Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 80.9% → 86.2%

Time: 29.0s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{\frac{d}{\frac{M}{\ell}}} \cdot \frac{D}{\frac{d}{M \cdot h}}\\ t_1 := \frac{M \cdot D}{d}\\ \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\frac{0.25}{\ell} \cdot \left(t_1 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* (/ D (/ d (/ M l))) (/ D (/ d (* M h))))) (t_1 (/ (* M D) d)))
   (if (<= (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)) -1e+26)
     (* w0 (sqrt (- 1.0 (* h (* (/ 0.25 l) (* t_1 t_1))))))
     (* w0 (+ 1.0 (* -0.125 (cbrt (* t_0 (* t_0 t_0)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D / (d / (M / l))) * (D / (d / (M * h)));
	double t_1 = (M * D) / d;
	double tmp;
	if ((pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+26) {
		tmp = w0 * sqrt((1.0 - (h * ((0.25 / l) * (t_1 * t_1)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * cbrt((t_0 * (t_0 * t_0)))));
	}
	return tmp;
}

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (D / (d / (M / l))) * (D / (d / (M * h)));
	double t_1 = (M * D) / d;
	double tmp;
	if ((Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l)) <= -1e+26) {
		tmp = w0 * Math.sqrt((1.0 - (h * ((0.25 / l) * (t_1 * t_1)))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * Math.cbrt((t_0 * (t_0 * t_0)))));
	}
	return tmp;
}

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(D / Float64(d / Float64(M / l))) * Float64(D / Float64(d / Float64(M * h))))
	t_1 = Float64(Float64(M * D) / d)
	tmp = 0.0
	if (Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)) <= -1e+26)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(0.25 / l) * Float64(t_1 * t_1))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * cbrt(Float64(t_0 * Float64(t_0 * t_0))))));
	end
	return tmp
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(N[(D / N[(d / N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision], -1e+26], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(0.25 / l), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $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)) < -1.00000000000000005e26

   1. Initial program 66.8%

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

      1. associate-/l* 66.7%

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

   3. Simplified 66.7%

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

      1. associate-/l* 66.8%

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

      2. clear-num 66.8%

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

      3. un-div-inv 66.8%

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

      4. associate-*l/ 62.5%

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

      5. *-commutative 62.5%

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

      6. associate-/r* 62.5%

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

      7. div-inv 62.5%

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

      8. metadata-eval 62.5%

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

   5. Applied egg-rr 62.5%

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

      1. associate-/r/ 64.1%

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

      2. *-commutative 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l/ 64.0%

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

      5. associate-*l* 69.3%

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

      6. *-commutative 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in M around 0 50.8%

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

      1. unpow2 50.8%

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

      2. associate-*l* 52.2%

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

      3. associate-*r/ 52.2%

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

      4. associate-*l* 50.8%

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

      5. unpow2 50.8%

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

      6. times-frac 52.2%

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

      7. unpow2 52.2%

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

      8. unpow2 52.2%

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

      9. unswap-sqr 59.7%

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

      10. unpow2 59.7%

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

   10. Simplified 59.7%

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

       1. times-frac 69.4%

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

       2. *-commutative 69.4%

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

       3. *-commutative 69.4%

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

   12. Applied egg-rr 69.4%

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

   if -1.00000000000000005e26 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

   1. Initial program 84.0%

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

      1. associate-/l* 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in M around 0 57.4%

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

      1. *-commutative 57.4%

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

      2. associate-/l* 58.5%

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

      3. unpow2 58.5%

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

      4. unpow2 58.5%

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

      5. *-commutative 58.5%

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

      6. unpow2 58.5%

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

   6. Simplified 58.5%

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

      1. div-inv 58.5%

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

      2. associate-*r* 62.9%

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

      3. associate-*l* 67.1%

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

   8. Applied egg-rr 67.1%

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

      1. div-inv 67.1%

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

      2. *-un-lft-identity 67.1%

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

      3. add-cbrt-cube 67.1%

         \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\sqrt[3]{\left(\frac{D \cdot D}{1 \cdot \frac{\left(\ell \cdot d\right) \cdot d}{M \cdot \left(M \cdot h\right)}} \cdot \frac{D \cdot D}{1 \cdot \frac{\left(\ell \cdot d\right) \cdot d}{M \cdot \left(M \cdot h\right)}}\right) \cdot \frac{D \cdot D}{1 \cdot \frac{\left(\ell \cdot d\right) \cdot d}{M \cdot \left(M \cdot h\right)}}}}\right) \]

