999999982e-314

   1. Initial program 76.7%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in M around 0 42.7%

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

      1. +-commutative 42.7%

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

      2. fma-def 42.7%

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

      3. associate-*r* 44.1%

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

      4. times-frac 45.5%

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

      5. *-commutative 45.5%

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

      6. unpow2 45.5%

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

      7. unpow2 45.5%

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

      8. swap-sqr 54.7%

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

      9. unpow2 54.7%

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

      10. *-commutative 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in D around 0 42.7%

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

      1. associate-*r* 44.1%

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

      2. unpow2 44.1%

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

      3. unpow2 44.1%

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

      4. swap-sqr 54.8%

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

      5. unpow2 54.8%

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

      6. times-frac 54.7%

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

      7. unpow2 54.7%

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

      8. unpow2 54.7%

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

      9. times-frac 62.1%

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

      10. unpow2 62.1%

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

      11. associate-*l/ 64.2%

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

      12. associate-*r* 68.4%

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

   9. Simplified 68.5%

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

       1. associate-*l/ 72.0%

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

       2. associate-/r/ 71.3%

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

   11. Applied egg-rr 71.3%

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

   if -4.99999999982e-314 < (/.f64 h l)

   1. Initial program 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in M around 0 92.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-314}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, w0 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ w0 \cdot \sqrt{1 + \left(h \cdot {\left(D_m \cdot \left(M_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right) \cdot \frac{-1}{\ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (*
  w0
  (sqrt (+ 1.0 (* (* h (pow (* D_m (* M_m (/ 0.5 d))) 2.0)) (/ -1.0 l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	return w0 * sqrt((1.0 + ((h * pow((D_m * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l))));
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    code = w0 * sqrt((1.0d0 + ((h * ((d_m * (m_m * (0.5d0 / d))) ** 2.0d0)) * ((-1.0d0) / l))))
end function

Java

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

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	return w0 * math.sqrt((1.0 + ((h * math.pow((D_m * (M_m * (0.5 / d))), 2.0)) * (-1.0 / l))))

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * (Float64(D_m * Float64(M_m * Float64(0.5 / d))) ^ 2.0)) * Float64(-1.0 / l)))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp = code(w0, M_m, D_m, h, l, d)
	tmp = w0 * sqrt((1.0 + ((h * ((D_m * (M_m * (0.5 / d))) ^ 2.0)) * (-1.0 / l))));
end

Wolfram

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

Derivation

1. Initial program 78.6%

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

3. Simplified 79.4%

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

   1. associate-*r/ 85.1%

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

   2. clear-num 85.1%

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

   3. associate-*l/ 84.7%

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

   4. div-inv 84.7%

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

   5. associate-*l* 84.6%

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

   6. associate-/r* 84.6%

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

   7. metadata-eval 84.6%

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

5. Applied egg-rr 84.6%

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

   1. associate-/r/ 84.7%

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

   2. *-commutative 84.7%

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

   3. *-commutative 84.7%

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

   4. associate-*r* 85.1%

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

   5. *-commutative 85.1%

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

7. Simplified 85.1%

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

8. Final simplification 85.1%

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

Alternative 4: 78.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-314}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, w0 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M_m \cdot \frac{D_m}{d}\right) \cdot \left(D_m \cdot M_m\right)}{d}\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= (/ h l) -5e-314)
   (fma -0.125 (* w0 (* (/ h l) (/ (* (* M_m (/ D_m d)) (* D_m M_m)) d))) w0)
   w0))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -5e-314) {
		tmp = fma(-0.125, (w0 * ((h / l) * (((M_m * (D_m / d)) * (D_m * M_m)) / d))), w0);
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -5e-314)
		tmp = fma(-0.125, Float64(w0 * Float64(Float64(h / l) * Float64(Float64(Float64(M_m * Float64(D_m / d)) * Float64(D_m * M_m)) / d))), w0);
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -5e-314], N[(-0.125 * N[(w0 * N[(N[(h / l), $MachinePrecision] * N[(N[(N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -4.99999999982e-314

