Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.3% → 90.5%

Time: 21.4s

Alternatives: 9

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% 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: 90.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot \left({\left(e^{0.16666666666666666}\right)}^{\log \left(-0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{M}{d}\right)\right)\right)} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(-2 \cdot \left(-\log D\right)\right)}\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

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 (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -2e+295)
   (pow
    (*
     (cbrt w0)
     (*
      (pow
       (exp 0.16666666666666666)
       (log (* -0.25 (* (/ M d) (* (/ h l) (/ M d))))))
      (pow (exp 0.16666666666666666) (* -2.0 (- (log D))))))
    3.0)
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* D (* M (/ 0.5 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) {
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+295) {
		tmp = pow((cbrt(w0) * (pow(exp(0.16666666666666666), log((-0.25 * ((M / d) * ((h / l) * (M / d)))))) * pow(exp(0.16666666666666666), (-2.0 * -log(D))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - (h * (pow((D * (M * (0.5 / d))), 2.0) / l))));
	}
	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 tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -2e+295) {
		tmp = Math.pow((Math.cbrt(w0) * (Math.pow(Math.exp(0.16666666666666666), Math.log((-0.25 * ((M / d) * ((h / l) * (M / d)))))) * Math.pow(Math.exp(0.16666666666666666), (-2.0 * -Math.log(D))))), 3.0);
	} else {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((D * (M * (0.5 / d))), 2.0) / l))));
	}
	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((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -2e+295)
		tmp = Float64(cbrt(w0) * Float64((exp(0.16666666666666666) ^ log(Float64(-0.25 * Float64(Float64(M / d) * Float64(Float64(h / l) * Float64(M / d)))))) * (exp(0.16666666666666666) ^ Float64(-2.0 * Float64(-log(D)))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)))));
	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[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -2e+295], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[(N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[Log[N[(-0.25 * N[(N[(M / d), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(-2.0 * (-N[Log[D], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -2e295

   1. Initial program 48.2%

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

   3. Simplified 48.2%

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

   5. Applied egg-rr 45.5%

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

   6. Taylor expanded in D around inf 12.8%

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

      1. *-commutative 12.8%

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

      2. unpow1/3 32.9%

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

      3. *-lft-identity 32.9%

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

      4. exp-prod 32.5%

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

      5. distribute-lft-neg-in 32.5%

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

      6. metadata-eval 32.5%

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

      7. unpow2 32.5%

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

      8. times-frac 31.2%

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

      9. unpow2 31.2%

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

      10. log-rec 31.2%

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

   8. Simplified 31.2%

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

      1. unpow-prod-up 31.2%

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

      2. times-frac 37.4%

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

   10. Applied egg-rr 37.4%

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

   12. Simplified 39.7%

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

   if -2e295 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 88.6%

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

   3. Simplified 89.2%

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

      1. frac-times 88.6%

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

      2. *-commutative 88.6%

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

      3. clear-num 88.6%

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

      4. un-div-inv 88.7%

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

      5. div-inv 88.7%

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

      6. associate-*l* 87.1%

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

      7. associate-/r* 87.1%

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

      8. metadata-eval 87.1%

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

   5. Applied egg-rr 87.1%

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

      1. associate-/r/ 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*r* 95.3%

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

      4. *-commutative 95.3%

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

   7. Simplified 95.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -2 \cdot 10^{+295}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot \left({\left(e^{0.16666666666666666}\right)}^{\log \left(-0.25 \cdot \left(\frac{M}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{M}{d}\right)\right)\right)} \cdot {\left(e^{0.16666666666666666}\right)}^{\left(-2 \cdot \left(-\log D\right)\right)}\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+168}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) - -2 \cdot \log D\right)}\right)}^{3}\\ \end{array} \end{array} \]

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 (<= (/ (* M D) (* 2.0 d)) 1e+168)
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* D (* M (/ 0.5 d))) 2.0) l)))))
   (pow
    (*
     (cbrt w0)
     (exp
      (*
       0.16666666666666666
       (- (log (* -0.25 (* (/ h l) (* (/ M d) (/ M d))))) (* -2.0 (log D))))))
    3.0)))

