Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.6% → 89.8%

Time: 16.6s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.6% 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: 89.8% accurate, 0.3× speedup

Math

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

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)) (- INFINITY))
   (pow
    (*
     (cbrt w0)
     (pow
      (exp 0.16666666666666666)
      (fma -2.0 (- (log D)) (log (* -0.25 (* (/ h l) (* (/ M d) (/ M d))))))))
    3.0)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (/ M d) (* D 0.5)) 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)) <= -((double) INFINITY)) {
		tmp = pow((cbrt(w0) * pow(exp(0.16666666666666666), fma(-2.0, -log(D), log((-0.25 * ((h / l) * ((M / d) * (M / d)))))))), 3.0);
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M / d) * (D * 0.5)), 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)) <= Float64(-Inf))
		tmp = Float64(cbrt(w0) * (exp(0.16666666666666666) ^ fma(-2.0, Float64(-log(D)), log(Float64(-0.25 * Float64(Float64(h / l) * Float64(Float64(M / d) * Float64(M / d)))))))) ^ 3.0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 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], (-Infinity)], N[Power[N[(N[Power[w0, 1/3], $MachinePrecision] * N[Power[N[Exp[0.16666666666666666], $MachinePrecision], N[(-2.0 * (-N[Log[D], $MachinePrecision]) + N[Log[N[(-0.25 * N[(N[(h / l), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $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)) < -inf.0

   1. Initial program 53.7%

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

      1. *-commutative 53.7%

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

      2. times-frac 57.0%

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

   3. Simplified 57.0%

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

   5. Applied egg-rr 56.9%

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

   6. Taylor expanded in D around inf 15.7%

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

      1. *-commutative 15.7%

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

      2. unpow1/3 19.6%

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

      3. *-lft-identity 19.6%

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

      4. exp-prod 19.6%

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

      5. +-commutative 19.6%

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

      6. fma-def 19.6%

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

      7. log-rec 19.6%

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

      8. distribute-lft-neg-in 19.6%

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

      9. metadata-eval 19.6%

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

      10. *-commutative 19.6%

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

      11. *-commutative 19.6%

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

      12. times-frac 21.3%

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

      13. unpow2 21.3%

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

      14. unpow2 21.3%

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

      15. times-frac 25.6%

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

   8. Simplified 25.6%

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

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

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

      2. times-frac 86.8%

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

   3. Simplified 86.8%

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

      1. *-commutative 86.8%

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

      2. frac-times 87.2%

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

      3. *-commutative 87.2%

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

      4. associate-*l/ 95.0%

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

      5. *-commutative 95.0%

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

      6. frac-times 95.1%

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

      7. div-inv 95.1%

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

      8. metadata-eval 95.1%

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

   5. Applied egg-rr 95.1%

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

4. Final simplification 79.1%

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

Alternative 2: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-170}:\\ \;\;\;\;w0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {t_0}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (if (<= t_0 2e-170)
     w0
     (if (<= t_0 2e+223)
       (* w0 (sqrt (- 1.0 (* (pow t_0 2.0) (/ h l)))))
       (*
        w0
        (sqrt
         (+ 1.0 (* -0.25 (* (* D (/ D l)) (/ (* h (/ M (/ d M))) d))))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	double tmp;
	if (t_0 <= 2e-170) {
		tmp = w0;
	} else if (t_0 <= 2e+223) {
		tmp = w0 * sqrt((1.0 - (pow(t_0, 2.0) * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (m * d) / (2.0d0 * d_1)
    if (t_0 <= 2d-170) then
        tmp = w0
    else if (t_0 <= 2d+223) then
        tmp = w0 * sqrt((1.0d0 - ((t_0 ** 2.0d0) * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d * (d / l)) * ((h * (m / (d_1 / m))) / d_1)))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / (2.0 * d);
	double tmp;
	if (t_0 <= 2e-170) {
		tmp = w0;
	} else if (t_0 <= 2e+223) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(t_0, 2.0) * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = (M * D) / (2.0 * d)
	tmp = 0
	if t_0 <= 2e-170:
		tmp = w0
	elif t_0 <= 2e+223:
		tmp = w0 * math.sqrt((1.0 - (math.pow(t_0, 2.0) * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d))
	tmp = 0.0
	if (t_0 <= 2e-170)
		tmp = w0;
	elseif (t_0 <= 2e+223)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((t_0 ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(h * Float64(M / Float64(d / M))) / d))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (M * D) / (2.0 * d);
	tmp = 0.0;
	if (t_0 <= 2e-170)
		tmp = w0;
	elseif (t_0 <= 2e+223)
		tmp = w0 * sqrt((1.0 - ((t_0 ^ 2.0) * (h / l))));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-170], w0, If[LessEqual[t$95$0, 2e+223], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 1.99999999999999997e-170

