Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.5% → 89.3%

Time: 17.5s

Alternatives: 17

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M \cdot D}{d}\\ w0 \cdot \sqrt{1 - \frac{t\_0 \cdot h}{\ell \cdot \frac{4}{t\_0}}} \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) d)))
   (* w0 (sqrt (- 1.0 (/ (* t_0 h) (* l (/ 4.0 t_0))))))))

C

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 - ((t_0 * h) / (l * (4.0 / t_0)))));
}

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
    real(8) :: t_0
    t_0 = (m * d) / d_1
    code = w0 * sqrt((1.0d0 - ((t_0 * h) / (l * (4.0d0 / t_0)))))
end function

Java

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 - ((t_0 * h) / (l * (4.0 / t_0)))));
}

Python

def code(w0, M, D, h, l, d):
	t_0 = (M * D) / d
	return w0 * math.sqrt((1.0 - ((t_0 * h) / (l * (4.0 / t_0)))))

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(Float64(M * D) / d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 * h) / Float64(l * Float64(4.0 / t_0))))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	t_0 = (M * D) / d;
	tmp = w0 * sqrt((1.0 - ((t_0 * h) / (l * (4.0 / t_0)))));
end

Wolfram

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[(N[(t$95$0 * h), $MachinePrecision] / N[(l * N[(4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.3%

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

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

   5. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d \cdot 2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{d}}{2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   11. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   12. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   13. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 88.4%

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

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\frac{\frac{M}{\frac{d}{D}}}{2}}{\frac{1}{h}}\right)\right)\right)\right) \]

   2. associate-/l/ N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\frac{M}{\frac{d}{D}}}{\frac{1}{h} \cdot 2}\right)\right)\right)\right) \]

   3. frac-times N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1 \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1 \cdot \frac{1}{\frac{\frac{d}{D}}{M}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

   5. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{\frac{\frac{d}{D}}{M}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}\right), \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 88.7%

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{0.5} \cdot \frac{2}{h}}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), \left(\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}\right)\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), \left(\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}\right)\right)\right)\right)\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot 2}{h}\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot 2\right), h\right)\right)\right)\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\left(\left(\frac{\ell}{\frac{M}{\frac{d}{D}}} \cdot \frac{1}{\frac{1}{2}}\right) \cdot 2\right), h\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\left(\left(\frac{\ell}{\frac{M}{\frac{d}{D}}} \cdot 2\right) \cdot 2\right), h\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{M}{\frac{d}{D}}} \cdot \left(2 \cdot 2\right)\right), h\right)\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{M}{\frac{d}{D}}} \cdot 4\right), h\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{M}{\frac{d}{D}}} \cdot \left(\mathsf{neg}\left(-4\right)\right)\right), h\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{\frac{M}{\frac{d}{D}}}\right), \left(\mathsf{neg}\left(-4\right)\right)\right), h\right)\right)\right)\right)\right) \]

   12. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{1}{\frac{M}{\frac{d}{D}}}\right), \left(\mathsf{neg}\left(-4\right)\right)\right), h\right)\right)\right)\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{\frac{d}{D}}{M}\right), \left(\mathsf{neg}\left(-4\right)\right)\right), h\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\frac{d}{D}}{M}\right)\right), \left(\mathsf{neg}\left(-4\right)\right)\right), h\right)\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{d}{D}\right), M\right)\right), \left(\mathsf{neg}\left(-4\right)\right)\right), h\right)\right)\right)\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), M\right)\right), \left(\mathsf{neg}\left(-4\right)\right)\right), h\right)\right)\right)\right)\right) \]

   17. metadata-eval 88.8%

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(d, D\right), M\right)\right), 4\right), h\right)\right)\right)\right)\right) \]

8. Applied egg-rr 88.8%

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

   1. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M}{\frac{d}{D}}}{\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4} \cdot h\right)\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M}{\frac{d}{D}} \cdot h}{\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4}\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}} \cdot h\right), \left(\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), h\right), \left(\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4\right)\right)\right)\right)\right) \]