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 tmp;
	if (((M * D) / (2.0 * d)) <= 1e+168) {
		tmp = w0 * sqrt((1.0 - (h * (pow((D * (M * (0.5 / d))), 2.0) / l))));
	} else {
		tmp = pow((cbrt(w0) * exp((0.16666666666666666 * (log((-0.25 * ((h / l) * ((M / d) * (M / d))))) - (-2.0 * log(D)))))), 3.0);
	}
	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 tmp;
	if (((M * D) / (2.0 * d)) <= 1e+168) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((D * (M * (0.5 / d))), 2.0) / l))));
	} else {
		tmp = Math.pow((Math.cbrt(w0) * Math.exp((0.16666666666666666 * (Math.log((-0.25 * ((h / l) * ((M / d) * (M / d))))) - (-2.0 * Math.log(D)))))), 3.0);
	}
	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(Float64(M * D) / Float64(2.0 * d)) <= 1e+168)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)))));
	else
		tmp = Float64(cbrt(w0) * exp(Float64(0.16666666666666666 * Float64(log(Float64(-0.25 * Float64(Float64(h / l) * Float64(Float64(M / d) * Float64(M / d))))) - Float64(-2.0 * log(D)))))) ^ 3.0;
	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_] := If[LessEqual[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 1e+168], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Exp[N[(0.16666666666666666 * N[(N[Log[N[(-0.25 * N[(N[(h / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(-2.0 * N[Log[D], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 9.9999999999999993e167

   1. Initial program 83.4%

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

   3. Simplified 83.4%

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

      1. frac-times 83.4%

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

      2. *-commutative 83.4%

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

      3. clear-num 83.4%

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

      4. un-div-inv 83.4%

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

      5. div-inv 83.4%

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

      6. associate-*l* 80.9%

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

      7. associate-/r* 80.9%

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

      8. metadata-eval 80.9%

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

   5. Applied egg-rr 80.9%

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

      1. associate-/r/ 86.6%

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

      2. *-commutative 86.6%

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

      3. associate-*r* 88.8%

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

      4. *-commutative 88.8%

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

   7. Simplified 88.8%

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

   if 9.9999999999999993e167 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 28.3%

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

   3. Simplified 32.0%

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

   5. Applied egg-rr 31.9%

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

   6. Taylor expanded in D around inf 11.0%

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

      1. *-commutative 11.0%

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

      2. unpow1/3 26.1%

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

      3. *-lft-identity 26.1%

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

      4. exp-prod 25.6%

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

      5. distribute-lft-neg-in 25.6%

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

      6. metadata-eval 25.6%

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

      7. unpow2 25.6%

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

      8. times-frac 22.7%

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

      9. unpow2 22.7%

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

      10. log-rec 22.7%

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

   8. Simplified 22.7%

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

      1. pow-exp 22.9%

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

      2. times-frac 26.8%

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

   10. Applied egg-rr 26.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 10^{+168}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{w0} \cdot e^{0.16666666666666666 \cdot \left(\log \left(-0.25 \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) - -2 \cdot \log D\right)}\right)}^{3}\\ \end{array} \]

Alternative 3: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+286}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot -0.25}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\\ \end{array} \end{array} \]

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
 (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 4e+286)
   (* w0 (sqrt (- 1.0 (* h (/ (pow (* D (* M (/ 0.5 d))) 2.0) l)))))
   (*
    w0
    (sqrt (+ 1.0 (* (/ (* D -0.25) (/ l D)) (* (/ M d) (/ M (/ d h)))))))))

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 (pow(((M * D) / (2.0 * d)), 2.0) <= 4e+286) {
		tmp = w0 * sqrt((1.0 - (h * (pow((D * (M * (0.5 / d))), 2.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	}
	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) :: tmp
    if ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 4d+286) then
        tmp = w0 * sqrt((1.0d0 - (h * (((d * (m * (0.5d0 / d_1))) ** 2.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 + (((d * (-0.25d0)) / (l / d)) * ((m / d_1) * (m / (d_1 / h))))))
    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 tmp;
	if (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 4e+286) {
		tmp = w0 * Math.sqrt((1.0 - (h * (Math.pow((D * (M * (0.5 / d))), 2.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 4e+286:
		tmp = w0 * math.sqrt((1.0 - (h * (math.pow((D * (M * (0.5 / d))), 2.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))))
	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(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 4e+286)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64((Float64(D * Float64(M * Float64(0.5 / d))) ^ 2.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D * -0.25) / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / Float64(d / h)))))));
	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)
	tmp = 0.0;
	if ((((M * D) / (2.0 * d)) ^ 2.0) <= 4e+286)
		tmp = w0 * sqrt((1.0 - (h * (((D * (M * (0.5 / d))) ^ 2.0) / l))));
	else
		tmp = w0 * sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	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_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 4e+286], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[Power[N[(D * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D * -0.25), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 4.00000000000000013e286