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. times-frac 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in M around 0 76.9%

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

   if 1.99999999999999997e-170 < (/.f64 (*.f64 M D) (*.f64 2 d)) < 2.00000000000000009e223

   1. Initial program 94.0%

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

   if 2.00000000000000009e223 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

      2. times-frac 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in w0 around 0 40.9%

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

      1. *-commutative 40.9%

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

      2. cancel-sign-sub-inv 40.9%

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

      3. *-commutative 40.9%

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

      4. distribute-lft-neg-in 40.9%

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

      5. *-commutative 40.9%

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

      6. distribute-rgt-neg-in 40.9%

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

   6. Simplified 53.3%

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

      1. associate-*r/ 53.4%

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

      2. associate-/l* 61.5%

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

   8. Applied egg-rr 61.5%

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

   9. Taylor expanded in D around 0 61.5%

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

       1. unpow2 37.2%

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

       2. associate-*l/ 37.3%

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

       3. *-commutative 37.3%

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

   11. Simplified 73.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{-170}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \end{array} \]

Alternative 3: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -8 \cdot 10^{-164}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

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 (<= (/ h l) (- INFINITY))
   (* w0 (sqrt (- 1.0 (* (/ 0.25 l) (* (/ h d) (/ (* D (* D (* M M))) d))))))
   (if (<= (/ h l) -8e-164)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
     (if (<= (/ h l) 0.0)
       (*
        w0
        (sqrt (+ 1.0 (* -0.25 (* (* D (/ D l)) (/ (* h (/ M (/ d M))) 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 ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((h / d) * ((D * (D * (M * M))) / d)))));
	} else if ((h / l) <= -8e-164) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else if ((h / l) <= 0.0) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	} else {
		tmp = w0;
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((h / l) <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / l) * ((h / d) * ((D * (D * (M * M))) / d)))));
	} else if ((h / l) <= -8e-164) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else if ((h / l) <= 0.0) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / 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 (h / l) <= -math.inf:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / l) * ((h / d) * ((D * (D * (M * M))) / d)))))
	elif (h / l) <= -8e-164:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	elif (h / l) <= 0.0:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / 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 (Float64(h / l) <= Float64(-Inf))
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / l) * Float64(Float64(h / d) * Float64(Float64(D * Float64(D * Float64(M * M))) / d))))));
	elseif (Float64(h / l) <= -8e-164)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	elseif (Float64(h / l) <= 0.0)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(h * Float64(M / Float64(d / M))) / 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 ((h / l) <= -Inf)
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((h / d) * ((D * (D * (M * M))) / d)))));
	elseif ((h / l) <= -8e-164)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	elseif ((h / l) <= 0.0)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / 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[N[(h / l), $MachinePrecision], (-Infinity)], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / l), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -8e-164], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], 0.0], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 h l) < -inf.0

   1. Initial program 23.8%

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

      1. *-commutative 23.8%

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

      2. times-frac 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in M around 0 56.6%

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

      1. associate-*r/ 56.6%

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

      2. *-commutative 56.6%

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

      3. times-frac 56.6%

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

      4. *-commutative 56.6%

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

      5. associate-*l* 56.6%

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

      6. unpow2 56.6%

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

      7. unpow2 56.6%

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

      8. swap-sqr 73.5%

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

      9. *-commutative 73.5%

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

      10. *-commutative 73.5%

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

      11. swap-sqr 56.6%

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

      12. unpow2 56.6%

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

   6. Simplified 56.6%

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

      1. times-frac 56.6%

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

      2. associate-*l* 63.3%

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

   8. Applied egg-rr 63.3%

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

   if -inf.0 < (/.f64 h l) < -7.99999999999999969e-164

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. times-frac 85.6%

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

   3. Simplified 85.6%

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

   if -7.99999999999999969e-164 < (/.f64 h l) < -0.0

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. times-frac 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in w0 around 0 48.9%

