Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.7% → 88.7%

Time: 16.9s

Alternatives: 8

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: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M = |M|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + \left(-2 \cdot \log d + 2 \cdot \log D\right)\right)}\right)}^{3}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (pow
      (*
       (cbrt w0)
       (exp
        (*
         0.16666666666666666
         (+
          (log (* -0.25 (/ (* h (pow M 2.0)) l)))
          (+ (* -2.0 (log d)) (* 2.0 (log D)))))))
      3.0)
     (if (<= t_0 0.5) (* w0 (sqrt (- 1.0 t_0))) w0))))

C

M = abs(M);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * (log((-0.25 * ((h * pow(M, 2.0)) / l))) + ((-2.0 * log(d)) + (2.0 * log(D))))))), 3.0);
	} else if (t_0 <= 0.5) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

M = Math.abs(M);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * (Math.log((-0.25 * ((h * Math.pow(M, 2.0)) / l))) + ((-2.0 * Math.log(d)) + (2.0 * Math.log(D))))))), 3.0);
	} else if (t_0 <= 0.5) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M = abs(M)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(log(Float64(-0.25 * Float64(Float64(h * (M ^ 2.0)) / l))) + Float64(Float64(-2.0 * log(d)) + Float64(2.0 * log(D))))))) ^ 3.0;
	elseif (t_0 <= 0.5)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[(-0.25 * N[(N[(h * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(-2.0 * N[Log[d], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Log[D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 57.3%

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

   3. Simplified 57.3%

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

   5. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}}\right)}^{3}} \]

   6. Taylor expanded in d around 0 24.8%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 35.0%

         \[\leadsto {\left(\color{blue}{\sqrt[3]{1 \cdot w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      2. *-lft-identity 35.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      3. exp-prod 34.8%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{{\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}}\right)}^{3} \]

      4. distribute-lft-neg-in 34.8%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \color{blue}{\left(\left(-0.25\right) \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right)} + -2 \cdot \log d\right)}\right)}^{3} \]

      5. metadata-eval 34.8%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(\color{blue}{-0.25} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      6. *-commutative 34.8%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \frac{\color{blue}{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      7. associate-*r/ 33.5%

         \[\leadsto {\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot \frac{{D}^{2}}{\ell}\right)}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

   8. Simplified 33.5%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{w0} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(\log \left(-0.25 \cdot \left(\left({M}^{2} \cdot h\right) \cdot \frac{{D}^{2}}{\ell}\right)\right) + -2 \cdot \log d\right)}\right)}}^{3} \]

   9. Taylor expanded in D around 0 15.0%

      \[\leadsto {\left(\sqrt[3]{w0} \cdot \color{blue}{e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{M}^{2} \cdot h}{\ell}\right) + \left(-2 \cdot \log d + 2 \cdot \log D\right)\right)}}\right)}^{3} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 0.5

   1. Initial program 99.9%

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

   if 0.5 < (*.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%

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

   4. Taylor expanded in D around 0 72.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {M}^{2}}{\ell}\right) + \left(-2 \cdot \log d + 2 \cdot \log D\right)\right)}\right)}^{3}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.5:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M = |M|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log d + \log \left(\frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell} \cdot \left(-0.25\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 (- INFINITY))
     (pow
      (*
       (cbrt w0)
       (exp
        (*
         0.16666666666666666
         (+
          (* -2.0 (log d))
          (log (* (/ (* h (pow (* M D) 2.0)) l) (- 0.25)))))))
      3.0)
     (if (<= t_0 0.5) (* w0 (sqrt (- 1.0 t_0))) w0))))

C

M = abs(M);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * ((-2.0 * log(d)) + log((((h * pow((M * D), 2.0)) / l) * -0.25)))))), 3.0);
	} else if (t_0 <= 0.5) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

M = Math.abs(M);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * ((-2.0 * Math.log(d)) + Math.log((((h * Math.pow((M * D), 2.0)) / l) * -0.25)))))), 3.0);
	} else if (t_0 <= 0.5) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M = abs(M)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(Float64(-2.0 * log(d)) + log(Float64(Float64(Float64(h * (Float64(M * D) ^ 2.0)) / l) * Float64(-0.25))))))) ^ 3.0;
	elseif (t_0 <= 0.5)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[(-2.0 * N[Log[d], $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(N[(h * N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * (-0.25)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -inf.0