   10. Applied egg-rr 76.8%

       \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \color{blue}{\sqrt[3]{\left(\frac{D \cdot D}{\frac{d \cdot \ell}{M} \cdot \frac{d}{h \cdot M}} \cdot \frac{D \cdot D}{\frac{d \cdot \ell}{M} \cdot \frac{d}{h \cdot M}}\right) \cdot \frac{D \cdot D}{\frac{d \cdot \ell}{M} \cdot \frac{d}{h \cdot M}}}}\right) \]
   11. Step-by-step derivation

       1. associate-*l* 76.8%

          \[\leadsto w0 \cdot \left(1 + -0.125 \cdot \sqrt[3]{\color{blue}{\frac{D \cdot D}{\frac{d \cdot \ell}{M} \cdot \frac{d}{h \cdot M}} \cdot \left(\frac{D \cdot D}{\frac{d \cdot \ell}{M} \cdot \frac{d}{h \cdot M}} \cdot \frac{D \cdot D}{\frac{d \cdot \ell}{M} \cdot \frac{d}{h \cdot M}}\right)}}\right) \]

   12. Simplified 84.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\frac{0.25}{\ell} \cdot \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \sqrt[3]{\left(\frac{D}{\frac{d}{\frac{M}{\ell}}} \cdot \frac{D}{\frac{d}{M \cdot h}}\right) \cdot \left(\left(\frac{D}{\frac{d}{\frac{M}{\ell}}} \cdot \frac{D}{\frac{d}{M \cdot h}}\right) \cdot \left(\frac{D}{\frac{d}{\frac{M}{\ell}}} \cdot \frac{D}{\frac{d}{M \cdot h}}\right)\right)}\right)\\ \end{array} \]

Alternative 2: 86.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ w0 \cdot \sqrt{1 - h \cdot \left(\frac{0.25}{\ell} \cdot \left(t_0 \cdot t_0\right)\right)} \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) d)))
   (* w0 (sqrt (- 1.0 (* h (* (/ 0.25 l) (* t_0 t_0))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / d;
	return w0 * sqrt((1.0 - (h * ((0.25 / l) * (t_0 * t_0)))));
}

Fortran

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

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = (M * D) / d;
	return w0 * Math.sqrt((1.0 - (h * ((0.25 / l) * (t_0 * t_0)))));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = (M * D) / d
	return w0 * math.sqrt((1.0 - (h * ((0.25 / l) * (t_0 * t_0)))))

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(0.25 / l) * Float64(t_0 * t_0))))))
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	t_0 = (M * D) / d;
	tmp = w0 * sqrt((1.0 - (h * ((0.25 / l) * (t_0 * t_0)))));
end