   5. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{M}{d} \cdot D\right), h\right), \left(\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4\right)\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{M \cdot D}{d}\right), h\right), \left(\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4\right)\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot D\right), d\right), h\right), \left(\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \left(\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \left(\ell \cdot \left(\frac{\frac{d}{D}}{M} \cdot 4\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \left(\frac{\frac{d}{D}}{M} \cdot 4\right)\right)\right)\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \left(\frac{1}{\frac{M}{\frac{d}{D}}} \cdot 4\right)\right)\right)\right)\right)\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \left(\frac{1 \cdot 4}{\frac{M}{\frac{d}{D}}}\right)\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \left(\frac{4}{\frac{M}{\frac{d}{D}}}\right)\right)\right)\right)\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(4, \left(\frac{M}{\frac{d}{D}}\right)\right)\right)\right)\right)\right)\right) \]

   15. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(4, \left(\frac{M}{d} \cdot D\right)\right)\right)\right)\right)\right)\right) \]

   16. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(4, \left(\frac{M \cdot D}{d}\right)\right)\right)\right)\right)\right)\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(4, \mathsf{/.f64}\left(\left(M \cdot D\right), d\right)\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 88.1%

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right), h\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, D\right), d\right)\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 88.1%

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{d} \cdot h}{\ell \cdot \frac{4}{\frac{M \cdot D}{d}}}}} \]
11. Add Preprocessing

Alternative 2: 83.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-64}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0
         (*
          w0
          (sqrt
           (+ 1.0 (/ (* (* M h) (/ (/ M (/ d D)) -4.0)) (* l (/ d D))))))))
   (if (<= l -1.2e-94)
     t_0
     (if (<= l 5e-64)
       (*
        w0
        (sqrt (+ 1.0 (/ (* h (/ (/ (* M (* M D)) (* d -4.0)) (/ d D))) l))))
       t_0))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = w0 * sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	double tmp;
	if (l <= -1.2e-94) {
		tmp = t_0;
	} else if (l <= 5e-64) {
		tmp = w0 * sqrt((1.0 + ((h * (((M * (M * D)) / (d * -4.0)) / (d / D))) / l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w0 * sqrt((1.0d0 + (((m * h) * ((m / (d_1 / d)) / (-4.0d0))) / (l * (d_1 / d)))))
    if (l <= (-1.2d-94)) then
        tmp = t_0
    else if (l <= 5d-64) then
        tmp = w0 * sqrt((1.0d0 + ((h * (((m * (m * d)) / (d_1 * (-4.0d0))) / (d_1 / d))) / l)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = w0 * Math.sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	double tmp;
	if (l <= -1.2e-94) {
		tmp = t_0;
	} else if (l <= 5e-64) {
		tmp = w0 * Math.sqrt((1.0 + ((h * (((M * (M * D)) / (d * -4.0)) / (d / D))) / l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = w0 * math.sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))))
	tmp = 0
	if l <= -1.2e-94:
		tmp = t_0
	elif l <= 5e-64:
		tmp = w0 * math.sqrt((1.0 + ((h * (((M * (M * D)) / (d * -4.0)) / (d / D))) / l)))
	else:
		tmp = t_0
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M * h) * Float64(Float64(M / Float64(d / D)) / -4.0)) / Float64(l * Float64(d / D))))))
	tmp = 0.0
	if (l <= -1.2e-94)
		tmp = t_0;
	elseif (l <= 5e-64)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * Float64(Float64(Float64(M * Float64(M * D)) / Float64(d * -4.0)) / Float64(d / D))) / l))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = w0 * sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	tmp = 0.0;
	if (l <= -1.2e-94)
		tmp = t_0;
	elseif (l <= 5e-64)
		tmp = w0 * sqrt((1.0 + ((h * (((M * (M * D)) / (d * -4.0)) / (d / D))) / l)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M * h), $MachinePrecision] * N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.2e-94], t$95$0, If[LessEqual[l, 5e-64], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[(N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * -4.0), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.2e-94 or 5.00000000000000033e-64 < l

   1. Initial program 85.9%

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

   3. Simplified 81.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right), \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right), \ell\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{1}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{M \cdot \left(M \cdot D\right)}{-4 \cdot d}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(D \cdot M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(d \cdot -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      17. /-lowering-/.f64 82.2%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \mathsf{/.f64}\left(d, D\right)\right)\right), \ell\right)\right)\right)\right) \]