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

      1. *-commutative 48.9%

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

      2. cancel-sign-sub-inv 48.9%

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

      3. *-commutative 48.9%

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

      4. distribute-lft-neg-in 48.9%

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

      5. *-commutative 48.9%

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

      6. distribute-rgt-neg-in 48.9%

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

   6. Simplified 55.8%

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

      1. associate-*r/ 55.9%

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

      2. associate-/l* 59.4%

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

   8. Applied egg-rr 59.4%

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

   9. Taylor expanded in D around 0 59.4%

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

       1. unpow2 45.6%

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

       2. associate-*l/ 59.4%

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

       3. *-commutative 59.4%

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

   11. Simplified 78.1%

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

   if -0.0 < (/.f64 h l)

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. times-frac 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in M around 0 100.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -8 \cdot 10^{-164}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \end{array} \end{array} \]

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 D) (* 2.0 d)) 2e+223)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (/ M d) (* D 0.5)) 2.0)) l))))
   (*
    w0
    (sqrt (+ 1.0 (* -0.25 (* (* D (/ D l)) (/ (* h (/ M (/ d M))) d))))))))

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)) <= 2e+223) {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	}
	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)) <= 2d+223) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((m / d_1) * (d * 0.5d0)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d * (d / l)) * ((h * (m / (d_1 / m))) / d_1)))))
    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 * D) / (2.0 * d)) <= 2e+223) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M / d) * (D * 0.5)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	}
	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 * D) / (2.0 * d)) <= 2e+223:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M / d) * (D * 0.5)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))))
	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)) <= 2e+223)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M / d) * Float64(D * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(h * Float64(M / Float64(d / M))) / d))))));
	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)) <= 2e+223)
		tmp = w0 * sqrt((1.0 - ((h * (((M / d) * (D * 0.5)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	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[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2e+223], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M / d), $MachinePrecision] * N[(D * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < 2.00000000000000009e223

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. times-frac 81.9%

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

   3. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. frac-times 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-*l/ 89.7%

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

      5. *-commutative 89.7%

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

      6. frac-times 88.9%

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

      7. div-inv 88.9%

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

      8. metadata-eval 88.9%

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

   5. Applied egg-rr 88.9%

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

   if 2.00000000000000009e223 < (/.f64 (*.f64 M D) (*.f64 2 d))

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

      2. times-frac 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in w0 around 0 40.9%

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

      1. *-commutative 40.9%

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

      2. cancel-sign-sub-inv 40.9%

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

      3. *-commutative 40.9%

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

      4. distribute-lft-neg-in 40.9%

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

      5. *-commutative 40.9%

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

      6. distribute-rgt-neg-in 40.9%

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

   6. Simplified 53.3%

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

      1. associate-*r/ 53.4%

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

      2. associate-/l* 61.5%

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

   8. Applied egg-rr 61.5%

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

   9. Taylor expanded in D around 0 61.5%

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

       1. unpow2 37.2%

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

       2. associate-*l/ 37.3%

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

       3. *-commutative 37.3%

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

   11. Simplified 73.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2 \cdot 10^{+223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \end{array} \]

Alternative 5: 78.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

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 (<= d 1.25e-53)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (* D (/ D l)) (/ (* h (/ M (/ d M))) d))))))
   (if (<= d 4e+154)
     (* w0 (sqrt (- 1.0 (* (/ 0.25 l) (* (/ h d) (/ (* D (* D (* M M))) d))))))
     (* w0 (+ 1.0 (* (* (pow (/ D d) 2.0) (/ (* h (* M M)) l)) -0.125))))))