   1. Initial program 57.3%

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

   3. Simplified 57.3%

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

   5. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{2}}}\right)}^{3}} \]

   6. Taylor expanded in d around 0 24.8%

      \[\leadsto {\color{blue}{\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell}\right) + -2 \cdot \log d\right)}\right)}}^{3} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 11.5%

         \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right)}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      2. expm1-udef 9.9%

         \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)} - 1}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      3. associate-*r* 9.9%

         \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}\right)} - 1}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      4. pow-prod-down 9.9%

         \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(D \cdot M\right)}^{2}} \cdot h\right)} - 1}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

   8. Applied egg-rr 9.9%

      \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(D \cdot M\right)}^{2} \cdot h\right)} - 1}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]
   9. Step-by-step derivation

      1. expm1-def 11.9%

         \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(D \cdot M\right)}^{2} \cdot h\right)\right)}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      2. expm1-log1p 25.2%

         \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{{\left(D \cdot M\right)}^{2} \cdot h}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

      3. *-commutative 25.2%

         \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{h \cdot {\left(D \cdot M\right)}^{2}}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

   10. Simplified 25.2%

       \[\leadsto {\left({\left(1 \cdot w0\right)}^{0.3333333333333333} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{\color{blue}{h \cdot {\left(D \cdot M\right)}^{2}}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 25.2%

          \[\leadsto {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(1 \cdot w0\right)}^{0.3333333333333333}\right)\right)} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

       2. expm1-udef 20.3%

          \[\leadsto {\left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left(1 \cdot w0\right)}^{0.3333333333333333}\right)} - 1\right)} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

       3. unpow1/3 20.5%

          \[\leadsto {\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt[3]{1 \cdot w0}}\right)} - 1\right) \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

       4. *-un-lft-identity 20.5%

          \[\leadsto {\left(\left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{w0}}\right)} - 1\right) \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

   12. Applied egg-rr 20.5%

       \[\leadsto {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{w0}\right)} - 1\right)} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]
   13. Step-by-step derivation

       1. expm1-def 26.8%

          \[\leadsto {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{w0}\right)\right)} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

       2. expm1-log1p 38.4%

          \[\leadsto {\left(\color{blue}{\sqrt[3]{w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

   14. Simplified 38.4%

       \[\leadsto {\left(\color{blue}{\sqrt[3]{w0}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}\right) + -2 \cdot \log d\right)}\right)}^{3} \]

   if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 0.5

   1. Initial program 99.9%

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

   if 0.5 < (*.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%

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

   4. Taylor expanded in D around 0 72.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(-2 \cdot \log d + \log \left(\frac{h \cdot {\left(M \cdot D\right)}^{2}}{\ell} \cdot \left(-0.25\right)\right)\right)}\right)}^{3}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.5:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 0.5:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))
   (if (<= t_0 0.5) (* w0 (sqrt (- 1.0 t_0))) w0)))

C

M = abs(M);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 0.5) {
		tmp = w0 * sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)
    if (t_0 <= 0.5d0) then
        tmp = w0 * sqrt((1.0d0 - t_0))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M = Math.abs(M);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double tmp;
	if (t_0 <= 0.5) {
		tmp = w0 * Math.sqrt((1.0 - t_0));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M = abs(M)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)
	tmp = 0
	if t_0 <= 0.5:
		tmp = w0 * math.sqrt((1.0 - t_0))
	else:
		tmp = w0
	return tmp

Julia

M = abs(M)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	tmp = 0.0
	if (t_0 <= 0.5)
		tmp = Float64(w0 * sqrt(Float64(1.0 - t_0)));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M = abs(M)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (((M * D) / (2.0 * d)) ^ 2.0) * (h / l);
	tmp = 0.0;
	if (t_0 <= 0.5)
		tmp = w0 * sqrt((1.0 - t_0));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.5], N[(w0 * N[Sqrt[N[(1.0 - t$95$0), $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)) < 0.5