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

      1. frac-times 88.3%

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

      2. *-commutative 88.3%

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

      3. clear-num 88.3%

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

      4. un-div-inv 88.3%

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

      5. div-inv 88.3%

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

      6. associate-*l* 85.4%

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

      7. associate-/r* 85.4%

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

      8. metadata-eval 85.4%

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

   5. Applied egg-rr 85.4%

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

      1. associate-/r/ 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r* 94.6%

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

      4. *-commutative 94.6%

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

   7. Simplified 94.6%

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

   if 4.00000000000000013e286 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 44.0%

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

   3. Simplified 45.7%

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

      1. clear-num 45.7%

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

      2. un-div-inv 45.7%

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

      3. div-inv 45.7%

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

      4. metadata-eval 45.7%

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

   5. Applied egg-rr 45.7%

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

   6. Taylor expanded in w0 around 0 40.3%

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

      1. *-commutative 40.3%

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

      2. cancel-sign-sub-inv 40.3%

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

      3. *-commutative 40.3%

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

      4. metadata-eval 40.3%

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

      5. times-frac 37.2%

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

      6. unpow2 37.2%

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

      7. associate-*l/ 40.3%

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

      8. associate-/r/ 40.4%

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

      9. unpow2 40.4%

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

      10. unpow2 40.4%

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

      11. associate-/r* 42.5%

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

      12. *-commutative 42.5%

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

      13. associate-*r* 47.2%

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

   8. Simplified 48.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 4 \cdot 10^{+286}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot -0.25}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\\ \end{array} \]

Alternative 4: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 5 \cdot 10^{-86}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot -0.25}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{+86}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= d 5e-86)
   (* w0 (sqrt (+ 1.0 (* (/ (* D -0.25) (/ l D)) (* (/ M d) (/ M (/ d h)))))))
   (if (<= d 1.18e+86)
     (* w0 (sqrt (- 1.0 (* 0.25 (/ (* D (* D (* M (* M h)))) (* l (* d d)))))))
     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 (d <= 5e-86) {
		tmp = w0 * sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	} else if (d <= 1.18e+86) {
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * (D * (M * (M * h)))) / (l * (d * d))))));
	} 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) :: tmp
    if (d_1 <= 5d-86) then
        tmp = w0 * sqrt((1.0d0 + (((d * (-0.25d0)) / (l / d)) * ((m / d_1) * (m / (d_1 / h))))))
    else if (d_1 <= 1.18d+86) then
        tmp = w0 * sqrt((1.0d0 - (0.25d0 * ((d * (d * (m * (m * h)))) / (l * (d_1 * d_1))))))
    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 tmp;
	if (d <= 5e-86) {
		tmp = w0 * Math.sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	} else if (d <= 1.18e+86) {
		tmp = w0 * Math.sqrt((1.0 - (0.25 * ((D * (D * (M * (M * h)))) / (l * (d * d))))));
	} 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):
	tmp = 0
	if d <= 5e-86:
		tmp = w0 * math.sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))))
	elif d <= 1.18e+86:
		tmp = w0 * math.sqrt((1.0 - (0.25 * ((D * (D * (M * (M * h)))) / (l * (d * d))))))
	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 (d <= 5e-86)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D * -0.25) / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / Float64(d / h)))))));
	elseif (d <= 1.18e+86)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(0.25 * Float64(Float64(D * Float64(D * Float64(M * Float64(M * h)))) / Float64(l * Float64(d * d)))))));
	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)
	tmp = 0.0;
	if (d <= 5e-86)
		tmp = w0 * sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	elseif (d <= 1.18e+86)
		tmp = w0 * sqrt((1.0 - (0.25 * ((D * (D * (M * (M * h)))) / (l * (d * d))))));
	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_] := If[LessEqual[d, 5e-86], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D * -0.25), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.18e+86], N[(w0 * N[Sqrt[N[(1.0 - N[(0.25 * N[(N[(D * N[(D * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if d < 4.9999999999999999e-86