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 80.1%

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

3. Simplified 80.1%

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

   1. associate-/l* 79.3%

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

   2. clear-num 79.3%

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

   3. un-div-inv 79.3%

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

   4. associate-*l/ 77.8%

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

   5. *-commutative 77.8%

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

   6. associate-/r* 77.8%

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

   7. div-inv 77.8%

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

   8. metadata-eval 77.8%

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

5. Applied egg-rr 77.8%

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

   1. associate-/r/ 83.5%

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

   2. *-commutative 83.5%

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

   3. *-commutative 83.5%

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

   4. associate-*l/ 83.5%

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

   5. associate-*l* 85.8%

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

   6. *-commutative 85.8%

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

7. Simplified 85.8%

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

8. Taylor expanded in M around 0 59.2%

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

   1. unpow2 59.2%

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

   2. associate-*l* 63.1%

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

   3. associate-*r/ 63.1%

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

   4. associate-*l* 59.2%

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

   5. unpow2 59.2%

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

   6. times-frac 60.8%

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

   7. unpow2 60.8%

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

   8. unpow2 60.8%

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

   9. unswap-sqr 73.8%

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

   10. unpow2 73.8%

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

10. Simplified 73.8%

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

    1. times-frac 85.0%

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

    2. *-commutative 85.0%

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

    3. *-commutative 85.0%

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

12. Applied egg-rr 85.0%

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

13. Final simplification 85.0%

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

Alternative 3: 80.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 + (w0 * ((-0.125d0) * (h * (((d * (m / d_1)) ** 2.0d0) / l))))
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 + (w0 * (-0.125 * (h * (Math.pow((D * (M / d)), 2.0) / l))));
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	return w0 + (w0 * (-0.125 * (h * (math.pow((D * (M / d)), 2.0) / l))))

Julia

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

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 + (w0 * (-0.125 * (h * (((D * (M / d)) ^ 2.0) / l))));
end

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 80.1%

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

3. Simplified 80.1%

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

4. Taylor expanded in M around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. associate-/l* 53.4%

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

   3. unpow2 53.4%

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

   4. unpow2 53.4%

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

   5. *-commutative 53.4%

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

   6. unpow2 53.4%

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

6. Simplified 53.4%

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

7. Taylor expanded in D around 0 52.7%

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

   1. unpow2 52.7%

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

   2. unpow2 52.7%

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

   3. associate-*r* 55.8%

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

   4. *-commutative 55.8%

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

   5. *-commutative 55.8%

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

   6. unpow2 55.8%

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

   7. associate-*l* 59.5%

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

   8. associate-*r/ 59.8%

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

   9. associate-*l* 64.6%

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

   10. times-frac 72.7%

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

9. Simplified 72.7%

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

10. Taylor expanded in w0 around 0 52.7%

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

12. Simplified 52.6%

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

13. Taylor expanded in D around 0 52.7%

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

    1. unpow2 52.7%

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

    2. unpow2 52.7%

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

    3. times-frac 53.4%

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

    4. unpow2 53.4%

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

    5. *-commutative 53.4%

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

    6. associate-/l* 55.3%

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

    7. times-frac 55.4%

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

    8. associate-/l* 55.8%

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

    9. associate-*l/ 55.3%

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

    10. associate-*r/ 57.0%

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

    11. associate-*l/ 57.0%

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

    12. unpow2 57.0%

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

    13. associate-/r/ 57.1%

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

15. Simplified 78.5%

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

16. Final simplification 78.5%

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

Alternative 4: 77.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{M}{\frac{d}{D}}\\ \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{+79}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(t_0 \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ M (/ d D))))
   (if (<= (* M D) 2e+79)
     w0
     (* w0 (+ 1.0 (* -0.125 (* (/ h l) (* t_0 t_0))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M / (d / D);
	double tmp;
	if ((M * D) <= 2e+79) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m / (d_1 / d)
    if ((m * d) <= 2d+79) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((h / l) * (t_0 * t_0))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M / (d / D);
	double tmp;
	if ((M * D) <= 2e+79) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = M / (d / D)
	tmp = 0
	if (M * D) <= 2e+79:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(M / Float64(d / D))
	tmp = 0.0
	if (Float64(M * D) <= 2e+79)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * Float64(t_0 * t_0)))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M / (d / D);
	tmp = 0.0;
	if ((M * D) <= 2e+79)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * ((h / l) * (t_0 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * D), $MachinePrecision], 2e+79], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.99999999999999993e79

   1. Initial program 81.4%

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

      1. associate-/l* 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in M around 0 76.4%

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

   if 1.99999999999999993e79 < (*.f64 M D)

   1. Initial program 65.7%

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

      1. associate-/l* 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in M around 0 42.4%