   1. Initial program 89.1%

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

   if 0.5 < (*.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%

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

   4. Taylor expanded in D around 0 72.4%

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

4. Final simplification 87.9%

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

Alternative 4: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+288}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}, 1\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-281}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= (/ h l) -2e+288)
   (* w0 (fma -0.125 (* h (/ (pow (* M (/ D d)) 2.0) l)) 1.0))
   (if (<= (/ h l) -1e-281)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ D 2.0) (/ M d)) 2.0)))))
     w0)))

C

M = abs(M);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -2e+288) {
		tmp = w0 * fma(-0.125, (h * (pow((M * (D / d)), 2.0) / l)), 1.0);
	} else if ((h / l) <= -1e-281) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((D / 2.0) * (M / d)), 2.0))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Julia

M = abs(M)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(h / l) <= -2e+288)
		tmp = Float64(w0 * fma(-0.125, Float64(h * Float64((Float64(M * Float64(D / d)) ^ 2.0) / l)), 1.0));
	elseif (Float64(h / l) <= -1e-281)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(D / 2.0) * Float64(M / d)) ^ 2.0)))));
	else
		tmp = w0;
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[N[(h / l), $MachinePrecision], -2e+288], N[(w0 * N[(-0.125 * N[(h * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1e-281], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / 2.0), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 h l) < -2e288

   1. Initial program 42.0%

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

   3. Simplified 42.0%

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

      1. associate-*r/ 74.7%

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

      2. frac-times 74.7%

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

      3. *-commutative 74.7%

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

      4. *-un-lft-identity 74.7%

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

      5. times-frac 74.7%

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

      6. metadata-eval 74.7%

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

      7. *-commutative 74.7%

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

   5. Applied egg-rr 74.7%

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

      1. add-sqr-sqrt 55.3%

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

      2. pow2 55.3%

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

      3. sqrt-prod 40.0%

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

      4. sqrt-pow1 44.3%

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

      5. metadata-eval 44.3%

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

      6. pow1 44.3%

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

      7. associate-/l* 44.5%

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

   7. Applied egg-rr 44.5%

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

   8. Taylor expanded in D around 0 41.9%

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

      1. +-commutative 41.9%

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

      2. fma-def 41.9%

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

      3. times-frac 45.5%

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

      4. associate-/l* 30.4%

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

      5. associate-*r/ 30.4%

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

      6. unpow2 30.4%

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

      7. unpow2 30.4%

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

      8. times-frac 41.9%

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

      9. unpow2 41.9%

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

      10. swap-sqr 42.2%

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

      11. unpow2 42.2%

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

      12. associate-*l/ 42.2%

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

      13. associate-/l* 42.2%

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

   10. Simplified 42.2%

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

       1. associate-/r/ 64.3%

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

       2. associate-/r/ 64.3%

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

   12. Applied egg-rr 64.3%

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

   if -2e288 < (/.f64 h l) < -1e-281

   1. Initial program 86.1%

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

   3. Simplified 84.5%

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

   if -1e-281 < (/.f64 h l)

   1. Initial program 88.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in D around 0 93.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+288}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}, 1\right)\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-281}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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 - ((h * ((0.5d0 * ((m * d) / d_1)) ** 2.0d0)) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 81.3%

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

   1. associate-*r/ 85.2%

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

   2. frac-times 86.4%

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

   3. *-commutative 86.4%

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

   4. *-un-lft-identity 86.4%

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

   5. times-frac 86.4%

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

   6. metadata-eval 86.4%

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

   7. *-commutative 86.4%

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

5. Applied egg-rr 86.4%

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

6. Final simplification 86.4%

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

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ w0 \cdot \mathsf{fma}\left(-0.125, h \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}, 1\right) \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (fma -0.125 (* h (/ (pow (* M (/ D d)) 2.0) l)) 1.0)))

C

M = abs(M);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * fma(-0.125, (h * (pow((M * (D / d)), 2.0) / l)), 1.0);
}

Julia

M = abs(M)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	return Float64(w0 * fma(-0.125, Float64(h * Float64((Float64(M * Float64(D / d)) ^ 2.0) / l)), 1.0))
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[(-0.125 * N[(h * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