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 <= 1.25e-53) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	} else if (d <= 4e+154) {
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((h / d) * ((D * (D * (M * M))) / d)))));
	} else {
		tmp = w0 * (1.0 + ((pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125));
	}
	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 <= 1.25d-53) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d * (d / l)) * ((h * (m / (d_1 / m))) / d_1)))))
    else if (d_1 <= 4d+154) then
        tmp = w0 * sqrt((1.0d0 - ((0.25d0 / l) * ((h / d_1) * ((d * (d * (m * m))) / d_1)))))
    else
        tmp = w0 * (1.0d0 + ((((d / d_1) ** 2.0d0) * ((h * (m * m)) / l)) * (-0.125d0)))
    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 <= 1.25e-53) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	} else if (d <= 4e+154) {
		tmp = w0 * Math.sqrt((1.0 - ((0.25 / l) * ((h / d) * ((D * (D * (M * M))) / d)))));
	} else {
		tmp = w0 * (1.0 + ((Math.pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125));
	}
	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 <= 1.25e-53:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))))
	elif d <= 4e+154:
		tmp = w0 * math.sqrt((1.0 - ((0.25 / l) * ((h / d) * ((D * (D * (M * M))) / d)))))
	else:
		tmp = w0 * (1.0 + ((math.pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125))
	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 <= 1.25e-53)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(h * Float64(M / Float64(d / M))) / d))))));
	elseif (d <= 4e+154)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(0.25 / l) * Float64(Float64(h / d) * Float64(Float64(D * Float64(D * Float64(M * M))) / d))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64((Float64(D / d) ^ 2.0) * Float64(Float64(h * Float64(M * M)) / l)) * -0.125)));
	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 <= 1.25e-53)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	elseif (d <= 4e+154)
		tmp = w0 * sqrt((1.0 - ((0.25 / l) * ((h / d) * ((D * (D * (M * M))) / d)))));
	else
		tmp = w0 * (1.0 + ((((D / d) ^ 2.0) * ((h * (M * M)) / l)) * -0.125));
	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, 1.25e-53], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e+154], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(0.25 / l), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.25e-53

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. times-frac 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in w0 around 0 51.5%

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

      1. *-commutative 51.5%

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

      2. cancel-sign-sub-inv 51.5%

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

      3. *-commutative 51.5%

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

      4. distribute-lft-neg-in 51.5%

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

      5. *-commutative 51.5%

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

      6. distribute-rgt-neg-in 51.5%

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

   6. Simplified 58.3%

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

      1. associate-*r/ 61.3%

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

      2. associate-/l* 63.1%

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

   8. Applied egg-rr 63.1%

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

   9. Taylor expanded in D around 0 63.1%

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

       1. unpow2 48.7%

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

       2. associate-*l/ 52.4%

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

       3. *-commutative 52.4%

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

   11. Simplified 67.4%

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

   if 1.25e-53 < d < 4.00000000000000015e154

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

      2. times-frac 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in M around 0 49.8%

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

      1. associate-*r/ 49.8%

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

      2. *-commutative 49.8%

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

      3. times-frac 50.4%

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

      4. *-commutative 50.4%

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

      5. associate-*l* 52.8%

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

      6. unpow2 52.8%

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

      7. unpow2 52.8%

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

      8. swap-sqr 75.0%

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

      9. *-commutative 75.0%

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

      10. *-commutative 75.0%

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

      11. swap-sqr 52.8%

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

      12. unpow2 52.8%

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

   6. Simplified 52.8%

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

      1. times-frac 56.6%

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

      2. associate-*l* 66.0%

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

   8. Applied egg-rr 66.0%

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

   if 4.00000000000000015e154 < d

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. times-frac 88.6%

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

   3. Simplified 88.6%

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

      1. *-commutative 88.6%

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

      2. frac-times 81.8%

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

      3. *-commutative 81.8%

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

      4. associate-*l/ 93.2%

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

      5. *-commutative 93.2%

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

      6. frac-times 97.8%

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

      7. div-inv 97.8%

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

      8. metadata-eval 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in h around 0 54.5%

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

      1. *-commutative 54.5%

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

      2. *-commutative 54.5%

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

      3. times-frac 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

      6. times-frac 70.8%

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

      7. unpow2 70.8%

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

      8. unpow2 70.8%

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

   8. Simplified 70.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+154}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25}{\ell} \cdot \left(\frac{h}{d} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot M\right)\right)}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 6: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.85 \cdot 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