   1. Initial program 76.7%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in M around 0 42.7%

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

      1. +-commutative 42.7%

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

      2. fma-def 42.7%

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

      3. associate-*r* 44.1%

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

      4. times-frac 45.5%

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

      5. *-commutative 45.5%

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

      6. unpow2 45.5%

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

      7. unpow2 45.5%

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

      8. swap-sqr 54.7%

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

      9. unpow2 54.7%

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

      10. *-commutative 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in D around 0 42.7%

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

      1. associate-*r* 44.1%

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

      2. unpow2 44.1%

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

      3. unpow2 44.1%

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

      4. swap-sqr 54.8%

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

      5. unpow2 54.8%

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

      6. times-frac 54.7%

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

      7. unpow2 54.7%

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

      8. unpow2 54.7%

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

      9. times-frac 62.1%

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

      10. unpow2 62.1%

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

      11. associate-*l/ 64.2%

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

      12. associate-*r* 68.4%

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

   9. Simplified 68.5%

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

       1. associate-/l* 68.4%

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

       2. pow2 68.4%

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

       3. associate-*r/ 68.4%

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

       4. associate-/l* 68.4%

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

       5. associate-/r/ 67.7%

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

   11. Applied egg-rr 67.7%

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

   if -4.99999999982e-314 < (/.f64 h l)

   1. Initial program 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in M around 0 92.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{-314}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, w0 \cdot \left(\frac{h}{\ell} \cdot \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(D \cdot M\right)}{d}\right), w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D_m}{\frac{d}{M_m}}\\ \mathbf{if}\;M_m \leq 3.3 \cdot 10^{-201}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{t_0}{\ell} \cdot \frac{t_0}{\frac{1}{w0 \cdot h}}, w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (/ D_m (/ d M_m))))
   (if (<= M_m 3.3e-201)
     w0
     (fma -0.125 (* (/ t_0 l) (/ t_0 (/ 1.0 (* w0 h)))) w0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m / (d / M_m);
	double tmp;
	if (M_m <= 3.3e-201) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, ((t_0 / l) * (t_0 / (1.0 / (w0 * h)))), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m / Float64(d / M_m))
	tmp = 0.0
	if (M_m <= 3.3e-201)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(Float64(t_0 / l) * Float64(t_0 / Float64(1.0 / Float64(w0 * h)))), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 3.3e-201], w0, N[(-0.125 * N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / N[(1.0 / N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 3.3000000000000003e-201

   1. Initial program 78.7%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in M around 0 69.3%

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

   if 3.3000000000000003e-201 < M

   1. Initial program 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in M around 0 49.2%

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

      1. +-commutative 49.2%

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

      2. fma-def 49.2%

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

      3. associate-*r* 51.1%

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

      4. times-frac 52.1%

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

      5. *-commutative 52.1%

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

      6. unpow2 52.1%

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

      7. unpow2 52.1%

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

      8. swap-sqr 61.8%

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

      9. unpow2 61.8%

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

      10. *-commutative 61.8%

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

   6. Simplified 61.8%

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

      1. associate-*r/ 63.7%

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

      2. add-sqr-sqrt 63.7%

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

      3. pow2 63.7%

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

      4. sqrt-div 63.7%

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

      5. unpow2 63.7%

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

      6. sqrt-prod 43.6%

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

      7. add-sqr-sqrt 70.4%

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

      8. unpow2 70.4%

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

      9. sqrt-prod 37.5%

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

      10. add-sqr-sqrt 74.3%

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

   8. Applied egg-rr 74.3%

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

      1. associate-/l* 72.4%

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

      2. associate-/l* 72.4%

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

   10. Simplified 72.4%

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

       1. associate-/r/ 72.4%

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

       2. unpow2 72.4%

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

       3. div-inv 72.4%

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

       4. times-frac 78.0%

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

       5. associate-/r/ 77.1%

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

       6. associate-/r/ 77.1%

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

   12. Applied egg-rr 77.1%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.3 \cdot 10^{-201}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{\frac{D}{\frac{d}{M}}}{\ell} \cdot \frac{\frac{D}{\frac{d}{M}}}{\frac{1}{w0 \cdot h}}, w0\right)\\ \end{array} \]

