Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.7% → 91.6%

Time: 14.6s

Alternatives: 8

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ t_1 := D \cdot \frac{M}{d}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+295}:\\ \;\;\;\;w0 \cdot \left(t_1 \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \mathbf{elif}\;t_0 \leq 0.002:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{\frac{M}{d}}{\frac{2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{h \cdot \left(w0 \cdot {t_1}^{2}\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) (t_1 (* D (/ M d))))
   (if (<= t_0 -5e+295)
     (* w0 (* t_1 (sqrt (* h (/ -0.25 l)))))
     (if (<= t_0 0.002)
       (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ (/ M d) (/ 2.0 D)) 2.0)))))
       (+ w0 (* -0.125 (/ (* h (* w0 (pow t_1 2.0))) l)))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double t_1 = D * (M / d);
	double tmp;
	if (t_0 <= -5e+295) {
		tmp = w0 * (t_1 * sqrt((h * (-0.25 / l))));
	} else if (t_0 <= 0.002) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) / (2.0 / D)), 2.0))));
	} else {
		tmp = w0 + (-0.125 * ((h * (w0 * pow(t_1, 2.0))) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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) :: t_1
    real(8) :: tmp
    t_0 = (((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)
    t_1 = d * (m / d_1)
    if (t_0 <= (-5d+295)) then
        tmp = w0 * (t_1 * sqrt((h * ((-0.25d0) / l))))
    else if (t_0 <= 0.002d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) / (2.0d0 / d)) ** 2.0d0))))
    else
        tmp = w0 + ((-0.125d0) * ((h * (w0 * (t_1 ** 2.0d0))) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double t_1 = D * (M / d);
	double tmp;
	if (t_0 <= -5e+295) {
		tmp = w0 * (t_1 * Math.sqrt((h * (-0.25 / l))));
	} else if (t_0 <= 0.002) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) / (2.0 / D)), 2.0))));
	} else {
		tmp = w0 + (-0.125 * ((h * (w0 * Math.pow(t_1, 2.0))) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)
	t_1 = D * (M / d)
	tmp = 0
	if t_0 <= -5e+295:
		tmp = w0 * (t_1 * math.sqrt((h * (-0.25 / l))))
	elif t_0 <= 0.002:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) / (2.0 / D)), 2.0))))
	else:
		tmp = w0 + (-0.125 * ((h * (w0 * math.pow(t_1, 2.0))) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	t_1 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (t_0 <= -5e+295)
		tmp = Float64(w0 * Float64(t_1 * sqrt(Float64(h * Float64(-0.25 / l)))));
	elseif (t_0 <= 0.002)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) / Float64(2.0 / D)) ^ 2.0)))));
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * Float64(w0 * (t_1 ^ 2.0))) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (((M * D) / (2.0 * d)) ^ 2.0) * (h / l);
	t_1 = D * (M / d);
	tmp = 0.0;
	if (t_0 <= -5e+295)
		tmp = w0 * (t_1 * sqrt((h * (-0.25 / l))));
	elseif (t_0 <= 0.002)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) / (2.0 / D)) ^ 2.0))));
	else
		tmp = w0 + (-0.125 * ((h * (w0 * (t_1 ^ 2.0))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+295], N[(w0 * N[(t$95$1 * N[Sqrt[N[(h * N[(-0.25 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / d), $MachinePrecision] / N[(2.0 / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 + N[(-0.125 * N[(N[(h * N[(w0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -4.99999999999999991e295

   1. Initial program 53.2%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in D around inf 41.8%

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

      1. associate-*r/ 41.8%

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

      2. *-commutative 41.8%

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

      3. times-frac 41.9%

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

      4. associate-*r* 43.6%

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

      5. unpow2 43.6%

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

      6. unpow2 43.6%

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

      7. swap-sqr 51.0%

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

      8. unpow2 51.0%

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

      9. associate-/l* 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in D around 0 41.9%

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

      1. associate-*r* 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. swap-sqr 51.0%

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

      5. unpow2 51.0%

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

      6. associate-*l/ 51.0%

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

      7. *-commutative 51.0%

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

      8. unpow2 51.0%

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

      9. unpow2 51.0%

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

      10. times-frac 54.9%

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

      11. associate-/l* 53.2%

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

      12. associate-/l* 53.2%

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

      13. unpow2 53.2%

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

      14. associate-/l* 54.9%

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

      15. *-rgt-identity 54.9%

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

      16. associate-*r/ 54.9%

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

      17. associate-*l* 54.9%

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

      18. associate-*r/ 54.9%

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

      19. associate-*l/ 54.9%

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

      20. *-rgt-identity 54.9%

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

   9. Simplified 54.9%

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

       1. pow1/2 54.9%

          \[\leadsto w0 \cdot \color{blue}{{\left(\frac{-0.25}{\ell} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}^{0.5}} \]

       2. associate-*r* 54.9%

          \[\leadsto w0 \cdot {\color{blue}{\left(\left(\frac{-0.25}{\ell} \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}^{0.5} \]

       3. unpow-prod-down 58.2%

          \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot {\left({\left(D \cdot \frac{M}{d}\right)}^{2}\right)}^{0.5}\right)} \]

       4. pow1/2 58.2%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \color{blue}{\sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}}\right) \]

       5. *-commutative 58.2%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}\right) \]

   11. Applied egg-rr 58.2%

       \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\left(\frac{M}{d} \cdot D\right)}^{2}}\right)} \]
   12. Step-by-step derivation

       1. unpow1/2 58.2%

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

       2. *-commutative 58.2%

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

       3. unpow2 58.2%

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

       4. rem-sqrt-square 76.6%

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

       5. *-commutative 76.6%

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

   13. Simplified 76.6%

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

       1. expm1-log1p-u 43.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left|D \cdot \frac{M}{d}\right|\right)\right)\right)} \]

       2. expm1-udef 33.4%

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

       3. *-commutative 33.4%

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

       4. add-sqr-sqrt 14.4%

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

       5. fabs-sqr 14.4%

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

       6. add-sqr-sqrt 14.6%

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

       7. *-commutative 14.6%

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

   15. Applied egg-rr 14.6%

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

       1. expm1-def 19.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)} \]