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

      1. *-commutative 42.4%

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

      2. associate-/l* 45.4%

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

      3. unpow2 45.4%

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

      4. unpow2 45.4%

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

      5. *-commutative 45.4%

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

      6. unpow2 45.4%

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

   6. Simplified 45.4%

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

   7. Taylor expanded in D around 0 42.4%

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

      1. unpow2 42.4%

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

      2. unpow2 42.4%

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

      3. associate-*r* 42.5%

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

      4. *-commutative 42.5%

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

      5. *-commutative 42.5%

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

      6. unpow2 42.5%

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

      7. associate-*l* 42.5%

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

      8. associate-*r/ 45.4%

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

      9. associate-*l* 54.3%

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

      10. times-frac 63.3%

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

   9. Simplified 63.3%

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

   10. Taylor expanded in M around 0 42.4%

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

       1. *-commutative 42.4%

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

       2. *-commutative 42.4%

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

       3. *-commutative 42.4%

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

       4. associate-*r* 48.3%

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

       5. times-frac 48.1%

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

       6. unpow2 48.1%

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

       7. *-commutative 48.1%

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

       8. unpow2 48.1%

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

       9. unpow2 48.1%

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

       10. swap-sqr 54.1%

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

       11. unpow2 54.1%

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

       12. *-commutative 54.1%

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

   12. Simplified 54.1%

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

       1. unpow2 54.1%

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

       2. *-commutative 54.1%

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

       3. *-commutative 54.1%

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

       4. frac-times 63.0%

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

       5. associate-/l* 63.0%

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

       6. associate-/l* 65.9%

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

   14. Applied egg-rr 65.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \leq 2 \cdot 10^{+79}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}\right)\right)\right)\\ \end{array} \]

Alternative 5: 78.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 3.4 \cdot 10^{-149}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \left(\frac{D}{d \cdot \ell} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.4e-149)
   w0
   (* w0 (+ 1.0 (* -0.125 (* M (* (* M h) (* (/ D (* d l)) (/ D d)))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.4e-149) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (M * ((M * h) * ((D / (d * l)) * (D / d))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 3.4d-149) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (m * ((m * h) * ((d / (d_1 * l)) * (d / d_1))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.4e-149) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (-0.125 * (M * ((M * h) * ((D / (d * l)) * (D / d))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 3.4e-149:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (-0.125 * (M * ((M * h) * ((D / (d * l)) * (D / d))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.4e-149)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(M * Float64(Float64(M * h) * Float64(Float64(D / Float64(d * l)) * Float64(D / d)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 3.4e-149)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (-0.125 * (M * ((M * h) * ((D / (d * l)) * (D / d))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.4e-149], w0, N[(w0 * N[(1.0 + N[(-0.125 * N[(M * N[(N[(M * h), $MachinePrecision] * N[(N[(D / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.3999999999999999e-149

   1. Initial program 80.8%

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

      1. associate-/l* 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in M around 0 70.2%

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

   if 3.3999999999999999e-149 < M

   1. Initial program 76.9%

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

      1. associate-/l* 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in M around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/l* 58.8%