Alternative 6: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D_m}{\frac{d}{M_m}}\\ \mathbf{if}\;M_m \leq 8 \cdot 10^{-185}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, t_0 \cdot \frac{t_0}{\frac{\ell}{w0 \cdot h}}, w0\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (/ D_m (/ d M_m))))
   (if (<= M_m 8e-185) w0 (fma -0.125 (* t_0 (/ t_0 (/ l (* w0 h)))) w0))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m / (d / M_m);
	double tmp;
	if (M_m <= 8e-185) {
		tmp = w0;
	} else {
		tmp = fma(-0.125, (t_0 * (t_0 / (l / (w0 * h)))), w0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m / Float64(d / M_m))
	tmp = 0.0
	if (M_m <= 8e-185)
		tmp = w0;
	else
		tmp = fma(-0.125, Float64(t_0 * Float64(t_0 / Float64(l / Float64(w0 * h)))), w0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 8e-185], w0, N[(-0.125 * N[(t$95$0 * N[(t$95$0 / N[(l / N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 7.9999999999999999e-185

   1. Initial program 78.1%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in M around 0 69.0%

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

   if 7.9999999999999999e-185 < M

   1. Initial program 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in M around 0 49.6%

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

      1. +-commutative 49.6%

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

      2. fma-def 49.6%

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

      3. associate-*r* 51.7%

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

      4. times-frac 52.7%

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

      5. *-commutative 52.7%

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

      6. unpow2 52.7%

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

      7. unpow2 52.7%

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

      8. swap-sqr 61.8%

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

      9. unpow2 61.8%

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

      10. *-commutative 61.8%

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

   6. Simplified 61.8%

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

      1. associate-*r/ 63.9%

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

      2. add-sqr-sqrt 63.9%

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

      3. pow2 63.9%

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

      4. sqrt-div 63.9%

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

      5. unpow2 63.9%

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

      6. sqrt-prod 43.8%

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

      7. add-sqr-sqrt 70.9%

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

      8. unpow2 70.9%

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

      9. sqrt-prod 38.3%

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

      10. add-sqr-sqrt 74.9%

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

   8. Applied egg-rr 74.9%

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

      1. associate-/l* 72.9%

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

      2. associate-/l* 72.9%

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

   10. Simplified 72.9%

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

       1. associate-/r/ 72.9%

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

       2. unpow2 72.9%

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

       3. *-un-lft-identity 72.9%

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

       4. times-frac 76.9%

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

       5. associate-/r/ 76.0%

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

       6. associate-/r/ 76.0%

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

   12. Applied egg-rr 76.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 8 \cdot 10^{-185}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \frac{D}{\frac{d}{M}} \cdot \frac{\frac{D}{\frac{d}{M}}}{\frac{\ell}{w0 \cdot h}}, w0\right)\\ \end{array} \]

Alternative 7: 72.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 3 \cdot 10^{+29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{h \cdot \left(-{\left(M_m \cdot \frac{D_m}{d}\right)}^{2}\right)}{\frac{-\ell}{w0}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 3e+29)
   w0
   (* -0.125 (/ (* h (- (pow (* M_m (/ D_m d)) 2.0))) (/ (- l) w0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 3e+29) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((h * -pow((M_m * (D_m / d)), 2.0)) / (-l / w0));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 3d+29) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((h * -((m_m * (d_m / d)) ** 2.0d0)) / (-l / w0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 3e+29) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((h * -Math.pow((M_m * (D_m / d)), 2.0)) / (-l / w0));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 3e+29:
		tmp = w0
	else:
		tmp = -0.125 * ((h * -math.pow((M_m * (D_m / d)), 2.0)) / (-l / w0))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 3e+29)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(h * Float64(-(Float64(M_m * Float64(D_m / d)) ^ 2.0))) / Float64(Float64(-l) / w0)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 3e+29)
		tmp = w0;
	else
		tmp = -0.125 * ((h * -((M_m * (D_m / d)) ^ 2.0)) / (-l / w0));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 3e+29], w0, N[(-0.125 * N[(N[(h * (-N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] / N[((-l) / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.9999999999999999e29