   6. Applied egg-rr 82.2%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\ell \cdot \frac{d}{D}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \left(M \cdot \frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(h \cdot M\right) \cdot \frac{D \cdot M}{d \cdot -4}\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot M\right), \left(\frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{D \cdot M}{d}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M \cdot D}{d}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M}{d} \cdot D}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M}{\frac{d}{D}}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \mathsf{*.f64}\left(\ell, \left(\frac{d}{D}\right)\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 87.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(d, D\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 87.6%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\left(h \cdot M\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}} \]

   if -1.2e-94 < l < 5.00000000000000033e-64

   1. Initial program 70.2%

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

   3. Simplified 65.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right), \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right), \ell\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{1}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{M \cdot \left(M \cdot D\right)}{-4 \cdot d}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(D \cdot M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(d \cdot -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      17. /-lowering-/.f64 76.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \mathsf{/.f64}\left(d, D\right)\right)\right), \ell\right)\right)\right)\right) \]

   6. Applied egg-rr 76.0%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{h \cdot \frac{\frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-94}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-64}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 3.6 \cdot 10^{-159}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 4.8 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{D \cdot \frac{h \cdot \left(-0.25 \cdot \left(M \cdot M\right)\right)}{d}}{\frac{\ell}{\frac{D}{d}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 3.6e-159)
   w0
   (if (<= M 4.8e+76)
     (fma D (* w0 (/ D (/ (/ (* d (* d l)) h) (* M (* M -0.125))))) w0)
     (* w0 (sqrt (/ (* D (/ (* h (* -0.25 (* M M))) d)) (/ l (/ D d))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 3.6e-159) {
		tmp = w0;
	} else if (M <= 4.8e+76) {
		tmp = fma(D, (w0 * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))), w0);
	} else {
		tmp = w0 * sqrt(((D * ((h * (-0.25 * (M * M))) / d)) / (l / (D / d))));
	}
	return tmp;
}

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 3.6e-159)
		tmp = w0;
	elseif (M <= 4.8e+76)
		tmp = fma(D, Float64(w0 * Float64(D / Float64(Float64(Float64(d * Float64(d * l)) / h) / Float64(M * Float64(M * -0.125))))), w0);
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(D * Float64(Float64(h * Float64(-0.25 * Float64(M * M))) / d)) / Float64(l / Float64(D / d)))));
	end
	return tmp
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 3.6e-159], w0, If[LessEqual[M, 4.8e+76], N[(D * N[(w0 * N[(D / N[(N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(M * N[(M * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + w0), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D * N[(N[(h * N[(-0.25 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / N[(l / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 3.60000000000000021e-159

   1. Initial program 84.8%

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

   3. Simplified 79.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 70.6%

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

   if 3.60000000000000021e-159 < M < 4.8e76

   1. Initial program 76.7%

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

   3. Simplified 76.5%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 60.7%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right)\right) \]

   7. Simplified 60.7%

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

      1. +-commutative N/A

         \[\leadsto w0 \cdot \left(\left(D \cdot D\right) \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell} + \color{blue}{1}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot w0 + \color{blue}{1 \cdot w0} \]

      3. associate-*l* N/A

         \[\leadsto \left(D \cdot \left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot w0 + 1 \cdot w0 \]

      4. associate-*l* N/A

         \[\leadsto D \cdot \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot w0\right) + \color{blue}{1} \cdot w0 \]

      5. *-lft-identity N/A

         \[\leadsto D \cdot \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot w0\right) + w0 \]

      6. fma-define N/A

         \[\leadsto \mathsf{fma}\left(D, \color{blue}{\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot w0}, w0\right) \]

      7. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(D, \color{blue}{\left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot w0\right)}, w0\right) \]

   9. Applied egg-rr 69.5%

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

   if 4.8e76 < M

   1. Initial program 70.7%

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

   3. Simplified 62.0%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{4} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4} \cdot \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-1}{4} \cdot {M}^{2}\right) \cdot h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4} \cdot {M}^{2}\right), h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({M}^{2}\right)\right), h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(M \cdot M\right)\right), h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left(D \cdot D\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \ell\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right) \]