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

      3. unpow2 58.8%

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

      4. unpow2 58.8%

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

      5. *-commutative 58.8%

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

      6. unpow2 58.8%

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

   6. Simplified 58.8%

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

   7. Taylor expanded in D around 0 56.8%

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

      1. unpow2 56.8%

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

      2. unpow2 56.8%

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

      3. associate-*r* 59.8%

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

      4. *-commutative 59.8%

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

      5. *-commutative 59.8%

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

      6. unpow2 59.8%

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

      7. associate-*l* 62.0%

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

      8. associate-*r/ 64.1%

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

      9. associate-*l* 69.3%

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

      10. times-frac 77.6%

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

   9. Simplified 77.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.4 \cdot 10^{-149}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(M \cdot \left(\left(M \cdot h\right) \cdot \left(\frac{D}{d \cdot \ell} \cdot \frac{D}{d}\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 80.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{M \cdot h}\\ \mathbf{if}\;D \leq 2 \cdot 10^{+140}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{d \cdot \left(t_0 \cdot \frac{\ell}{M}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \frac{D}{t_0 \cdot \frac{d \cdot \ell}{M}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ d (* M h))))
   (if (<= D 2e+140)
     (* w0 (+ 1.0 (* -0.125 (/ (* D D) (* d (* t_0 (/ l M)))))))
     (* w0 (+ 1.0 (* -0.125 (* D (/ D (* t_0 (/ (* d l) M))))))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = d / (M * h);
	double tmp;
	if (D <= 2e+140) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / (d * (t_0 * (l / M))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * (D / (t_0 * ((d * l) / M))))));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_1 / (m * h)
    if (d <= 2d+140) then
        tmp = w0 * (1.0d0 + ((-0.125d0) * ((d * d) / (d_1 * (t_0 * (l / m))))))
    else
        tmp = w0 * (1.0d0 + ((-0.125d0) * (d * (d / (t_0 * ((d_1 * l) / m))))))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = d / (M * h);
	double tmp;
	if (D <= 2e+140) {
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / (d * (t_0 * (l / M))))));
	} else {
		tmp = w0 * (1.0 + (-0.125 * (D * (D / (t_0 * ((d * l) / M))))));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	t_0 = d / (M * h)
	tmp = 0
	if D <= 2e+140:
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / (d * (t_0 * (l / M))))))
	else:
		tmp = w0 * (1.0 + (-0.125 * (D * (D / (t_0 * ((d * l) / M))))))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	t_0 = Float64(d / Float64(M * h))
	tmp = 0.0
	if (D <= 2e+140)
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(Float64(D * D) / Float64(d * Float64(t_0 * Float64(l / M)))))));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(-0.125 * Float64(D * Float64(D / Float64(t_0 * Float64(Float64(d * l) / M)))))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = d / (M * h);
	tmp = 0.0;
	if (D <= 2e+140)
		tmp = w0 * (1.0 + (-0.125 * ((D * D) / (d * (t_0 * (l / M))))));
	else
		tmp = w0 * (1.0 + (-0.125 * (D * (D / (t_0 * ((d * l) / M))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(d / N[(M * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 2e+140], N[(w0 * N[(1.0 + N[(-0.125 * N[(N[(D * D), $MachinePrecision] / N[(d * N[(t$95$0 * N[(l / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(-0.125 * N[(D * N[(D / N[(t$95$0 * N[(N[(d * l), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 2.00000000000000012e140

   1. Initial program 82.0%

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

      1. associate-/l* 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in M around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-/l* 55.7%

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

      3. unpow2 55.7%

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

      4. unpow2 55.7%

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

      5. *-commutative 55.7%

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

      6. unpow2 55.7%

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

   6. Simplified 55.7%

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

      1. div-inv 55.7%

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

      2. associate-*r* 59.2%

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

      3. associate-*l* 62.8%

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

   8. Applied egg-rr 62.8%

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

   9. Taylor expanded in l around 0 55.7%

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

       1. *-commutative 55.7%

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

       2. unpow2 55.7%

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

       3. associate-*r* 59.2%

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

       4. *-rgt-identity 59.2%

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

       5. *-commutative 59.2%

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

       6. unpow2 59.2%

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

       7. associate-*r* 62.8%

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

       8. *-commutative 62.8%

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

       9. associate-*r/ 62.8%

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

       10. associate-*l* 66.8%

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

       11. associate-*r/ 66.8%

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

       12. *-rgt-identity 66.8%

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

       13. *-commutative 66.8%

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

       14. times-frac 71.2%

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

   11. Simplified 71.2%

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

   if 2.00000000000000012e140 < D

   1. Initial program 56.8%

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

      1. associate-/l* 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in M around 0 34.7%