       2. expm1-log1p 36.1%

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

       3. *-commutative 36.1%

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

   17. Simplified 36.1%

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

   if -4.99999999999999991e295 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 2e-3

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. clear-num 99.4%

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

      3. un-div-inv 99.4%

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

   5. Applied egg-rr 99.4%

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

   if 2e-3 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in D around 0 53.0%

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

      1. times-frac 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in D around 0 53.0%

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

      1. associate-*r* 53.0%

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

      2. times-frac 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

      5. swap-sqr 47.4%

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

      6. unpow2 47.4%

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

      7. times-frac 47.4%

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

      8. associate-/l* 47.4%

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

      9. associate-/l* 47.4%

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

      10. unpow2 47.4%

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

      11. associate-/l* 47.4%

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

      12. *-rgt-identity 47.4%

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

      13. associate-*r/ 47.4%

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

      14. associate-*l* 47.4%

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

      15. associate-*r/ 47.4%

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

      16. associate-*l/ 47.4%

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

      17. *-rgt-identity 47.4%

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

   9. Simplified 47.4%

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

       1. expm1-log1p-u 36.8%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\right)} \]

       2. expm1-udef 36.8%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right)} \]

       3. *-commutative 36.8%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right) \]

       4. associate-/l* 36.8%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \color{blue}{\frac{h}{\frac{\ell}{w0}}}\right)} - 1\right) \]

   11. Applied egg-rr 36.8%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 36.8%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\right)} \]

       2. expm1-log1p 47.4%

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

       3. *-commutative 47.4%

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

       4. associate-/r/ 0.0%

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

       5. associate-*l* 0.0%

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

       6. *-commutative 0.0%

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

   13. Simplified 0.0%

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

       1. associate-*l/ 79.9%

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

   15. Applied egg-rr 79.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+295}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.002:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{\frac{M}{d}}{\frac{2}{D}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{h \cdot \left(w0 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}\\ \end{array} \]

Alternative 2: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ t_1 := D \cdot \frac{M}{d}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+295}:\\ \;\;\;\;w0 \cdot \left(t_1 \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \mathbf{elif}\;t_0 \leq 0.002:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{h \cdot \left(w0 \cdot {t_1}^{2}\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) (t_1 (* D (/ M d))))
   (if (<= t_0 -5e+295)
     (* w0 (* t_1 (sqrt (* h (/ -0.25 l)))))
     (if (<= t_0 0.002)
       (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M d) (/ D 2.0)) 2.0)))))
       (+ w0 (* -0.125 (/ (* h (* w0 (pow t_1 2.0))) l)))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double t_1 = D * (M / d);
	double tmp;
	if (t_0 <= -5e+295) {
		tmp = w0 * (t_1 * sqrt((h * (-0.25 / l))));
	} else if (t_0 <= 0.002) {
		tmp = w0 * sqrt((1.0 - ((h / l) * pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 + (-0.125 * ((h * (w0 * pow(t_1, 2.0))) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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) :: t_1
    real(8) :: tmp
    t_0 = (((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)
    t_1 = d * (m / d_1)
    if (t_0 <= (-5d+295)) then
        tmp = w0 * (t_1 * sqrt((h * ((-0.25d0) / l))))
    else if (t_0 <= 0.002d0) then
        tmp = w0 * sqrt((1.0d0 - ((h / l) * (((m / d_1) * (d / 2.0d0)) ** 2.0d0))))
    else
        tmp = w0 + ((-0.125d0) * ((h * (w0 * (t_1 ** 2.0d0))) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l);
	double t_1 = D * (M / d);
	double tmp;
	if (t_0 <= -5e+295) {
		tmp = w0 * (t_1 * Math.sqrt((h * (-0.25 / l))));
	} else if (t_0 <= 0.002) {
		tmp = w0 * Math.sqrt((1.0 - ((h / l) * Math.pow(((M / d) * (D / 2.0)), 2.0))));
	} else {
		tmp = w0 + (-0.125 * ((h * (w0 * Math.pow(t_1, 2.0))) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)
	t_1 = D * (M / d)
	tmp = 0
	if t_0 <= -5e+295:
		tmp = w0 * (t_1 * math.sqrt((h * (-0.25 / l))))
	elif t_0 <= 0.002:
		tmp = w0 * math.sqrt((1.0 - ((h / l) * math.pow(((M / d) * (D / 2.0)), 2.0))))
	else:
		tmp = w0 + (-0.125 * ((h * (w0 * math.pow(t_1, 2.0))) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))
	t_1 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (t_0 <= -5e+295)
		tmp = Float64(w0 * Float64(t_1 * sqrt(Float64(h * Float64(-0.25 / l)))));
	elseif (t_0 <= 0.002)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / l) * (Float64(Float64(M / d) * Float64(D / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * Float64(w0 * (t_1 ^ 2.0))) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = (((M * D) / (2.0 * d)) ^ 2.0) * (h / l);
	t_1 = D * (M / d);
	tmp = 0.0;
	if (t_0 <= -5e+295)
		tmp = w0 * (t_1 * sqrt((h * (-0.25 / l))));
	elseif (t_0 <= 0.002)
		tmp = w0 * sqrt((1.0 - ((h / l) * (((M / d) * (D / 2.0)) ^ 2.0))));
	else
		tmp = w0 + (-0.125 * ((h * (w0 * (t_1 ^ 2.0))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+295], N[(w0 * N[(t$95$1 * N[Sqrt[N[(h * N[(-0.25 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], 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], N[(w0 + N[(-0.125 * N[(N[(h * N[(w0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -4.99999999999999991e295

   1. Initial program 53.2%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in D around inf 41.8%

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

      1. associate-*r/ 41.8%

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

      2. *-commutative 41.8%

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

      3. times-frac 41.9%

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

      4. associate-*r* 43.6%

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

      5. unpow2 43.6%

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

      6. unpow2 43.6%

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

      7. swap-sqr 51.0%

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

      8. unpow2 51.0%

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

      9. associate-/l* 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in D around 0 41.9%