   1. Initial program 72.9%

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

   3. Simplified 73.6%

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

      1. clear-num 73.6%

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

      2. un-div-inv 73.6%

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

      3. div-inv 73.6%

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

      4. metadata-eval 73.6%

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

   5. Applied egg-rr 73.6%

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

   6. Taylor expanded in w0 around 0 43.1%

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

      1. *-commutative 43.1%

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

      2. cancel-sign-sub-inv 43.1%

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

      3. *-commutative 43.1%

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

      4. metadata-eval 43.1%

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

      5. times-frac 49.1%

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

      6. unpow2 49.1%

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

      7. associate-*l/ 51.0%

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

      8. associate-/r/ 51.0%

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

      9. unpow2 51.0%

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

      10. unpow2 51.0%

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

      11. associate-/r* 58.2%

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

      12. *-commutative 58.2%

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

      13. associate-*r* 61.8%

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

   8. Simplified 63.3%

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

   if 4.9999999999999999e-86 < d < 1.18e86

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in D around 0 64.7%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}} \]
   5. Step-by-step derivation

      1. unpow2 64.7%

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

      2. unpow2 64.7%

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

      3. unpow2 64.7%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in D around 0 64.7%

      \[\leadsto w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}}{\ell \cdot \left(d \cdot d\right)}} \]
   8. Step-by-step derivation

      1. unpow2 61.6%

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

      2. associate-*r* 64.1%

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

      3. unpow2 64.1%

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

      4. associate-*r* 77.1%

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

   9. Simplified 80.2%

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

   if 1.18e86 < d

   1. Initial program 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in D around 0 94.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5 \cdot 10^{-86}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot -0.25}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{+86}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.9 \cdot 10^{+52}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot -0.25}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= d 1.9e+52)
   (* w0 (sqrt (+ 1.0 (* (/ (* D -0.25) (/ l D)) (* (/ M d) (/ M (/ d h)))))))
   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 (d <= 1.9e+52) {
		tmp = w0 * sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	} 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) :: tmp
    if (d_1 <= 1.9d+52) then
        tmp = w0 * sqrt((1.0d0 + (((d * (-0.25d0)) / (l / d)) * ((m / d_1) * (m / (d_1 / h))))))
    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 tmp;
	if (d <= 1.9e+52) {
		tmp = w0 * Math.sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	} 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):
	tmp = 0
	if d <= 1.9e+52:
		tmp = w0 * math.sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))))
	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 (d <= 1.9e+52)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(D * -0.25) / Float64(l / D)) * Float64(Float64(M / d) * Float64(M / Float64(d / h)))))));
	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)
	tmp = 0.0;
	if (d <= 1.9e+52)
		tmp = w0 * sqrt((1.0 + (((D * -0.25) / (l / D)) * ((M / d) * (M / (d / h))))));
	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_] := If[LessEqual[d, 1.9e+52], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(D * -0.25), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.9e52