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 (<= d 1.85e+84)
   (* w0 (sqrt (+ 1.0 (* -0.25 (* (* D (/ D l)) (/ (* h (/ M (/ d M))) d))))))
   (* w0 (+ 1.0 (* (* (pow (/ D d) 2.0) (/ (* h (* M M)) l)) -0.125)))))

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 <= 1.85e+84) {
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	} else {
		tmp = w0 * (1.0 + ((pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125));
	}
	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 <= 1.85d+84) then
        tmp = w0 * sqrt((1.0d0 + ((-0.25d0) * ((d * (d / l)) * ((h * (m / (d_1 / m))) / d_1)))))
    else
        tmp = w0 * (1.0d0 + ((((d / d_1) ** 2.0d0) * ((h * (m * m)) / l)) * (-0.125d0)))
    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 <= 1.85e+84) {
		tmp = w0 * Math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	} else {
		tmp = w0 * (1.0 + ((Math.pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125));
	}
	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 <= 1.85e+84:
		tmp = w0 * math.sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))))
	else:
		tmp = w0 * (1.0 + ((math.pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125))
	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 <= 1.85e+84)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(-0.25 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(h * Float64(M / Float64(d / M))) / d))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64((Float64(D / d) ^ 2.0) * Float64(Float64(h * Float64(M * M)) / l)) * -0.125)));
	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 <= 1.85e+84)
		tmp = w0 * sqrt((1.0 + (-0.25 * ((D * (D / l)) * ((h * (M / (d / M))) / d)))));
	else
		tmp = w0 * (1.0 + ((((D / d) ^ 2.0) * ((h * (M * M)) / l)) * -0.125));
	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, 1.85e+84], N[(w0 * N[Sqrt[N[(1.0 + N[(-0.25 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.85e84

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. times-frac 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in w0 around 0 53.0%

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

      1. *-commutative 53.0%

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

      2. cancel-sign-sub-inv 53.0%

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

      3. *-commutative 53.0%

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

      4. distribute-lft-neg-in 53.0%

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

      5. *-commutative 53.0%

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

      6. distribute-rgt-neg-in 53.0%

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

   6. Simplified 58.8%

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

      1. associate-*r/ 61.4%

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

      2. associate-/l* 63.4%

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

   8. Applied egg-rr 63.4%

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

   9. Taylor expanded in D around 0 63.4%

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

       1. unpow2 50.5%

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

       2. associate-*l/ 54.2%

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

       3. *-commutative 54.2%

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

   11. Simplified 68.2%

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

   if 1.85e84 < d

   1. Initial program 76.6%

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

      1. *-commutative 76.6%

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

      2. times-frac 81.3%

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

   3. Simplified 81.3%

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

      1. *-commutative 81.3%

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

      2. frac-times 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-*l/ 87.7%

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

      5. *-commutative 87.7%

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

      6. frac-times 90.9%

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

      7. div-inv 90.9%

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

      8. metadata-eval 90.9%

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

   5. Applied egg-rr 90.9%

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

   6. Taylor expanded in h around 0 48.0%

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

      1. *-commutative 48.0%

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

      2. *-commutative 48.0%

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

      3. times-frac 46.7%

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

      4. unpow2 46.7%

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

      5. unpow2 46.7%

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

      6. times-frac 64.5%

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

      7. unpow2 64.5%

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

      8. unpow2 64.5%

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

   8. Simplified 64.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.85 \cdot 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{h \cdot \frac{M}{\frac{d}{M}}}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 7: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{-152}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

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 5.8e-152)
   w0
   (* w0 (+ 1.0 (* (* (pow (/ D d) 2.0) (/ (* h (* M M)) l)) -0.125)))))