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

      1. associate-*r* 43.6%

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

      2. unpow2 43.6%

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

      3. unpow2 43.6%

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

      4. swap-sqr 51.0%

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

      5. unpow2 51.0%

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

      6. associate-*l/ 51.0%

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

      7. *-commutative 51.0%

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

      8. unpow2 51.0%

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

      9. unpow2 51.0%

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

      10. times-frac 54.9%

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

      11. associate-/l* 53.2%

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

      12. associate-/l* 53.2%

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

      13. unpow2 53.2%

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

      14. associate-/l* 54.9%

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

      15. *-rgt-identity 54.9%

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

      16. associate-*r/ 54.9%

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

      17. associate-*l* 54.9%

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

      18. associate-*r/ 54.9%

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

      19. associate-*l/ 54.9%

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

      20. *-rgt-identity 54.9%

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

   9. Simplified 54.9%

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

       1. pow1/2 54.9%

          \[\leadsto w0 \cdot \color{blue}{{\left(\frac{-0.25}{\ell} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}^{0.5}} \]

       2. associate-*r* 54.9%

          \[\leadsto w0 \cdot {\color{blue}{\left(\left(\frac{-0.25}{\ell} \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}^{0.5} \]

       3. unpow-prod-down 58.2%

          \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot {\left({\left(D \cdot \frac{M}{d}\right)}^{2}\right)}^{0.5}\right)} \]

       4. pow1/2 58.2%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \color{blue}{\sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}}\right) \]

       5. *-commutative 58.2%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}\right) \]

   11. Applied egg-rr 58.2%

       \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\left(\frac{M}{d} \cdot D\right)}^{2}}\right)} \]
   12. Step-by-step derivation

       1. unpow1/2 58.2%

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

       2. *-commutative 58.2%

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

       3. unpow2 58.2%

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

       4. rem-sqrt-square 76.6%

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

       5. *-commutative 76.6%

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

   13. Simplified 76.6%

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

       1. expm1-log1p-u 43.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left|D \cdot \frac{M}{d}\right|\right)\right)\right)} \]

       2. expm1-udef 33.4%

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

       3. *-commutative 33.4%

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

       4. add-sqr-sqrt 14.4%

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

       5. fabs-sqr 14.4%

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

       6. add-sqr-sqrt 14.6%

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

       7. *-commutative 14.6%

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

   15. Applied egg-rr 14.6%

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

       1. expm1-def 19.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)} \]

       2. expm1-log1p 36.1%

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

       3. *-commutative 36.1%

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

   17. Simplified 36.1%

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

   if -4.99999999999999991e295 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < 2e-3

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   if 2e-3 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in D around 0 53.0%

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

      1. times-frac 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in D around 0 53.0%

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

      1. associate-*r* 53.0%

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

      2. times-frac 42.1%

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

      3. unpow2 42.1%

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

      4. unpow2 42.1%

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

      5. swap-sqr 47.4%

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

      6. unpow2 47.4%

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

      7. times-frac 47.4%

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

      8. associate-/l* 47.4%

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

      9. associate-/l* 47.4%

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

      10. unpow2 47.4%

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

      11. associate-/l* 47.4%

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

      12. *-rgt-identity 47.4%

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

      13. associate-*r/ 47.4%

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

      14. associate-*l* 47.4%

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

      15. associate-*r/ 47.4%

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

      16. associate-*l/ 47.4%

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

      17. *-rgt-identity 47.4%

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

   9. Simplified 47.4%

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

       1. expm1-log1p-u 36.8%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\right)} \]

       2. expm1-udef 36.8%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right)} \]

       3. *-commutative 36.8%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right) \]

       4. associate-/l* 36.8%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \color{blue}{\frac{h}{\frac{\ell}{w0}}}\right)} - 1\right) \]

   11. Applied egg-rr 36.8%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 36.8%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\right)} \]

       2. expm1-log1p 47.4%

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

       3. *-commutative 47.4%

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

       4. associate-/r/ 0.0%

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

       5. associate-*l* 0.0%

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

       6. *-commutative 0.0%

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

   13. Simplified 0.0%

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

       1. associate-*l/ 79.9%

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

   15. Applied egg-rr 79.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+295}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 0.002:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{h \cdot \left(w0 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}\\ \end{array} \]

Alternative 3: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+72}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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)) -5e+72)
   (* w0 (* (* D (/ M d)) (sqrt (* h (/ -0.25 l)))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (/ (* M D) d)) 2.0)) l))))))

C

M = abs(M);
D = abs(D);
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)) <= -5e+72) {
		tmp = w0 * ((D * (M / d)) * sqrt((h * (-0.25 / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * ((M * D) / d)), 2.0)) / l)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: d should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l)) <= (-5d+72)) then
        tmp = w0 * ((d * (m / d_1)) * sqrt((h * ((-0.25d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((h * ((0.5d0 * ((m * d) / d_1)) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
d = Math.abs(d);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+72) {
		tmp = w0 * ((D * (M / d)) * Math.sqrt((h * (-0.25 / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * ((M * D) / d)), 2.0)) / l)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -5e+72:
		tmp = w0 * ((D * (M / d)) * math.sqrt((h * (-0.25 / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((0.5 * ((M * D) / d)), 2.0)) / l)))
	return tmp

Julia

M = abs(M)
D = abs(D)
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)) <= -5e+72)
		tmp = Float64(w0 * Float64(Float64(D * Float64(M / d)) * sqrt(Float64(h * Float64(-0.25 / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (((((M * D) / (2.0 * d)) ^ 2.0) * (h / l)) <= -5e+72)
		tmp = w0 * ((D * (M / d)) * sqrt((h * (-0.25 / l))));
	else
		tmp = w0 * sqrt((1.0 - ((h * ((0.5 * ((M * D) / d)) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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], -5e+72], N[(w0 * N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h * N[(-0.25 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -4.99999999999999992e72

   1. Initial program 62.8%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in D around inf 37.4%

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

      1. associate-*r/ 37.4%

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

      2. *-commutative 37.4%

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

      3. times-frac 37.4%

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

      4. associate-*r* 38.8%

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

      5. unpow2 38.8%

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

      6. unpow2 38.8%

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

      7. swap-sqr 48.8%

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

      8. unpow2 48.8%

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

      9. associate-/l* 48.9%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in D around 0 37.4%