   1. Initial program 80.4%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in M around 0 72.6%

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

   if 2.9999999999999999e29 < M

   1. Initial program 71.6%

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

   3. Simplified 71.7%

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

      1. associate-*r/ 78.1%

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

      2. clear-num 78.1%

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

      3. associate-*l/ 78.1%

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

      4. div-inv 78.1%

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

      5. associate-*l* 78.1%

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

      6. associate-/r* 78.1%

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

      7. metadata-eval 78.1%

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

   5. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.1%

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

      2. *-commutative 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*r* 78.0%

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

      5. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in l around inf 43.4%

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

      1. associate-*r* 43.5%

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

      2. unpow2 43.5%

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

      3. unpow2 43.5%

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

      4. swap-sqr 60.9%

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

      5. unpow2 60.9%

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

      6. *-commutative 60.9%

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

   10. Simplified 60.9%

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

   11. Taylor expanded in h around inf 20.8%

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

       1. associate-*r* 20.7%

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

       2. times-frac 22.6%

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

       3. unpow2 22.6%

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

       4. unpow2 22.6%

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

       5. swap-sqr 25.3%

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

       6. unpow2 25.3%

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

       7. times-frac 25.8%

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

       8. associate-*l/ 25.8%

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

       9. associate-*l/ 25.8%

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

       10. unpow2 25.8%

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

       11. *-commutative 25.8%

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

       12. associate-*l/ 23.8%

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

       13. *-commutative 23.8%

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

   13. Simplified 23.8%

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

       1. *-commutative 23.8%

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

       2. unpow2 23.8%

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

       3. associate-*l/ 23.8%

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

       4. associate-*l/ 23.8%

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

       5. frac-times 23.2%

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

       6. unpow2 23.2%

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

       7. unpow2 23.2%

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

       8. *-commutative 23.2%

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

       9. associate-/r/ 25.4%

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

       10. frac-2neg 25.4%

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

       11. frac-times 25.3%

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

   15. Applied egg-rr 25.3%

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

       1. times-frac 25.4%

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

       2. associate-*r/ 25.6%

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

       3. unpow2 25.6%

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

       4. associate-*r/ 25.9%

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

       5. associate-/l* 25.9%

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

       6. unpow2 25.9%

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

       7. associate-*r/ 26.0%

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

       8. associate-*r/ 25.9%

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

       9. times-frac 26.0%

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

       10. associate-/l* 26.0%

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

       11. unpow2 26.0%

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

       12. associate-/r/ 26.0%

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

       13. distribute-neg-frac 26.0%

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

   17. Simplified 26.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{h \cdot \left(-{\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\frac{-\ell}{w0}}\\ \end{array} \]

Alternative 8: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 3 \cdot 10^{+29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(M_m \cdot \frac{D_m}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 3e+29)
   w0
   (* -0.125 (* (pow (* M_m (/ D_m d)) 2.0) (/ h (/ l w0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 3e+29) {
		tmp = w0;
	} else {
		tmp = -0.125 * (pow((M_m * (D_m / d)), 2.0) * (h / (l / w0)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 3d+29) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((m_m * (d_m / d)) ** 2.0d0) * (h / (l / w0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 3e+29) {
		tmp = w0;
	} else {
		tmp = -0.125 * (Math.pow((M_m * (D_m / d)), 2.0) * (h / (l / w0)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 3e+29:
		tmp = w0
	else:
		tmp = -0.125 * (math.pow((M_m * (D_m / d)), 2.0) * (h / (l / w0)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 3e+29)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64((Float64(M_m * Float64(D_m / d)) ^ 2.0) * Float64(h / Float64(l / w0))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 3e+29)
		tmp = w0;
	else
		tmp = -0.125 * (((M_m * (D_m / d)) ^ 2.0) * (h / (l / w0)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 3e+29], w0, N[(-0.125 * N[(N[Power[N[(M$95$m * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / N[(l / w0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.9999999999999999e29