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

      1. *-commutative 34.7%

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

      2. associate-/l* 34.7%

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

      3. unpow2 34.7%

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

      4. unpow2 34.7%

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

      5. *-commutative 34.7%

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

      6. unpow2 34.7%

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

   6. Simplified 34.7%

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

      1. div-inv 34.7%

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

      2. associate-*r* 34.7%

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

      3. associate-*l* 34.7%

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

   8. Applied egg-rr 34.7%

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

      1. div-inv 34.7%

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

      2. *-un-lft-identity 34.7%

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

      3. times-frac 55.5%

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

      4. times-frac 59.4%

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

      5. *-commutative 59.4%

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

      6. *-commutative 59.4%

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

   10. Applied egg-rr 59.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2 \cdot 10^{+140}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \frac{D \cdot D}{d \cdot \left(\frac{d}{M \cdot h} \cdot \frac{\ell}{M}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + -0.125 \cdot \left(D \cdot \frac{D}{\frac{d}{M \cdot h} \cdot \frac{d \cdot \ell}{M}}\right)\right)\\ \end{array} \]

Alternative 7: 70.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4e+21)
   w0
   (* -0.125 (* (/ (* D D) l) (/ (* (* M M) (* h w0)) (* d d))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+21) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / l) * (((M * M) * (h * w0)) / (d * d)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 4d+21) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / l) * (((m * m) * (h * w0)) / (d_1 * d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+21) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / l) * (((M * M) * (h * w0)) / (d * d)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4e+21:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / l) * (((M * M) * (h * w0)) / (d * d)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4e+21)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / l) * Float64(Float64(Float64(M * M) * Float64(h * w0)) / Float64(d * d))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4e+21)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / l) * (((M * M) * (h * w0)) / (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4e+21], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4e21

   1. Initial program 80.1%

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

      1. associate-/l* 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in M around 0 73.3%

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

   if 4e21 < M

   1. Initial program 75.6%

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

      1. associate-/l* 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-/l* 48.6%

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

      3. unpow2 48.6%

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

      4. unpow2 48.6%

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

      5. *-commutative 48.6%

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

      6. unpow2 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in D around 0 48.6%

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

      1. unpow2 48.6%

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

      2. unpow2 48.6%

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

      3. associate-*r* 55.4%

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

      4. *-commutative 55.4%

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

      5. *-commutative 55.4%

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

      6. unpow2 55.4%

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

      7. associate-*l* 55.6%

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

      8. associate-*r/ 55.6%

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

      9. associate-*l* 64.7%

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

      10. times-frac 69.4%

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

   9. Simplified 69.4%

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

   10. Taylor expanded in M around inf 33.0%

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

       1. associate-/r* 33.0%

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

       2. *-commutative 33.0%

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

       3. associate-/r* 33.0%

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

       4. times-frac 32.8%

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

       5. unpow2 32.8%

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

       6. unpow2 32.8%

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

       7. *-commutative 32.8%

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

       8. associate-*r* 30.4%

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

       9. unpow2 30.4%

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

   12. Simplified 30.4%

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

   13. Taylor expanded in D around 0 33.0%

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

       1. unpow2 33.0%

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

       2. unpow2 33.0%

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

       3. times-frac 33.0%

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

       4. *-commutative 33.0%

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

       5. unpow2 33.0%

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

       6. *-commutative 33.0%

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

       7. *-commutative 33.0%

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

       8. associate-*r* 30.5%

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

       9. *-commutative 30.5%

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

   15. Simplified 30.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{\ell} \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{d \cdot d}\right)\\ \end{array} \]

Alternative 8: 70.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4e+21)
   w0
   (* -0.125 (* (/ (* D D) (* d d)) (/ (* (* M M) (* h w0)) l)))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+21) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * (((M * M) * (h * w0)) / l));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 4d+21) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) / (d_1 * d_1)) * (((m * m) * (h * w0)) / l))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+21) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) / (d * d)) * (((M * M) * (h * w0)) / l));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4e+21:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) / (d * d)) * (((M * M) * (h * w0)) / l))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4e+21)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) / Float64(d * d)) * Float64(Float64(Float64(M * M) * Float64(h * w0)) / l)));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4e+21)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) / (d * d)) * (((M * M) * (h * w0)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4e+21], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * N[(h * w0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4e21