   7. Simplified 31.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\left(\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\left(\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D}{d} \cdot \frac{D}{d \cdot \ell}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D}{d}\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(h \cdot \left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(h \cdot \left(\left(\frac{-1}{4} \cdot M\right) \cdot M\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(h \cdot \left(\frac{-1}{4} \cdot M\right)\right) \cdot M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(h \cdot \left(\frac{-1}{4} \cdot M\right)\right), M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{-1}{4} \cdot M\right)\right), M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \frac{-1}{4}\right)\right), M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \frac{-1}{4}\right)\right), M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \frac{-1}{4}\right)\right), M\right)\right), d\right), \mathsf{/.f64}\left(D, \left(d \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 34.8%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \frac{-1}{4}\right)\right), M\right)\right), d\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 34.8%

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

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{D \cdot \left(\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M\right)}{d} \cdot \frac{1}{\frac{d \cdot \ell}{D}}\right)\right)\right) \]

       2. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{D \cdot \left(\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M\right)}{d}}{\frac{d \cdot \ell}{D}}\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{D \cdot \left(\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M\right)}{d}\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \frac{\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M}{d}\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\frac{\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M}{d}\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(h \cdot \left(\left(M \cdot \frac{-1}{4}\right) \cdot M\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(M \cdot \frac{-1}{4}\right) \cdot M\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{-1}{4} \cdot M\right) \cdot M\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \left(M \cdot M\right)\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\frac{\ell \cdot d}{D}\right)\right)\right)\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right) \]

       15. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\ell \cdot \frac{1}{\frac{D}{d}}\right)\right)\right)\right) \]

       16. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\frac{\ell}{\frac{D}{d}}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \mathsf{/.f64}\left(\ell, \left(\frac{D}{d}\right)\right)\right)\right)\right) \]

       18. /-lowering-/.f64 36.6%

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 36.6%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.6 \cdot 10^{-159}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 4.8 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(D, w0 \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}, w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{D \cdot \frac{h \cdot \left(-0.25 \cdot \left(M \cdot M\right)\right)}{d}}{\frac{\ell}{\frac{D}{d}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.6 \cdot 10^{-155}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 6 \cdot 10^{+79}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{D \cdot \frac{h \cdot \left(-0.25 \cdot \left(M \cdot M\right)\right)}{d}}{\frac{\ell}{\frac{D}{d}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.6e-155)
   w0
   (if (<= M 6e+79)
     (* w0 (+ 1.0 (* D (/ D (/ (/ (* d (* d l)) h) (* M (* M -0.125)))))))
     (* w0 (sqrt (/ (* D (/ (* h (* -0.25 (* M M))) d)) (/ l (/ D d))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.6e-155) {
		tmp = w0;
	} else if (M <= 6e+79) {
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	} else {
		tmp = w0 * sqrt(((D * ((h * (-0.25 * (M * M))) / d)) / (l / (D / d))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 1.6d-155) then
        tmp = w0
    else if (m <= 6d+79) then
        tmp = w0 * (1.0d0 + (d * (d / (((d_1 * (d_1 * l)) / h) / (m * (m * (-0.125d0)))))))
    else
        tmp = w0 * sqrt(((d * ((h * ((-0.25d0) * (m * m))) / d_1)) / (l / (d / d_1))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.6e-155) {
		tmp = w0;
	} else if (M <= 6e+79) {
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	} else {
		tmp = w0 * Math.sqrt(((D * ((h * (-0.25 * (M * M))) / d)) / (l / (D / d))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.6e-155:
		tmp = w0
	elif M <= 6e+79:
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))))
	else:
		tmp = w0 * math.sqrt(((D * ((h * (-0.25 * (M * M))) / d)) / (l / (D / d))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.6e-155)
		tmp = w0;
	elseif (M <= 6e+79)
		tmp = Float64(w0 * Float64(1.0 + Float64(D * Float64(D / Float64(Float64(Float64(d * Float64(d * l)) / h) / Float64(M * Float64(M * -0.125)))))));
	else
		tmp = Float64(w0 * sqrt(Float64(Float64(D * Float64(Float64(h * Float64(-0.25 * Float64(M * M))) / d)) / Float64(l / Float64(D / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.6e-155)
		tmp = w0;
	elseif (M <= 6e+79)
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	else
		tmp = w0 * sqrt(((D * ((h * (-0.25 * (M * M))) / d)) / (l / (D / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.6e-155], w0, If[LessEqual[M, 6e+79], N[(w0 * N[(1.0 + N[(D * N[(D / N[(N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(M * N[(M * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(N[(D * N[(N[(h * N[(-0.25 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / N[(l / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.60000000000000006e-155