   1. Initial program 80.4%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in M around 0 72.6%

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

   if 2.9999999999999999e29 < M

   1. Initial program 71.6%

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

   3. Simplified 71.7%

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

      1. associate-*r/ 78.1%

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

      2. clear-num 78.1%

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

      3. associate-*l/ 78.1%

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

      4. div-inv 78.1%

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

      5. associate-*l* 78.1%

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

      6. associate-/r* 78.1%

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

      7. metadata-eval 78.1%

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

   5. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.1%

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

      2. *-commutative 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*r* 78.0%

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

      5. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in l around inf 43.4%

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

      1. associate-*r* 43.5%

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

      2. unpow2 43.5%

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

      3. unpow2 43.5%

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

      4. swap-sqr 60.9%

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

      5. unpow2 60.9%

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

      6. *-commutative 60.9%

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

   10. Simplified 60.9%

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

   11. Taylor expanded in h around inf 20.8%

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

       1. associate-*r* 20.7%

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

       2. times-frac 22.6%

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

       3. unpow2 22.6%

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

       4. unpow2 22.6%

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

       5. swap-sqr 25.3%

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

       6. unpow2 25.3%

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

       7. times-frac 25.8%

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

       8. associate-*l/ 25.8%

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

       9. associate-*l/ 25.8%

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

       10. unpow2 25.8%

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

       11. *-commutative 25.8%

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

       12. associate-*l/ 23.8%

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

       13. *-commutative 23.8%

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

   13. Simplified 23.8%

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

   14. Taylor expanded in w0 around 0 25.8%

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

       1. associate-/l* 26.0%

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

   16. Simplified 26.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left({\left(M \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\\ \end{array} \]

Alternative 9: 72.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} \mathbf{if}\;M_m \leq 2 \cdot 10^{+31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(w0 \cdot h\right) \cdot {\left(\frac{D_m}{\frac{d}{M_m}}\right)}^{2}}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (if (<= M_m 2e+31)
   w0
   (* -0.125 (/ (* (* w0 h) (pow (/ D_m (/ d M_m)) 2.0)) l))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2e+31) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((w0 * h) * pow((D_m / (d / M_m)), 2.0)) / l);
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: tmp
    if (m_m <= 2d+31) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((w0 * h) * ((d_m / (d / m_m)) ** 2.0d0)) / l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double tmp;
	if (M_m <= 2e+31) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((w0 * h) * Math.pow((D_m / (d / M_m)), 2.0)) / l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	tmp = 0
	if M_m <= 2e+31:
		tmp = w0
	else:
		tmp = -0.125 * (((w0 * h) * math.pow((D_m / (d / M_m)), 2.0)) / l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	tmp = 0.0
	if (M_m <= 2e+31)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(w0 * h) * (Float64(D_m / Float64(d / M_m)) ^ 2.0)) / l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	tmp = 0.0;
	if (M_m <= 2e+31)
		tmp = w0;
	else
		tmp = -0.125 * (((w0 * h) * ((D_m / (d / M_m)) ^ 2.0)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := If[LessEqual[M$95$m, 2e+31], w0, N[(-0.125 * N[(N[(N[(w0 * h), $MachinePrecision] * N[Power[N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.9999999999999999e31

   1. Initial program 80.4%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in M around 0 72.6%

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

   if 1.9999999999999999e31 < M

   1. Initial program 71.6%

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

   3. Simplified 71.7%

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

      1. associate-*r/ 78.1%

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

      2. clear-num 78.1%

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

      3. associate-*l/ 78.1%

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

      4. div-inv 78.1%

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

      5. associate-*l* 78.1%

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

      6. associate-/r* 78.1%

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

      7. metadata-eval 78.1%

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

   5. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.1%

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

      2. *-commutative 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*r* 78.0%

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

      5. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in l around inf 43.4%

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

      1. associate-*r* 43.5%

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

      2. unpow2 43.5%

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

      3. unpow2 43.5%

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

      4. swap-sqr 60.9%

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

      5. unpow2 60.9%

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

      6. *-commutative 60.9%

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

   10. Simplified 60.9%

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

   11. Taylor expanded in h around inf 20.8%

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

       1. associate-*r* 20.7%

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

       2. times-frac 22.6%

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

       3. unpow2 22.6%

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

       4. unpow2 22.6%

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

       5. swap-sqr 25.3%

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

       6. unpow2 25.3%

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

       7. times-frac 25.8%

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

       8. associate-*l/ 25.8%

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

       9. associate-*l/ 25.8%

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

       10. unpow2 25.8%

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

       11. *-commutative 25.8%

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

       12. associate-*l/ 23.8%

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

       13. *-commutative 23.8%

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

   13. Simplified 23.8%

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

       1. *-commutative 23.8%

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

       2. metadata-eval 23.8%

          \[\leadsto -0.125 \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{\color{blue}{\left(0.3333333333333333 \cdot 6\right)}} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right) \]