   1. Initial program 80.1%

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

      1. associate-/l* 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in M around 0 73.3%

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

   if 4e21 < M

   1. Initial program 75.6%

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

      1. associate-/l* 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-/l* 48.6%

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

      3. unpow2 48.6%

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

      4. unpow2 48.6%

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

      5. *-commutative 48.6%

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

      6. unpow2 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in D around 0 48.6%

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

      1. unpow2 48.6%

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

      2. unpow2 48.6%

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

      3. associate-*r* 55.4%

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

      4. *-commutative 55.4%

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

      5. *-commutative 55.4%

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

      6. unpow2 55.4%

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

      7. associate-*l* 55.6%

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

      8. associate-*r/ 55.6%

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

      9. associate-*l* 64.7%

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

      10. times-frac 69.4%

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

   9. Simplified 69.4%

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

   10. Taylor expanded in M around inf 33.0%

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

       1. associate-/r* 33.0%

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

       2. *-commutative 33.0%

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

       3. associate-/r* 33.0%

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

       4. times-frac 32.8%

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

       5. unpow2 32.8%

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

       6. unpow2 32.8%

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

       7. *-commutative 32.8%

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

       8. associate-*r* 30.4%

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

       9. unpow2 30.4%

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

   12. Simplified 30.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \left(\frac{D \cdot D}{d \cdot d} \cdot \frac{\left(M \cdot M\right) \cdot \left(h \cdot w0\right)}{\ell}\right)\\ \end{array} \]

Alternative 9: 70.9% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4e+21)
   w0
   (* -0.125 (/ (* (* D D) (* w0 (* h (* M M)))) (* l (* d d))))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+21) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) * (w0 * (h * (M * M)))) / (l * (d * d)));
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 4d+21) then
        tmp = w0
    else
        tmp = (-0.125d0) * (((d * d) * (w0 * (h * (m * m)))) / (l * (d_1 * d_1)))
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+21) {
		tmp = w0;
	} else {
		tmp = -0.125 * (((D * D) * (w0 * (h * (M * M)))) / (l * (d * d)));
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4e+21:
		tmp = w0
	else:
		tmp = -0.125 * (((D * D) * (w0 * (h * (M * M)))) / (l * (d * d)))
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4e+21)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(D * D) * Float64(w0 * Float64(h * Float64(M * M)))) / Float64(l * Float64(d * d))));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4e+21)
		tmp = w0;
	else
		tmp = -0.125 * (((D * D) * (w0 * (h * (M * M)))) / (l * (d * d)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4e+21], w0, N[(-0.125 * N[(N[(N[(D * D), $MachinePrecision] * N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4e21

   1. Initial program 80.1%

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

      1. associate-/l* 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in M around 0 73.3%

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

   if 4e21 < M

   1. Initial program 75.6%

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

      1. associate-/l* 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in M around 0 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-/l* 48.6%

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

      3. unpow2 48.6%

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

      4. unpow2 48.6%

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

      5. *-commutative 48.6%

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

      6. unpow2 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in D around 0 48.6%

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

      1. unpow2 48.6%

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

      2. unpow2 48.6%

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

      3. associate-*r* 55.4%

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

      4. *-commutative 55.4%

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

      5. *-commutative 55.4%

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

      6. unpow2 55.4%

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

      7. associate-*l* 55.6%

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

      8. associate-*r/ 55.6%

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

      9. associate-*l* 64.7%

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

      10. times-frac 69.4%

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

   9. Simplified 69.4%

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

   10. Taylor expanded in M around inf 33.0%

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

       1. associate-/r* 33.0%

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

       2. *-commutative 33.0%

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

       3. associate-/r* 33.0%

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

       4. times-frac 32.8%

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

       5. unpow2 32.8%

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

       6. unpow2 32.8%

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

       7. *-commutative 32.8%

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

       8. associate-*r* 30.4%

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

       9. unpow2 30.4%

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

   12. Simplified 30.4%

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

       1. frac-times 30.5%

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

       2. associate-*l* 33.0%

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

   14. Applied egg-rr 33.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+21}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(D \cdot D\right) \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell \cdot \left(d \cdot d\right)}\\ \end{array} \]