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 <= 5.8e-152) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125));
	}
	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 <= 5.8d-152) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((((d / d_1) ** 2.0d0) * ((h * (m * m)) / l)) * (-0.125d0)))
    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 <= 5.8e-152) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((Math.pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125));
	}
	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 <= 5.8e-152:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((math.pow((D / d), 2.0) * ((h * (M * M)) / l)) * -0.125))
	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 <= 5.8e-152)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64((Float64(D / d) ^ 2.0) * Float64(Float64(h * Float64(M * M)) / l)) * -0.125)));
	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 <= 5.8e-152)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((((D / d) ^ 2.0) * ((h * (M * M)) / l)) * -0.125));
	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, 5.8e-152], w0, N[(w0 * N[(1.0 + N[(N[(N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 5.8000000000000003e-152

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      2. times-frac 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in M around 0 72.0%

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

   if 5.8000000000000003e-152 < M

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. times-frac 83.3%

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

   3. Simplified 83.3%

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

      1. *-commutative 83.3%

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

      2. frac-times 83.3%

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

      3. *-commutative 83.3%

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

      4. associate-*l/ 89.7%

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

      5. *-commutative 89.7%

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

      6. frac-times 89.8%

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

      7. div-inv 89.8%

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

      8. metadata-eval 89.8%

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

   5. Applied egg-rr 89.8%

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

   6. Taylor expanded in h around 0 56.6%

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

      1. *-commutative 56.6%

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

      2. *-commutative 56.6%

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

      3. times-frac 55.5%

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

      4. unpow2 55.5%

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

      5. unpow2 55.5%

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

      6. times-frac 66.5%

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

      7. unpow2 66.5%

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

      8. unpow2 66.5%

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

   8. Simplified 66.5%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5.8 \cdot 10^{-152}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{h \cdot \left(M \cdot M\right)}{\ell}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 8: 77.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.32 \cdot 10^{-65}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 1.32e-65)
   w0
   (* w0 (+ 1.0 (* -0.125 (* D (* (/ D l) (* (/ M (/ d M)) (/ h d)))))))))

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 <= 1.32e-65) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * ((M / (d / M)) * (h / d))))));
	}
	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 <= 1.32d-65) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * ((d / l) * ((m / (d_1 / m)) * (h / d_1))))))
    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 <= 1.32e-65) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * ((M / (d / M)) * (h / d))))));
	}
	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 <= 1.32e-65:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * ((M / (d / M)) * (h / d))))))
	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 <= 1.32e-65)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(Float64(D / l) * Float64(Float64(M / Float64(d / M)) * Float64(h / d)))))));
	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 <= 1.32e-65)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (D * ((D / l) * ((M / (d / M)) * (h / d))))));
	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, 1.32e-65], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(N[(D / l), $MachinePrecision] * N[(N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.32e-65

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. times-frac 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in M around 0 73.3%

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

   if 1.32e-65 < M

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in M around 0 50.8%

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

      1. associate-*r/ 50.8%

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

      2. *-commutative 50.8%

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

      3. associate-*r/ 50.8%

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

      4. *-commutative 50.8%

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

      5. times-frac 49.4%

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

      6. unpow2 49.4%

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

      7. *-commutative 49.4%

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

      8. unpow2 49.4%

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

      9. unpow2 49.4%

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

   6. Simplified 49.4%

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

      1. associate-*r/ 50.8%

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

      2. associate-/l* 54.9%

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

      3. *-commutative 54.9%

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

   8. Applied egg-rr 54.9%

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

      1. associate-/l* 50.8%

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

      2. associate-*r/ 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-*r/ 53.5%

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

      5. frac-times 62.0%

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

      6. associate-*l* 63.5%

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

      7. associate-/l* 72.9%

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

   10. Applied egg-rr 72.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.32 \cdot 10^{-65}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \left(\frac{D}{\ell} \cdot \left(\frac{M}{\frac{d}{M}} \cdot \frac{h}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 72.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.06 \cdot 10^{+48}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}\right)\\ \end{array} \end{array} \]

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 1.06e+48)
   w0
   (* -0.125 (* (* D (/ D l)) (/ (* (* M M) (* h w0)) (* d d))))))