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

      1. associate-*r* 38.8%

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

      2. unpow2 38.8%

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

      3. unpow2 38.8%

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

      4. swap-sqr 48.8%

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

      5. unpow2 48.8%

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

      6. associate-*l/ 48.9%

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

      7. *-commutative 48.9%

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

      8. unpow2 48.9%

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

      9. unpow2 48.9%

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

      10. times-frac 61.6%

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

      11. associate-/l* 60.2%

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

      12. associate-/l* 60.2%

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

      13. unpow2 60.2%

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

      14. associate-/l* 61.6%

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

      15. *-rgt-identity 61.6%

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

      16. associate-*r/ 61.6%

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

      17. associate-*l* 61.5%

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

      18. associate-*r/ 61.6%

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

      19. associate-*l/ 61.6%

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

      20. *-rgt-identity 61.6%

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

   9. Simplified 61.6%

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

       1. pow1/2 61.6%

          \[\leadsto w0 \cdot \color{blue}{{\left(\frac{-0.25}{\ell} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}^{0.5}} \]

       2. associate-*r* 62.9%

          \[\leadsto w0 \cdot {\color{blue}{\left(\left(\frac{-0.25}{\ell} \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}^{0.5} \]

       3. unpow-prod-down 65.4%

          \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot {\left({\left(D \cdot \frac{M}{d}\right)}^{2}\right)}^{0.5}\right)} \]

       4. pow1/2 65.4%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \color{blue}{\sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}}\right) \]

       5. *-commutative 65.4%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}\right) \]

   11. Applied egg-rr 65.4%

       \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\left(\frac{M}{d} \cdot D\right)}^{2}}\right)} \]
   12. Step-by-step derivation

       1. unpow1/2 65.4%

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

       2. *-commutative 65.4%

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

       3. unpow2 65.4%

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

       4. rem-sqrt-square 80.0%

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

       5. *-commutative 80.0%

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

   13. Simplified 80.0%

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

       1. expm1-log1p-u 49.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left|D \cdot \frac{M}{d}\right|\right)\right)\right)} \]

       2. expm1-udef 31.9%

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

       3. *-commutative 31.9%

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

       4. add-sqr-sqrt 15.5%

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

       5. fabs-sqr 15.5%

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

       6. add-sqr-sqrt 15.8%

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

       7. *-commutative 15.8%

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

   15. Applied egg-rr 15.8%

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

       1. expm1-def 25.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)} \]

       2. expm1-log1p 41.2%

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

       3. *-commutative 41.2%

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

   17. Simplified 41.2%

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

   if -4.99999999999999992e72 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 89.6%

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

   3. Simplified 89.7%

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

      1. associate-*r/ 97.9%

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

      2. frac-times 97.9%

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

      3. *-commutative 97.9%

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

      4. *-un-lft-identity 97.9%

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

      5. times-frac 97.9%

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

      6. metadata-eval 97.9%

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

      7. *-commutative 97.9%

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

   5. Applied egg-rr 97.9%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -5 \cdot 10^{+72}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 4: 78.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-10}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+188}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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) 1e-10)
   w0
   (if (<= (* M D) 5e+188)
     (+ w0 (* -0.125 (* (pow (/ (* M D) d) 2.0) (/ (* h w0) l))))
     (* w0 (* (* D (/ M d)) (sqrt (* h (/ -0.25 l))))))))

C

M = abs(M);
D = abs(D);
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) <= 1e-10) {
		tmp = w0;
	} else if ((M * D) <= 5e+188) {
		tmp = w0 + (-0.125 * (pow(((M * D) / d), 2.0) * ((h * w0) / l)));
	} else {
		tmp = w0 * ((D * (M / d)) * sqrt((h * (-0.25 / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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) <= 1d-10) then
        tmp = w0
    else if ((m * d) <= 5d+188) then
        tmp = w0 + ((-0.125d0) * ((((m * d) / d_1) ** 2.0d0) * ((h * w0) / l)))
    else
        tmp = w0 * ((d * (m / d_1)) * sqrt((h * ((-0.25d0) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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) <= 1e-10) {
		tmp = w0;
	} else if ((M * D) <= 5e+188) {
		tmp = w0 + (-0.125 * (Math.pow(((M * D) / d), 2.0) * ((h * w0) / l)));
	} else {
		tmp = w0 * ((D * (M / d)) * Math.sqrt((h * (-0.25 / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (M * D) <= 1e-10:
		tmp = w0
	elif (M * D) <= 5e+188:
		tmp = w0 + (-0.125 * (math.pow(((M * D) / d), 2.0) * ((h * w0) / l)))
	else:
		tmp = w0 * ((D * (M / d)) * math.sqrt((h * (-0.25 / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (Float64(M * D) <= 1e-10)
		tmp = w0;
	elseif (Float64(M * D) <= 5e+188)
		tmp = Float64(w0 + Float64(-0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) * Float64(Float64(h * w0) / l))));
	else
		tmp = Float64(w0 * Float64(Float64(D * Float64(M / d)) * sqrt(Float64(h * Float64(-0.25 / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
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) <= 1e-10)
		tmp = w0;
	elseif ((M * D) <= 5e+188)
		tmp = w0 + (-0.125 * ((((M * D) / d) ^ 2.0) * ((h * w0) / l)));
	else
		tmp = w0 * ((D * (M / d)) * sqrt((h * (-0.25 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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[(M * D), $MachinePrecision], 1e-10], w0, If[LessEqual[N[(M * D), $MachinePrecision], 5e+188], N[(w0 + N[(-0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * w0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h * N[(-0.25 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 1.00000000000000004e-10

   1. Initial program 85.8%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in D around 0 79.4%

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

   if 1.00000000000000004e-10 < (*.f64 M D) < 5.0000000000000001e188

   1. Initial program 84.9%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in D around 0 26.1%

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

      1. times-frac 25.9%

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

   6. Simplified 25.9%

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

   7. Taylor expanded in D around 0 26.1%

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

      1. associate-*r* 26.1%

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

      2. times-frac 26.0%

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

      3. unpow2 26.0%

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

      4. unpow2 26.0%

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

      5. swap-sqr 48.7%

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

      6. unpow2 48.7%

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

      7. times-frac 61.4%

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

      8. associate-/l* 58.3%

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

      9. associate-/l* 58.3%

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

      10. unpow2 58.3%

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

      11. associate-/l* 61.4%

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

      12. *-rgt-identity 61.4%

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

      13. associate-*r/ 61.4%

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

      14. associate-*l* 61.3%

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

      15. associate-*r/ 61.3%

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

      16. associate-*l/ 61.3%

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

      17. *-rgt-identity 61.3%

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

   9. Simplified 61.3%

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

       1. associate-*r/ 61.4%

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

       2. *-commutative 61.4%

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

   11. Applied egg-rr 61.4%

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

   if 5.0000000000000001e188 < (*.f64 M D)