Alternative 10: 72.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} M = |M|\\ D = |D|\\ [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 2.1e+19)
   w0
   (* -0.125 (/ (* (* (/ D d) (/ D d)) (* w0 (* h (* M M)))) l))))

C

M = abs(M);
D = abs(D);
assert(M < D);
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e+19) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((D / d) * (D / d)) * (w0 * (h * (M * M)))) / l);
	}
	return tmp;
}

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (m <= 2.1d+19) then
        tmp = w0
    else
        tmp = (-0.125d0) * ((((d / d_1) * (d / d_1)) * (w0 * (h * (m * m)))) / l)
    end if
    code = tmp
end function

Java

M = Math.abs(M);
D = Math.abs(D);
assert M < D;
public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 2.1e+19) {
		tmp = w0;
	} else {
		tmp = -0.125 * ((((D / d) * (D / d)) * (w0 * (h * (M * M)))) / l);
	}
	return tmp;
}

Python

M = abs(M)
D = abs(D)
[M, D] = sort([M, D])
def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 2.1e+19:
		tmp = w0
	else:
		tmp = -0.125 * ((((D / d) * (D / d)) * (w0 * (h * (M * M)))) / l)
	return tmp

Julia

M = abs(M)
D = abs(D)
M, D = sort([M, D])
function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 2.1e+19)
		tmp = w0;
	else
		tmp = Float64(-0.125 * Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(w0 * Float64(h * Float64(M * M)))) / l));
	end
	return tmp
end

MATLAB

M = abs(M)
D = abs(D)
M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 2.1e+19)
		tmp = w0;
	else
		tmp = -0.125 * ((((D / d) * (D / d)) * (w0 * (h * (M * M)))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 2.1e+19], w0, N[(-0.125 * N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(w0 * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.1e19

   1. Initial program 79.9%

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

      1. associate-/l* 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in M around 0 73.5%

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

   if 2.1e19 < M

   1. Initial program 76.7%

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

      1. associate-/l* 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in M around 0 46.5%

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

      1. *-commutative 46.5%

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

      2. associate-/l* 46.5%

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

      3. unpow2 46.5%

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

      4. unpow2 46.5%

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

      5. *-commutative 46.5%

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

      6. unpow2 46.5%

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

   6. Simplified 46.5%

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

   7. Taylor expanded in D around 0 46.5%

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

      1. unpow2 46.5%

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

      2. unpow2 46.5%

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

      3. associate-*r* 53.0%

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

      4. *-commutative 53.0%

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

      5. *-commutative 53.0%

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

      6. unpow2 53.0%

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

      7. associate-*l* 53.3%

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

      8. associate-*r/ 53.3%

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

      9. associate-*l* 64.2%

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

      10. times-frac 68.8%

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

   9. Simplified 68.8%

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

   10. Taylor expanded in M around inf 31.5%

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

       1. associate-/r* 31.6%

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

       2. *-commutative 31.6%

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

       3. associate-/r* 31.5%

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

       4. times-frac 31.3%

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

       5. unpow2 31.3%

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

       6. unpow2 31.3%

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

       7. *-commutative 31.3%

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

       8. associate-*r* 29.2%

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

       9. unpow2 29.2%

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

   12. Simplified 29.2%

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

       1. associate-*r/ 29.3%

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

       2. times-frac 32.0%

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

       3. associate-*l* 34.1%

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

   14. Applied egg-rr 34.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(w0 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\ell}\\ \end{array} \]

Alternative 11: 67.7% accurate, 216.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: M should be positive before calling this function
NOTE: D should be positive before calling this function
NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-/l* 80.1%

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

3. Simplified 80.1%

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

4. Taylor expanded in M around 0 69.0%

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

5. Final simplification 69.0%

   \[\leadsto w0 \]

Reproduce

herbie shell --seed 2023187 
(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))))))