   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 74.6%

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

      1. clear-num 74.6%

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

      2. un-div-inv 74.7%

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

      3. div-inv 74.7%

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

      4. metadata-eval 74.7%

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

   5. Applied egg-rr 74.7%

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

   6. Taylor expanded in w0 around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. cancel-sign-sub-inv 46.3%

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

      3. *-commutative 46.3%

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

      4. metadata-eval 46.3%

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

      5. times-frac 51.4%

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

      6. unpow2 51.4%

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

      7. associate-*l/ 54.5%

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

      8. associate-/r/ 54.5%

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

      9. unpow2 54.5%

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

      10. unpow2 54.5%

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

      11. associate-/r* 60.5%

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

      12. *-commutative 60.5%

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

      13. associate-*r* 64.2%

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

   8. Simplified 65.4%

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

   if 1.9e52 < d

   1. Initial program 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in D around 0 90.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.9 \cdot 10^{+52}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D \cdot -0.25}{\frac{\ell}{D}} \cdot \left(\frac{M}{d} \cdot \frac{M}{\frac{d}{h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 6: 73.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-66}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+85}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= d 9.5e-66)
   (* w0 (+ 1.0 (* (* (/ D (/ l D)) (* h (* (/ M d) (/ M d)))) -0.125)))
   (if (<= d 9e+85)
     (* w0 (+ 1.0 (* (/ (* D (* D (* M (* M h)))) (* l (* d d))) -0.125)))
     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 (d <= 9.5e-66) {
		tmp = w0 * (1.0 + (((D / (l / D)) * (h * ((M / d) * (M / d)))) * -0.125));
	} else if (d <= 9e+85) {
		tmp = w0 * (1.0 + (((D * (D * (M * (M * h)))) / (l * (d * d))) * -0.125));
	} 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) :: tmp
    if (d_1 <= 9.5d-66) then
        tmp = w0 * (1.0d0 + (((d / (l / d)) * (h * ((m / d_1) * (m / d_1)))) * (-0.125d0)))
    else if (d_1 <= 9d+85) then
        tmp = w0 * (1.0d0 + (((d * (d * (m * (m * h)))) / (l * (d_1 * d_1))) * (-0.125d0)))
    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 tmp;
	if (d <= 9.5e-66) {
		tmp = w0 * (1.0 + (((D / (l / D)) * (h * ((M / d) * (M / d)))) * -0.125));
	} else if (d <= 9e+85) {
		tmp = w0 * (1.0 + (((D * (D * (M * (M * h)))) / (l * (d * d))) * -0.125));
	} 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):
	tmp = 0
	if d <= 9.5e-66:
		tmp = w0 * (1.0 + (((D / (l / D)) * (h * ((M / d) * (M / d)))) * -0.125))
	elif d <= 9e+85:
		tmp = w0 * (1.0 + (((D * (D * (M * (M * h)))) / (l * (d * d))) * -0.125))
	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 (d <= 9.5e-66)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D / Float64(l / D)) * Float64(h * Float64(Float64(M / d) * Float64(M / d)))) * -0.125)));
	elseif (d <= 9e+85)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D * Float64(D * Float64(M * Float64(M * h)))) / Float64(l * Float64(d * d))) * -0.125)));
	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)
	tmp = 0.0;
	if (d <= 9.5e-66)
		tmp = w0 * (1.0 + (((D / (l / D)) * (h * ((M / d) * (M / d)))) * -0.125));
	elseif (d <= 9e+85)
		tmp = w0 * (1.0 + (((D * (D * (M * (M * h)))) / (l * (d * d))) * -0.125));
	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_] := If[LessEqual[d, 9.5e-66], N[(w0 * N[(1.0 + N[(N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9e+85], N[(w0 * N[(1.0 + N[(N[(N[(D * N[(D * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]]

Derivation

1. Split input into 3 regimes

2. if d < 9.5000000000000004e-66

   1. Initial program 73.2%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in D around 0 42.7%

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

      1. *-commutative 42.7%

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

      2. unpow2 42.7%

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

      3. unpow2 42.7%

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

      4. unpow2 42.7%

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

   6. Simplified 42.7%

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

   7. Taylor expanded in D around 0 42.7%

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

      1. associate-/r* 49.8%

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

      2. unpow2 49.8%

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

      3. associate-*r* 52.3%

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

      4. associate-*l/ 51.7%

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

      5. unpow2 51.7%

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

      6. associate-*r/ 51.2%

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

      7. associate-/l* 53.0%

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

      8. associate-*r* 49.9%

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

      9. associate-*l/ 49.4%

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

      10. unpow2 49.4%

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

      11. *-commutative 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in M around 0 49.4%