   1. Initial program 47.3%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in D around inf 43.1%

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

      1. associate-*r/ 43.1%

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

      2. *-commutative 43.1%

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

      3. times-frac 47.4%

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

      4. associate-*r* 47.4%

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

      5. unpow2 47.4%

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

      6. unpow2 47.4%

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

      7. swap-sqr 47.4%

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

      8. unpow2 47.4%

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

      9. associate-/l* 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in D around 0 47.4%

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

      1. associate-*r* 47.4%

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

      2. unpow2 47.4%

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

      3. unpow2 47.4%

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

      4. swap-sqr 47.4%

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

      5. unpow2 47.4%

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

      6. associate-*l/ 47.4%

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

      7. *-commutative 47.4%

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

      8. unpow2 47.4%

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

      9. unpow2 47.4%

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

      10. times-frac 51.8%

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

      11. associate-/l* 51.8%

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

      12. associate-/l* 51.8%

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

      13. unpow2 51.8%

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

      14. associate-/l* 51.8%

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

      15. *-rgt-identity 51.8%

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

      16. associate-*r/ 51.8%

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

      17. associate-*l* 51.8%

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

      18. associate-*r/ 51.8%

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

      19. associate-*l/ 51.8%

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

      20. *-rgt-identity 51.8%

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

   9. Simplified 51.8%

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

       1. pow1/2 51.8%

          \[\leadsto w0 \cdot \color{blue}{{\left(\frac{-0.25}{\ell} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}^{0.5}} \]

       2. associate-*r* 43.3%

          \[\leadsto w0 \cdot {\color{blue}{\left(\left(\frac{-0.25}{\ell} \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}^{0.5} \]

       3. unpow-prod-down 47.2%

          \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot {\left({\left(D \cdot \frac{M}{d}\right)}^{2}\right)}^{0.5}\right)} \]

       4. pow1/2 47.2%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \color{blue}{\sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}}\right) \]

       5. *-commutative 47.2%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}\right) \]

   11. Applied egg-rr 47.2%

       \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\left(\frac{M}{d} \cdot D\right)}^{2}}\right)} \]
   12. Step-by-step derivation

       1. unpow1/2 47.2%

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

       2. *-commutative 47.2%

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

       3. unpow2 47.2%

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

       4. rem-sqrt-square 63.4%

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

       5. *-commutative 63.4%

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

   13. Simplified 63.4%

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

       1. expm1-log1p-u 28.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left|D \cdot \frac{M}{d}\right|\right)\right)\right)} \]

       2. expm1-udef 16.8%

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

       3. *-commutative 16.8%

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

       4. add-sqr-sqrt 16.3%

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

       5. fabs-sqr 16.3%

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

       6. add-sqr-sqrt 16.8%

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

       7. *-commutative 16.8%

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

   15. Applied egg-rr 16.8%

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

       1. expm1-def 24.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)} \]

       2. expm1-log1p 46.8%

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

       3. *-commutative 46.8%

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

   17. Simplified 46.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 10^{-10}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \cdot D \leq 5 \cdot 10^{+188}:\\ \;\;\;\;w0 + -0.125 \cdot \left({\left(\frac{M \cdot D}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \end{array} \]

Alternative 5: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;M \leq 4 \cdot 10^{-128}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 9 \cdot 10^{+29}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(w0 \cdot {t_0}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(t_0 \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (* D (/ M d))))
   (if (<= M 4e-128)
     w0
     (if (<= M 9e+29)
       (+ w0 (* -0.125 (* (/ h l) (* w0 (pow t_0 2.0)))))
       (* w0 (* t_0 (sqrt (* h (/ -0.25 l)))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / d);
	double tmp;
	if (M <= 4e-128) {
		tmp = w0;
	} else if (M <= 9e+29) {
		tmp = w0 + (-0.125 * ((h / l) * (w0 * pow(t_0, 2.0))));
	} else {
		tmp = w0 * (t_0 * sqrt((h * (-0.25 / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 = d * (m / d_1)
    if (m <= 4d-128) then
        tmp = w0
    else if (m <= 9d+29) then
        tmp = w0 + ((-0.125d0) * ((h / l) * (w0 * (t_0 ** 2.0d0))))
    else
        tmp = w0 * (t_0 * sqrt((h * ((-0.25d0) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 = D * (M / d);
	double tmp;
	if (M <= 4e-128) {
		tmp = w0;
	} else if (M <= 9e+29) {
		tmp = w0 + (-0.125 * ((h / l) * (w0 * Math.pow(t_0, 2.0))));
	} else {
		tmp = w0 * (t_0 * Math.sqrt((h * (-0.25 / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = D * (M / d)
	tmp = 0
	if M <= 4e-128:
		tmp = w0
	elif M <= 9e+29:
		tmp = w0 + (-0.125 * ((h / l) * (w0 * math.pow(t_0, 2.0))))
	else:
		tmp = w0 * (t_0 * math.sqrt((h * (-0.25 / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (M <= 4e-128)
		tmp = w0;
	elseif (M <= 9e+29)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h / l) * Float64(w0 * (t_0 ^ 2.0)))));
	else
		tmp = Float64(w0 * Float64(t_0 * sqrt(Float64(h * Float64(-0.25 / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = D * (M / d);
	tmp = 0.0;
	if (M <= 4e-128)
		tmp = w0;
	elseif (M <= 9e+29)
		tmp = w0 + (-0.125 * ((h / l) * (w0 * (t_0 ^ 2.0))));
	else
		tmp = w0 * (t_0 * sqrt((h * (-0.25 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, 4e-128], w0, If[LessEqual[M, 9e+29], N[(w0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(w0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(t$95$0 * N[Sqrt[N[(h * N[(-0.25 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 4.00000000000000022e-128

   1. Initial program 83.6%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in D around 0 73.2%

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

   if 4.00000000000000022e-128 < M < 9.0000000000000005e29

   1. Initial program 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in D around 0 57.3%

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

      1. times-frac 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in D around 0 57.3%