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 <= 1.06e+48) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (((M * M) * (h * w0)) / (d * d)));
	}
	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 <= 1.06d+48) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d * (d / l)) * (((m * m) * (h * w0)) / (d_1 * d_1)))
    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 <= 1.06e+48) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D * (D / l)) * (((M * M) * (h * w0)) / (d * d)));
	}
	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 <= 1.06e+48:
		tmp = w0
	else:
		tmp = -0.125 * ((D * (D / l)) * (((M * M) * (h * w0)) / (d * d)))
	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 <= 1.06e+48)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D * Float64(D / l)) * Float64(Float64(Float64(M * M) * Float64(h * w0)) / Float64(d * d))));
	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 <= 1.06e+48)
		tmp = w0;
	else
		tmp = -0.125 * ((D * (D / l)) * (((M * M) * (h * w0)) / (d * d)));
	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, 1.06e+48], w0, N[(-0.125 * N[(N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.06e48

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. times-frac 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in M around 0 74.0%

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

   if 1.06e48 < M

   1. Initial program 79.1%

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

      1. *-commutative 79.1%

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

      2. times-frac 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in M around 0 43.9%

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

      1. associate-*r/ 43.9%

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

      2. *-commutative 43.9%

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

      3. associate-*r/ 43.9%

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

      4. *-commutative 43.9%

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

      5. times-frac 43.9%

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

      6. unpow2 43.9%

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

      7. *-commutative 43.9%

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

      8. unpow2 43.9%

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

      9. unpow2 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in D around 0 43.9%

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

      1. unpow2 43.9%

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

      2. associate-*l/ 45.9%

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

      3. *-commutative 45.9%

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

   9. Simplified 45.9%

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

   10. Taylor expanded in D around inf 25.0%

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

       1. unpow2 25.0%

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

       2. *-commutative 25.0%

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

       3. times-frac 24.9%

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

       4. unpow2 24.9%

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

       5. associate-/l* 26.9%

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

       6. unpow2 26.9%

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

       7. associate-*r* 27.0%

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

   12. Simplified 27.0%

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

       1. associate-/r/ 27.0%

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

   14. Applied egg-rr 27.0%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.06 \cdot 10^{+48}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\left(D \cdot \frac{D}{\ell}\right) \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}\right)\\ \end{array} \]

Alternative 10: 72.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{+47}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{w0 \cdot \left(M \cdot \frac{M}{d}\right)}{\frac{d}{h}}\right)\\ \end{array} \end{array} \]

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 (<= M 1.75e+47)
   w0
   (* -0.125 (* (/ D (/ l D)) (/ (* w0 (* M (/ M d))) (/ 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 (M <= 1.75e+47) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D / (l / D)) * ((w0 * (M * (M / d))) / (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 <= 1.75d+47) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((d / (l / d)) * ((w0 * (m * (m / d_1))) / (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 (M <= 1.75e+47) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((D / (l / D)) * ((w0 * (M * (M / d))) / (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 M <= 1.75e+47:
		tmp = w0
	else:
		tmp = -0.125 * ((D / (l / D)) * ((w0 * (M * (M / d))) / (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 (M <= 1.75e+47)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(D / Float64(l / D)) * Float64(Float64(w0 * Float64(M * Float64(M / d))) / 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 <= 1.75e+47)
		tmp = w0;
	else
		tmp = -0.125 * ((D / (l / D)) * ((w0 * (M * (M / d))) / (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[M, 1.75e+47], w0, N[(-0.125 * N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(N[(w0 * N[(M * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.75000000000000008e47

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. times-frac 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in M around 0 74.0%

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

   if 1.75000000000000008e47 < M

   1. Initial program 79.1%

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

      1. *-commutative 79.1%

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

      2. times-frac 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in M around 0 43.9%

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

      1. associate-*r/ 43.9%

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

      2. *-commutative 43.9%

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

      3. associate-*r/ 43.9%

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

      4. *-commutative 43.9%

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

      5. times-frac 43.9%

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

      6. unpow2 43.9%

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

      7. *-commutative 43.9%

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

      8. unpow2 43.9%

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

      9. unpow2 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in D around 0 43.9%