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

3. Simplified 81.3%

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

   1. associate-*r/ 85.2%

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

   2. frac-times 86.4%

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

   3. *-commutative 86.4%

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

   4. *-un-lft-identity 86.4%

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

   5. times-frac 86.4%

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

   6. metadata-eval 86.4%

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

   7. *-commutative 86.4%

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

5. Applied egg-rr 86.4%

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

   1. add-sqr-sqrt 67.6%

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

   2. pow2 67.6%

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

   3. sqrt-prod 48.0%

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

   4. sqrt-pow1 49.2%

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

   5. metadata-eval 49.2%

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

   6. pow1 49.2%

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

   7. associate-/l* 49.3%

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

7. Applied egg-rr 49.3%

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

8. Taylor expanded in D around 0 54.2%

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

   1. +-commutative 54.2%

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

   2. fma-def 54.2%

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

   3. times-frac 55.9%

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

   4. associate-/l* 55.2%

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

   5. associate-*r/ 55.6%

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

   6. unpow2 55.6%

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

   7. unpow2 55.6%

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

   8. times-frac 66.5%

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

   9. unpow2 66.5%

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

   10. swap-sqr 75.4%

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

   11. unpow2 75.4%

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

   12. associate-*l/ 75.8%

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

   13. associate-/l* 75.4%

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

10. Simplified 75.4%

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

    1. associate-/r/ 79.3%

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

    2. associate-/r/ 79.2%

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

12. Applied egg-rr 79.2%

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

13. Final simplification 79.2%

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

Alternative 7: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;D \leq 1.55 \cdot 10^{+113}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{t_0 \cdot t_0}{\frac{\ell}{h}}, 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* M (/ D d))))
   (if (<= D 1.55e+113) w0 (* w0 (fma -0.125 (/ (* t_0 t_0) (/ l h)) 1.0)))))

C

M = abs(M);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D / d);
	double tmp;
	if (D <= 1.55e+113) {
		tmp = w0;
	} else {
		tmp = w0 * fma(-0.125, ((t_0 * t_0) / (l / h)), 1.0);
	}
	return tmp;
}

Julia

M = abs(M)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (D <= 1.55e+113)
		tmp = w0;
	else
		tmp = Float64(w0 * fma(-0.125, Float64(Float64(t_0 * t_0) / Float64(l / h)), 1.0));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 1.55e+113], w0, N[(w0 * N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.54999999999999996e113

   1. Initial program 84.0%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in D around 0 73.5%

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

   if 1.54999999999999996e113 < D

   1. Initial program 74.1%

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

   3. Simplified 73.5%

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

      1. associate-*r/ 73.8%

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

      2. frac-times 74.5%

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

      3. *-commutative 74.5%

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

      4. *-un-lft-identity 74.5%

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

      5. times-frac 74.5%

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

      6. metadata-eval 74.5%

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

      7. *-commutative 74.5%

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

   5. Applied egg-rr 74.5%

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

      1. add-sqr-sqrt 35.6%

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

      2. pow2 35.6%

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

      3. sqrt-prod 32.6%

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

      4. sqrt-pow1 35.3%

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

      5. metadata-eval 35.3%

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

      6. pow1 35.3%

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

      7. associate-/l* 35.3%

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

   7. Applied egg-rr 35.3%

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

   8. Taylor expanded in D around 0 41.8%

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

      1. +-commutative 41.8%

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

      2. fma-def 41.8%

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

      3. times-frac 44.7%

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

      4. associate-/l* 44.6%

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

      5. associate-*r/ 41.9%

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

      6. unpow2 41.9%

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

      7. unpow2 41.9%

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

      8. times-frac 56.0%

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

      9. unpow2 56.0%

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

      10. swap-sqr 61.5%

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

      11. unpow2 61.5%

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

      12. associate-*l/ 61.7%

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

      13. associate-/l* 61.7%

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

   10. Simplified 61.7%

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

       1. unpow2 61.7%

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

       2. associate-/r/ 61.5%

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

       3. associate-/r/ 61.5%

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

   12. Applied egg-rr 61.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.55 \cdot 10^{+113}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \mathsf{fma}\left(-0.125, \frac{\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{D}{d}\right)}{\frac{\ell}{h}}, 1\right)\\ \end{array} \]

Alternative 8: 68.2% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

M = abs(M)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0

Julia

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

MATLAB

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

Wolfram

NOTE: M should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 82.5%

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

3. Simplified 81.3%

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

4. Taylor expanded in D around 0 67.7%

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

5. Final simplification 67.7%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023318 
(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))))))