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

      1. associate-*r* 57.3%

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

      2. times-frac 54.0%

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

      3. unpow2 54.0%

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

      4. unpow2 54.0%

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

      5. swap-sqr 57.1%

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

      6. unpow2 57.1%

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

      7. times-frac 63.4%

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

      8. associate-/l* 63.4%

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

      9. associate-/l* 63.4%

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

      10. unpow2 63.4%

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

      11. associate-/l* 63.4%

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

      12. *-rgt-identity 63.4%

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

      13. associate-*r/ 63.4%

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

      14. associate-*l* 63.4%

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

      15. associate-*r/ 63.4%

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

      16. associate-*l/ 63.4%

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

      17. *-rgt-identity 63.4%

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

   9. Simplified 63.4%

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

       1. expm1-log1p-u 62.6%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\right)} \]

       2. expm1-udef 62.6%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right)} \]

       3. *-commutative 62.6%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right) \]

       4. associate-/l* 62.6%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \color{blue}{\frac{h}{\frac{\ell}{w0}}}\right)} - 1\right) \]

   11. Applied egg-rr 62.6%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 62.6%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\right)} \]

       2. expm1-log1p 63.3%

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

       3. *-commutative 63.3%

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

       4. associate-/r/ 69.7%

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

       5. associate-*l* 70.1%

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

       6. *-commutative 70.1%

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

   13. Simplified 70.1%

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

   if 9.0000000000000005e29 < M

   1. Initial program 79.7%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in D around inf 25.5%

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

      1. associate-*r/ 25.5%

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

      2. *-commutative 25.5%

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

      3. times-frac 25.5%

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

      4. associate-*r* 25.8%

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

      5. unpow2 25.8%

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

      6. unpow2 25.8%

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

      7. swap-sqr 28.1%

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

      8. unpow2 28.1%

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

      9. associate-/l* 28.1%

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

   6. Simplified 28.1%

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

   7. Taylor expanded in D around 0 25.5%

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

      1. associate-*r* 25.8%

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

      2. unpow2 25.8%

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

      3. unpow2 25.8%

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

      4. swap-sqr 28.1%

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

      5. unpow2 28.1%

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

      6. associate-*l/ 28.1%

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

      7. *-commutative 28.1%

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

      8. unpow2 28.1%

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

      9. unpow2 28.1%

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

      10. times-frac 31.6%

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

      11. associate-/l* 31.6%

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

      12. associate-/l* 31.6%

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

      13. unpow2 31.6%

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

      14. associate-/l* 31.6%

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

      15. *-rgt-identity 31.6%

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

      16. associate-*r/ 31.6%

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

      17. associate-*l* 31.5%

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

      18. associate-*r/ 31.6%

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

      19. associate-*l/ 31.6%

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

      20. *-rgt-identity 31.6%

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

   9. Simplified 31.6%

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

       1. pow1/2 31.6%

          \[\leadsto w0 \cdot \color{blue}{{\left(\frac{-0.25}{\ell} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}^{0.5}} \]

       2. associate-*r* 28.0%

          \[\leadsto w0 \cdot {\color{blue}{\left(\left(\frac{-0.25}{\ell} \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}^{0.5} \]

       3. unpow-prod-down 27.6%

          \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot {\left({\left(D \cdot \frac{M}{d}\right)}^{2}\right)}^{0.5}\right)} \]

       4. pow1/2 27.6%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \color{blue}{\sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}}\right) \]

       5. *-commutative 27.6%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}\right) \]

   11. Applied egg-rr 27.6%

       \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\left(\frac{M}{d} \cdot D\right)}^{2}}\right)} \]
   12. Step-by-step derivation

       1. unpow1/2 27.6%

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

       2. *-commutative 27.6%

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

       3. unpow2 27.6%

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

       4. rem-sqrt-square 32.6%

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

       5. *-commutative 32.6%

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

   13. Simplified 32.6%

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

       1. expm1-log1p-u 20.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left|D \cdot \frac{M}{d}\right|\right)\right)\right)} \]

       2. expm1-udef 13.3%

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

       3. *-commutative 13.3%

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

       4. add-sqr-sqrt 8.1%

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

       5. fabs-sqr 8.1%

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

       6. add-sqr-sqrt 8.2%

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

       7. *-commutative 8.2%

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

   15. Applied egg-rr 8.2%

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

       1. expm1-def 13.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)} \]

       2. expm1-log1p 17.1%

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

       3. *-commutative 17.1%

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

   17. Simplified 17.1%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{-128}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 9 \cdot 10^{+29}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(w0 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \end{array} \]

Alternative 6: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{+188}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{h \cdot \left(w0 \cdot {t_0}^{2}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(t_0 \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (* D (/ M d))))
   (if (<= (* M D) 5e+188)
     (+ w0 (* -0.125 (/ (* h (* w0 (pow t_0 2.0))) l)))
     (* w0 (* t_0 (sqrt (* h (/ -0.25 l))))))))

C

M = abs(M);
D = abs(D);
d = abs(d);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = D * (M / d);
	double tmp;
	if ((M * D) <= 5e+188) {
		tmp = w0 + (-0.125 * ((h * (w0 * pow(t_0, 2.0))) / l));
	} else {
		tmp = w0 * (t_0 * sqrt((h * (-0.25 / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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 = d * (m / d_1)
    if ((m * d) <= 5d+188) then
        tmp = w0 + ((-0.125d0) * ((h * (w0 * (t_0 ** 2.0d0))) / l))
    else
        tmp = w0 * (t_0 * sqrt((h * ((-0.25d0) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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 = D * (M / d);
	double tmp;
	if ((M * D) <= 5e+188) {
		tmp = w0 + (-0.125 * ((h * (w0 * Math.pow(t_0, 2.0))) / l));
	} else {
		tmp = w0 * (t_0 * Math.sqrt((h * (-0.25 / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = D * (M / d)
	tmp = 0
	if (M * D) <= 5e+188:
		tmp = w0 + (-0.125 * ((h * (w0 * math.pow(t_0, 2.0))) / l))
	else:
		tmp = w0 * (t_0 * math.sqrt((h * (-0.25 / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(D * Float64(M / d))
	tmp = 0.0
	if (Float64(M * D) <= 5e+188)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(h * Float64(w0 * (t_0 ^ 2.0))) / l)));
	else
		tmp = Float64(w0 * Float64(t_0 * sqrt(Float64(h * Float64(-0.25 / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
d = abs(d)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = D * (M / d);
	tmp = 0.0;
	if ((M * D) <= 5e+188)
		tmp = w0 + (-0.125 * ((h * (w0 * (t_0 ^ 2.0))) / l));
	else
		tmp = w0 * (t_0 * sqrt((h * (-0.25 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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[(D * N[(M / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 5e+188], N[(w0 + N[(-0.125 * N[(N[(h * N[(w0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(t$95$0 * N[Sqrt[N[(h * N[(-0.25 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 5.0000000000000001e188