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

      1. unpow2 43.9%

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

      2. associate-*l/ 45.9%

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

      3. *-commutative 45.9%

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

   9. Simplified 45.9%

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

   10. Taylor expanded in D around inf 25.0%

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

       1. unpow2 25.0%

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

       2. *-commutative 25.0%

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

       3. times-frac 24.9%

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

       4. unpow2 24.9%

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

       5. associate-/l* 26.9%

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

       6. unpow2 26.9%

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

       7. associate-*r* 27.0%

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

   12. Simplified 27.0%

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

   13. Taylor expanded in w0 around 0 26.9%

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

       1. *-commutative 26.9%

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

       2. associate-*r* 27.0%

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

       3. unpow2 27.0%

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

       4. times-frac 33.6%

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

       5. unpow2 33.6%

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

       6. associate-*r/ 33.9%

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

       7. *-commutative 33.9%

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

       8. associate-/l* 33.8%

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

       9. associate-*r/ 33.8%

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

   15. Simplified 33.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.75 \cdot 10^{+47}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \frac{w0 \cdot \left(M \cdot \frac{M}{d}\right)}{\frac{d}{h}}\right)\\ \end{array} \]

Alternative 11: 73.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.85 \cdot 10^{+47}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\frac{D}{\frac{d}{M}} \cdot \frac{h \cdot w0}{\frac{d}{M}}}{\frac{\ell}{D}}\\ \end{array} \end{array} \]

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 1.85e+47)
   w0
   (* -0.125 (/ (* (/ D (/ d M)) (/ (* h w0) (/ d M))) (/ l D)))))

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 <= 1.85e+47) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / (d / M)) * ((h * w0) / (d / M))) / (l / D));
	}
	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 <= 1.85d+47) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d / (d_1 / m)) * ((h * w0) / (d_1 / m))) / (l / d))
    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 <= 1.85e+47) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D / (d / M)) * ((h * w0) / (d / M))) / (l / D));
	}
	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 <= 1.85e+47:
		tmp = w0
	else:
		tmp = -0.125 * (((D / (d / M)) * ((h * w0) / (d / M))) / (l / D))
	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 <= 1.85e+47)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D / Float64(d / M)) * Float64(Float64(h * w0) / Float64(d / M))) / Float64(l / D)));
	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 <= 1.85e+47)
		tmp = w0;
	else
		tmp = -0.125 * (((D / (d / M)) * ((h * w0) / (d / M))) / (l / D));
	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, 1.85e+47], w0, N[(-0.125 * N[(N[(N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.8500000000000002e47

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. times-frac 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in M around 0 74.0%

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

   if 1.8500000000000002e47 < M

   1. Initial program 79.1%

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

      1. *-commutative 79.1%

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

      2. times-frac 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in M around 0 43.9%

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

      1. associate-*r/ 43.9%

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

      2. *-commutative 43.9%

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

      3. associate-*r/ 43.9%

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

      4. *-commutative 43.9%

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

      5. times-frac 43.9%

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

      6. unpow2 43.9%

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

      7. *-commutative 43.9%

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

      8. unpow2 43.9%

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

      9. unpow2 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in D around 0 43.9%

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

      1. unpow2 43.9%

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

      2. associate-*l/ 45.9%

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

      3. *-commutative 45.9%

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

   9. Simplified 45.9%

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

   10. Taylor expanded in D around inf 25.0%

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

       1. unpow2 25.0%

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

       2. *-commutative 25.0%

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

       3. times-frac 24.9%

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

       4. unpow2 24.9%

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

       5. associate-/l* 26.9%

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

       6. unpow2 26.9%

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

       7. associate-*r* 27.0%

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

   12. Simplified 27.0%

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

       1. associate-*l/ 27.4%

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

       2. associate-/l* 27.6%

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

   14. Applied egg-rr 27.6%

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

       1. associate-*r/ 27.6%

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

       2. times-frac 34.5%

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

       3. times-frac 34.5%

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

   16. Simplified 34.5%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.85 \cdot 10^{+47}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\frac{D}{\frac{d}{M}} \cdot \frac{h \cdot w0}{\frac{d}{M}}}{\frac{\ell}{D}}\\ \end{array} \]

Alternative 12: 68.3% 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 79.5%

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

   1. *-commutative 79.5%

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

   2. times-frac 79.9%

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

3. Simplified 79.9%

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

4. Taylor expanded in M around 0 68.1%

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

5. Final simplification 68.1%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023256 
(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))))))