       3. metadata-eval 23.8%

          \[\leadsto -0.125 \cdot \left({\left(\frac{D}{d} \cdot M\right)}^{\left(0.3333333333333333 \cdot \color{blue}{\left(2 \cdot 3\right)}\right)} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right) \]

       4. pow-pow 20.5%

          \[\leadsto -0.125 \cdot \left(\color{blue}{{\left({\left(\frac{D}{d} \cdot M\right)}^{0.3333333333333333}\right)}^{\left(2 \cdot 3\right)}} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right) \]

       5. pow1/3 23.8%

          \[\leadsto -0.125 \cdot \left({\color{blue}{\left(\sqrt[3]{\frac{D}{d} \cdot M}\right)}}^{\left(2 \cdot 3\right)} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right) \]

       6. pow-pow 23.8%

          \[\leadsto -0.125 \cdot \left(\color{blue}{{\left({\left(\sqrt[3]{\frac{D}{d} \cdot M}\right)}^{2}\right)}^{3}} \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right) \]

       7. associate-*r/ 25.8%

          \[\leadsto -0.125 \cdot \left({\left({\left(\sqrt[3]{\frac{D}{d} \cdot M}\right)}^{2}\right)}^{3} \cdot \color{blue}{\frac{w0 \cdot h}{\ell}}\right) \]

       8. *-commutative 25.8%

          \[\leadsto -0.125 \cdot \left({\left({\left(\sqrt[3]{\frac{D}{d} \cdot M}\right)}^{2}\right)}^{3} \cdot \frac{\color{blue}{h \cdot w0}}{\ell}\right) \]

       9. associate-*r/ 25.9%

          \[\leadsto -0.125 \cdot \color{blue}{\frac{{\left({\left(\sqrt[3]{\frac{D}{d} \cdot M}\right)}^{2}\right)}^{3} \cdot \left(h \cdot w0\right)}{\ell}} \]

       10. pow-pow 25.9%

           \[\leadsto -0.125 \cdot \frac{\color{blue}{{\left(\sqrt[3]{\frac{D}{d} \cdot M}\right)}^{\left(2 \cdot 3\right)}} \cdot \left(h \cdot w0\right)}{\ell} \]

       11. pow1/3 22.6%

           \[\leadsto -0.125 \cdot \frac{{\color{blue}{\left({\left(\frac{D}{d} \cdot M\right)}^{0.3333333333333333}\right)}}^{\left(2 \cdot 3\right)} \cdot \left(h \cdot w0\right)}{\ell} \]

       12. pow-pow 25.9%

           \[\leadsto -0.125 \cdot \frac{\color{blue}{{\left(\frac{D}{d} \cdot M\right)}^{\left(0.3333333333333333 \cdot \left(2 \cdot 3\right)\right)}} \cdot \left(h \cdot w0\right)}{\ell} \]

       13. associate-/r/ 25.9%

           \[\leadsto -0.125 \cdot \frac{{\color{blue}{\left(\frac{D}{\frac{d}{M}}\right)}}^{\left(0.3333333333333333 \cdot \left(2 \cdot 3\right)\right)} \cdot \left(h \cdot w0\right)}{\ell} \]

       14. metadata-eval 25.9%

           \[\leadsto -0.125 \cdot \frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{\left(0.3333333333333333 \cdot \color{blue}{6}\right)} \cdot \left(h \cdot w0\right)}{\ell} \]

       15. metadata-eval 25.9%

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

   15. Applied egg-rr 25.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{+31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(w0 \cdot h\right) \cdot {\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}\\ \end{array} \]