   1. Initial program 85.7%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in D around 0 47.4%

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

      1. times-frac 47.3%

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

   6. Simplified 47.3%

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

   7. Taylor expanded in D around 0 47.4%

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

      1. associate-*r* 49.5%

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

      2. times-frac 49.1%

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

      3. unpow2 49.1%

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

      4. unpow2 49.1%

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

      5. swap-sqr 63.1%

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

      6. unpow2 63.1%

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

      7. times-frac 72.2%

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

      8. associate-/l* 71.8%

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

      9. associate-/l* 71.8%

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

      10. unpow2 71.8%

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

      11. associate-/l* 72.2%

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

      12. *-rgt-identity 72.2%

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

      13. associate-*r/ 72.2%

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

      14. associate-*l* 72.2%

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

      15. associate-*r/ 72.2%

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

      16. associate-*l/ 72.2%

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

      17. *-rgt-identity 72.2%

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

   9. Simplified 72.2%

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

       1. expm1-log1p-u 65.7%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)\right)} \]

       2. expm1-udef 65.5%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(D \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right)} \]

       3. *-commutative 65.5%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2} \cdot \frac{h \cdot w0}{\ell}\right)} - 1\right) \]

       4. associate-/l* 67.6%

          \[\leadsto w0 + -0.125 \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \color{blue}{\frac{h}{\frac{\ell}{w0}}}\right)} - 1\right) \]

   11. Applied egg-rr 67.6%

       \[\leadsto w0 + -0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 67.8%

          \[\leadsto w0 + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{M}{d} \cdot D\right)}^{2} \cdot \frac{h}{\frac{\ell}{w0}}\right)\right)} \]

       2. expm1-log1p 74.4%

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

       3. *-commutative 74.4%

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

       4. associate-/r/ 74.8%

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

       5. associate-*l* 77.4%

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

       6. *-commutative 77.4%

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

   13. Simplified 77.4%

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

       1. associate-*l/ 82.6%

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

   15. Applied egg-rr 82.6%

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

   if 5.0000000000000001e188 < (*.f64 M D)

   1. Initial program 47.3%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in D around inf 43.1%

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

      1. associate-*r/ 43.1%

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

      2. *-commutative 43.1%

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

      3. times-frac 47.4%

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

      4. associate-*r* 47.4%

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

      5. unpow2 47.4%

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

      6. unpow2 47.4%

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

      7. swap-sqr 47.4%

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

      8. unpow2 47.4%

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

      9. associate-/l* 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in D around 0 47.4%

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

      1. associate-*r* 47.4%

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

      2. unpow2 47.4%

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

      3. unpow2 47.4%

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

      4. swap-sqr 47.4%

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

      5. unpow2 47.4%

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

      6. associate-*l/ 47.4%

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

      7. *-commutative 47.4%

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

      8. unpow2 47.4%

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

      9. unpow2 47.4%

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

      10. times-frac 51.8%

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

      11. associate-/l* 51.8%

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

      12. associate-/l* 51.8%

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

      13. unpow2 51.8%

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

      14. associate-/l* 51.8%

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

      15. *-rgt-identity 51.8%

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

      16. associate-*r/ 51.8%

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

      17. associate-*l* 51.8%

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

      18. associate-*r/ 51.8%

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

      19. associate-*l/ 51.8%

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

      20. *-rgt-identity 51.8%

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

   9. Simplified 51.8%

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

       1. pow1/2 51.8%

          \[\leadsto w0 \cdot \color{blue}{{\left(\frac{-0.25}{\ell} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}^{0.5}} \]

       2. associate-*r* 43.3%

          \[\leadsto w0 \cdot {\color{blue}{\left(\left(\frac{-0.25}{\ell} \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}^{0.5} \]

       3. unpow-prod-down 47.2%

          \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot {\left({\left(D \cdot \frac{M}{d}\right)}^{2}\right)}^{0.5}\right)} \]

       4. pow1/2 47.2%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \color{blue}{\sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}}\right) \]

       5. *-commutative 47.2%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}\right) \]

   11. Applied egg-rr 47.2%

       \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\left(\frac{M}{d} \cdot D\right)}^{2}}\right)} \]
   12. Step-by-step derivation

       1. unpow1/2 47.2%

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

       2. *-commutative 47.2%

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

       3. unpow2 47.2%

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

       4. rem-sqrt-square 63.4%

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

       5. *-commutative 63.4%

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

   13. Simplified 63.4%

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

       1. expm1-log1p-u 28.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left|D \cdot \frac{M}{d}\right|\right)\right)\right)} \]

       2. expm1-udef 16.8%

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

       3. *-commutative 16.8%

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

       4. add-sqr-sqrt 16.3%

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

       5. fabs-sqr 16.3%

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

       6. add-sqr-sqrt 16.8%

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

       7. *-commutative 16.8%

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

   15. Applied egg-rr 16.8%

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

       1. expm1-def 24.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)} \]

       2. expm1-log1p 46.8%

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

       3. *-commutative 46.8%

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

   17. Simplified 46.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 5 \cdot 10^{+188}:\\ \;\;\;\;w0 + -0.125 \cdot \frac{h \cdot \left(w0 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \end{array} \]

Alternative 7: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ d = |d|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 1.32 \cdot 10^{-30} \lor \neg \left(M \leq 8 \cdot 10^{-12}\right) \land M \leq 3.15 \cdot 10^{+29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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 (or (<= M 1.32e-30) (and (not (<= M 8e-12)) (<= M 3.15e+29)))
   w0
   (* w0 (* (* D (/ M d)) (sqrt (* h (/ -0.25 l)))))))