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

       1. unpow2 49.4%

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

       2. unpow2 49.4%

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

       3. times-frac 60.4%

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

   12. Simplified 60.4%

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

   if 9.5000000000000004e-66 < d < 9.00000000000000013e85

   1. Initial program 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in D around 0 59.6%

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

      1. *-commutative 59.6%

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

      2. unpow2 59.6%

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

      3. unpow2 59.6%

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

      4. unpow2 59.6%

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

   6. Simplified 59.6%

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

   7. Taylor expanded in D around 0 59.6%

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

      1. unpow2 59.6%

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

      2. associate-*r* 62.2%

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

      3. unpow2 62.2%

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

      4. associate-*r* 75.9%

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

   9. Simplified 75.9%

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

   if 9.00000000000000013e85 < d

   1. Initial program 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in D around 0 94.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9.5 \cdot 10^{-66}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+85}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot h\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 7: 72.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= d 2.8e+53)
   (* w0 (+ 1.0 (* (* (/ D (/ l D)) (* h (* (/ M d) (/ M d)))) -0.125)))
   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 (d <= 2.8e+53) {
		tmp = w0 * (1.0 + (((D / (l / D)) * (h * ((M / d) * (M / d)))) * -0.125));
	} 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) :: tmp
    if (d_1 <= 2.8d+53) then
        tmp = w0 * (1.0d0 + (((d / (l / d)) * (h * ((m / d_1) * (m / d_1)))) * (-0.125d0)))
    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 tmp;
	if (d <= 2.8e+53) {
		tmp = w0 * (1.0 + (((D / (l / D)) * (h * ((M / d) * (M / d)))) * -0.125));
	} 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):
	tmp = 0
	if d <= 2.8e+53:
		tmp = w0 * (1.0 + (((D / (l / D)) * (h * ((M / d) * (M / d)))) * -0.125))
	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 (d <= 2.8e+53)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(Float64(D / Float64(l / D)) * Float64(h * Float64(Float64(M / d) * Float64(M / d)))) * -0.125)));
	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)
	tmp = 0.0;
	if (d <= 2.8e+53)
		tmp = w0 * (1.0 + (((D / (l / D)) * (h * ((M / d) * (M / d)))) * -0.125));
	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_] := If[LessEqual[d, 2.8e+53], N[(w0 * N[(1.0 + N[(N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 2.8e53

   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 74.6%

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

   4. Taylor expanded in D around 0 44.7%

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

      1. *-commutative 44.7%

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

      2. unpow2 44.7%

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

      3. unpow2 44.7%

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

      4. unpow2 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in D around 0 44.7%

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

      1. associate-/r* 50.8%

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

      2. unpow2 50.8%

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

      3. associate-*r* 53.4%

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

      4. associate-*l/ 52.9%

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

      5. unpow2 52.9%

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

      6. associate-*r/ 52.5%

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

      7. associate-/l* 55.6%

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

      8. associate-*r* 52.5%

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

      9. associate-*l/ 52.0%

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

      10. unpow2 52.0%

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

      11. *-commutative 52.0%

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

   9. Simplified 52.0%

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

   10. Taylor expanded in M around 0 52.0%

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

       1. unpow2 52.0%

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

       2. unpow2 52.0%

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

       3. times-frac 61.5%

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

   12. Simplified 61.5%

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

   if 2.8e53 < d

   1. Initial program 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in D around 0 90.2%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;w0 \cdot \left(1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 8: 72.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 6 \cdot 10^{+31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \end{array} \]

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
 (if (<= M 6e+31)
   w0
   (* -0.125 (* (* (/ D d) (/ D d)) (/ (* w0 (* M (* M h))) 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) {
	double tmp;
	if (M <= 6e+31) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (M * (M * h))) / l));
	}
	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) :: tmp
    if (m <= 6d+31) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / d_1) * (d / d_1)) * ((w0 * (m * (m * h))) / l))
    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 tmp;
	if (M <= 6e+31) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (M * (M * h))) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 6e+31:
		tmp = w0
	else:
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (M * (M * h))) / l))
	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 (M <= 6e+31)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(w0 * Float64(M * Float64(M * h))) / l)));
	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)
	tmp = 0.0;
	if (M <= 6e+31)
		tmp = w0;
	else
		tmp = -0.125 * (((D / d) * (D / d)) * ((w0 * (M * (M * h))) / l));
	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_] := If[LessEqual[M, 6e+31], w0, N[(-0.125 * N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.99999999999999978e31

   1. Initial program 81.7%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in D around 0 71.8%

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

   if 5.99999999999999978e31 < M

   1. Initial program 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in D around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

      3. unpow2 36.2%

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

      4. unpow2 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in D around 0 36.2%

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

      1. associate-/r* 36.3%

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

      2. unpow2 36.3%

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

      3. associate-*r* 44.4%

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

      4. associate-*l/ 46.4%

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

      5. unpow2 46.4%

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

      6. associate-*r/ 44.3%

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

      7. associate-/l* 50.7%

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

      8. associate-*r* 42.6%

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

      9. associate-*l/ 44.7%

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

      10. unpow2 44.7%

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

      11. *-commutative 44.7%

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

   9. Simplified 44.7%

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

   10. Taylor expanded in D around inf 20.7%

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

       1. times-frac 20.4%

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

       2. unpow2 20.4%

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

       3. unpow2 20.4%

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

       4. unpow2 20.4%

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

       5. *-commutative 20.4%

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

       6. associate-*r* 20.6%

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

   12. Simplified 20.6%

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

       1. frac-times 21.3%

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

   14. Applied egg-rr 21.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 6 \cdot 10^{+31}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w0 \cdot \left(M \cdot \left(M \cdot h\right)\right)}{\ell}\right)\\ \end{array} \]

Alternative 9: 67.7% 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 77.6%

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

3. Simplified 78.0%

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

4. Taylor expanded in D around 0 66.7%

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

5. Final simplification 66.7%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023274 
(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))))))