Alternative 10: 72.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [w0, M_m, D_m, h, l, d] = \mathsf{sort}([w0, M_m, D_m, h, l, d])\\ \\ \begin{array}{l} t_0 := \frac{D_m}{\frac{d}{M_m}}\\ \mathbf{if}\;M_m \leq 3 \cdot 10^{+29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
(FPCore (w0 M_m D_m h l d)
 :precision binary64
 (let* ((t_0 (/ D_m (/ d M_m))))
   (if (<= M_m 3e+29) w0 (* -0.125 (* (* t_0 t_0) (* w0 (/ h l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(w0 < M_m && M_m < D_m && D_m < h && h < l && l < d);
double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m / (d / M_m);
	double tmp;
	if (M_m <= 3e+29) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((t_0 * t_0) * (w0 * (h / l)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
D_m = abs(D)
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
real(8) function code(w0, m_m, d_m, h, l, d)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_m / (d / m_m)
    if (m_m <= 3d+29) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((t_0 * t_0) * (w0 * (h / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert w0 < M_m && M_m < D_m && D_m < h && h < l && l < d;
public static double code(double w0, double M_m, double D_m, double h, double l, double d) {
	double t_0 = D_m / (d / M_m);
	double tmp;
	if (M_m <= 3e+29) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((t_0 * t_0) * (w0 * (h / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[w0, M_m, D_m, h, l, d] = sort([w0, M_m, D_m, h, l, d])
def code(w0, M_m, D_m, h, l, d):
	t_0 = D_m / (d / M_m)
	tmp = 0
	if M_m <= 3e+29:
		tmp = w0
	else:
		tmp = -0.125 * ((t_0 * t_0) * (w0 * (h / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
w0, M_m, D_m, h, l, d = sort([w0, M_m, D_m, h, l, d])
function code(w0, M_m, D_m, h, l, d)
	t_0 = Float64(D_m / Float64(d / M_m))
	tmp = 0.0
	if (M_m <= 3e+29)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(t_0 * t_0) * Float64(w0 * Float64(h / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
w0, M_m, D_m, h, l, d = num2cell(sort([w0, M_m, D_m, h, l, d])){:}
function tmp_2 = code(w0, M_m, D_m, h, l, d)
	t_0 = D_m / (d / M_m);
	tmp = 0.0;
	if (M_m <= 3e+29)
		tmp = w0;
	else
		tmp = -0.125 * ((t_0 * t_0) * (w0 * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: w0, M_m, D_m, h, l, and d should be sorted in increasing order before calling this function.
code[w0_, M$95$m_, D$95$m_, h_, l_, d_] := Block[{t$95$0 = N[(D$95$m / N[(d / M$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 3e+29], w0, N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(w0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.9999999999999999e29

   1. Initial program 80.4%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in M around 0 72.6%

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

   if 2.9999999999999999e29 < M

   1. Initial program 71.6%

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

   3. Simplified 71.7%

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

      1. associate-*r/ 78.1%

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

      2. clear-num 78.1%

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

      3. associate-*l/ 78.1%

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

      4. div-inv 78.1%

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

      5. associate-*l* 78.1%

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

      6. associate-/r* 78.1%

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

      7. metadata-eval 78.1%

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

   5. Applied egg-rr 78.1%

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

      1. associate-/r/ 78.1%

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

      2. *-commutative 78.1%

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

      3. *-commutative 78.1%

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

      4. associate-*r* 78.0%

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

      5. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in l around inf 43.4%

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

      1. associate-*r* 43.5%

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

      2. unpow2 43.5%

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

      3. unpow2 43.5%

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

      4. swap-sqr 60.9%

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

      5. unpow2 60.9%

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

      6. *-commutative 60.9%

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

   10. Simplified 60.9%

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

   11. Taylor expanded in h around inf 20.8%

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

       1. associate-*r* 20.7%

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

       2. times-frac 22.6%

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

       3. unpow2 22.6%

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

       4. unpow2 22.6%

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

       5. swap-sqr 25.3%

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

       6. unpow2 25.3%

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

       7. times-frac 25.8%

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

       8. associate-*l/ 25.8%

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

       9. associate-*l/ 25.8%

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

       10. unpow2 25.8%

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

       11. *-commutative 25.8%

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

       12. associate-*l/ 23.8%

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

       13. *-commutative 23.8%

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

   13. Simplified 23.8%

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

       1. *-commutative 23.8%

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

       2. associate-/r/ 23.8%

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

       3. unpow2 23.8%

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

   15. Applied egg-rr 23.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{+29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{\frac{d}{M}} \cdot \frac{D}{\frac{d}{M}}\right) \cdot \left(w0 \cdot \frac{h}{\ell}\right)\right)\\ \end{array} \]

Alternative 11: 67.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

3. Simplified 79.4%

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

4. Taylor expanded in M around 0 69.5%

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

5. Final simplification 69.5%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023334 
(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))))))