C

M = abs(M);
D = abs(D);
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-30) || (!(M <= 8e-12) && (M <= 3.15e+29))) {
		tmp = w0;
	} else {
		tmp = w0 * ((D * (M / d)) * sqrt((h * (-0.25 / l))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D 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-30) .or. (.not. (m <= 8d-12)) .and. (m <= 3.15d+29)) then
        tmp = w0
    else
        tmp = w0 * ((d * (m / d_1)) * sqrt((h * ((-0.25d0) / l))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
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-30) || (!(M <= 8e-12) && (M <= 3.15e+29))) {
		tmp = w0;
	} else {
		tmp = w0 * ((D * (M / d)) * Math.sqrt((h * (-0.25 / l))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
d = abs(d)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if (M <= 1.32e-30) or (not (M <= 8e-12) and (M <= 3.15e+29)):
		tmp = w0
	else:
		tmp = w0 * ((D * (M / d)) * math.sqrt((h * (-0.25 / l))))
	return tmp

Julia

M = abs(M)
D = abs(D)
d = abs(d)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if ((M <= 1.32e-30) || (!(M <= 8e-12) && (M <= 3.15e+29)))
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(D * Float64(M / d)) * sqrt(Float64(h * Float64(-0.25 / l)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
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-30) || (~((M <= 8e-12)) && (M <= 3.15e+29)))
		tmp = w0;
	else
		tmp = w0 * ((D * (M / d)) * sqrt((h * (-0.25 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: 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[Or[LessEqual[M, 1.32e-30], And[N[Not[LessEqual[M, 8e-12]], $MachinePrecision], LessEqual[M, 3.15e+29]]], w0, N[(w0 * N[(N[(D * N[(M / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h * N[(-0.25 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.32e-30 or 7.99999999999999984e-12 < M < 3.1499999999999999e29

   1. Initial program 83.6%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in D around 0 73.9%

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

   if 1.32e-30 < M < 7.99999999999999984e-12 or 3.1499999999999999e29 < M

   1. Initial program 77.3%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in D around inf 24.8%

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

      1. associate-*r/ 24.8%

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

      2. *-commutative 24.8%

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

      3. times-frac 24.8%

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

      4. associate-*r* 25.1%

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

      5. unpow2 25.1%

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

      6. unpow2 25.1%

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

      7. swap-sqr 27.4%

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

      8. unpow2 27.4%

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

      9. associate-/l* 27.4%

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

   6. Simplified 27.4%

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

   7. Taylor expanded in D around 0 24.8%

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

      1. associate-*r* 25.1%

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

      2. unpow2 25.1%

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

      3. unpow2 25.1%

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

      4. swap-sqr 27.4%

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

      5. unpow2 27.4%

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

      6. associate-*l/ 27.4%

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

      7. *-commutative 27.4%

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

      8. unpow2 27.4%

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

      9. unpow2 27.4%

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

      10. times-frac 30.7%

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

      11. associate-/l* 30.7%

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

      12. associate-/l* 30.7%

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

      13. unpow2 30.7%

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

      14. associate-/l* 30.7%

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

      15. *-rgt-identity 30.7%

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

      16. associate-*r/ 30.7%

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

      17. associate-*l* 30.7%

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

      18. associate-*r/ 30.7%

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

      19. associate-*l/ 30.7%

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

      20. *-rgt-identity 30.7%

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

   9. Simplified 30.7%

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

       1. pow1/2 30.7%

          \[\leadsto w0 \cdot \color{blue}{{\left(\frac{-0.25}{\ell} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)}^{0.5}} \]

       2. associate-*r* 27.3%

          \[\leadsto w0 \cdot {\color{blue}{\left(\left(\frac{-0.25}{\ell} \cdot h\right) \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)}}^{0.5} \]

       3. unpow-prod-down 28.4%

          \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot {\left({\left(D \cdot \frac{M}{d}\right)}^{2}\right)}^{0.5}\right)} \]

       4. pow1/2 28.4%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \color{blue}{\sqrt{{\left(D \cdot \frac{M}{d}\right)}^{2}}}\right) \]

       5. *-commutative 28.4%

          \[\leadsto w0 \cdot \left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\color{blue}{\left(\frac{M}{d} \cdot D\right)}}^{2}}\right) \]

   11. Applied egg-rr 28.4%

       \[\leadsto w0 \cdot \color{blue}{\left({\left(\frac{-0.25}{\ell} \cdot h\right)}^{0.5} \cdot \sqrt{{\left(\frac{M}{d} \cdot D\right)}^{2}}\right)} \]
   12. Step-by-step derivation

       1. unpow1/2 28.4%

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

       2. *-commutative 28.4%

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

       3. unpow2 28.4%

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

       4. rem-sqrt-square 33.3%

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

       5. *-commutative 33.3%

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

   13. Simplified 33.3%

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

       1. expm1-log1p-u 20.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left|D \cdot \frac{M}{d}\right|\right)\right)\right)} \]

       2. expm1-udef 14.3%

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

       3. *-commutative 14.3%

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

       4. add-sqr-sqrt 7.8%

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

       5. fabs-sqr 7.8%

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

       6. add-sqr-sqrt 7.9%

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

       7. *-commutative 7.9%

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

   15. Applied egg-rr 7.9%

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

       1. expm1-def 12.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(w0 \cdot \left(\sqrt{h \cdot \frac{-0.25}{\ell}} \cdot \left(D \cdot \frac{M}{d}\right)\right)\right)\right)} \]

       2. expm1-log1p 16.6%

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

       3. *-commutative 16.6%

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

   17. Simplified 16.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.32 \cdot 10^{-30} \lor \neg \left(M \leq 8 \cdot 10^{-12}\right) \land M \leq 3.15 \cdot 10^{+29}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(D \cdot \frac{M}{d}\right) \cdot \sqrt{h \cdot \frac{-0.25}{\ell}}\right)\\ \end{array} \]

Alternative 8: 68.1% accurate, 216.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.1%

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

3. Simplified 82.1%

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

4. Taylor expanded in D around 0 69.2%

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

5. Final simplification 69.2%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023300 
(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))))))