   1. Initial program 84.8%

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

   3. Simplified 79.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 70.6%

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

   if 1.60000000000000006e-155 < M < 5.99999999999999948e79

   1. Initial program 77.4%

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

   3. Simplified 77.2%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 62.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right)\right) \]

   7. Simplified 62.0%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(D \cdot \color{blue}{\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \color{blue}{D}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right), \color{blue}{D}\right)\right)\right) \]

   9. Applied egg-rr 70.5%

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

   if 5.99999999999999948e79 < M

   1. Initial program 69.3%

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

   3. Simplified 60.3%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{4} \cdot \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-1}{4} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4} \cdot \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{-1}{4} \cdot {M}^{2}\right) \cdot h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4} \cdot {M}^{2}\right), h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({M}^{2}\right)\right), h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(M \cdot M\right)\right), h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left(D \cdot D\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \ell\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right) \]

      15. *-lowering-*.f64 30.5%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right) \]

   7. Simplified 30.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\left(\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\left(\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D\right) \cdot D}{d \cdot \left(d \cdot \ell\right)}\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D}{d} \cdot \frac{D}{d \cdot \ell}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D}{d}\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right) \cdot D\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right) \cdot h\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(h \cdot \left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(h \cdot \left(\left(\frac{-1}{4} \cdot M\right) \cdot M\right)\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\left(h \cdot \left(\frac{-1}{4} \cdot M\right)\right) \cdot M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(h \cdot \left(\frac{-1}{4} \cdot M\right)\right), M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{-1}{4} \cdot M\right)\right), M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \left(M \cdot \frac{-1}{4}\right)\right), M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \frac{-1}{4}\right)\right), M\right)\right), d\right), \left(\frac{D}{d \cdot \ell}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \frac{-1}{4}\right)\right), M\right)\right), d\right), \mathsf{/.f64}\left(D, \left(d \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 34.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(M, \frac{-1}{4}\right)\right), M\right)\right), d\right), \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \ell\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 34.0%

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

       1. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{D \cdot \left(\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M\right)}{d} \cdot \frac{1}{\frac{d \cdot \ell}{D}}\right)\right)\right) \]

       2. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\left(\frac{\frac{D \cdot \left(\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M\right)}{d}}{\frac{d \cdot \ell}{D}}\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{D \cdot \left(\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M\right)}{d}\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \frac{\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M}{d}\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \left(\frac{\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M}{d}\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(h \cdot \left(M \cdot \frac{-1}{4}\right)\right) \cdot M\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(h \cdot \left(\left(M \cdot \frac{-1}{4}\right) \cdot M\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(M \cdot \frac{-1}{4}\right) \cdot M\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\left(\frac{-1}{4} \cdot M\right) \cdot M\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{-1}{4} \cdot \left(M \cdot M\right)\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \left(M \cdot M\right)\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\frac{d \cdot \ell}{D}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\frac{\ell \cdot d}{D}\right)\right)\right)\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right) \]

       15. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\ell \cdot \frac{1}{\frac{D}{d}}\right)\right)\right)\right) \]

       16. un-div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \left(\frac{\ell}{\frac{D}{d}}\right)\right)\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \mathsf{/.f64}\left(\ell, \left(\frac{D}{d}\right)\right)\right)\right)\right) \]

       18. /-lowering-/.f64 35.9%

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(M, M\right)\right)\right), d\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(D, d\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 35.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.6 \cdot 10^{-155}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 6 \cdot 10^{+79}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{\frac{D \cdot \frac{h \cdot \left(-0.25 \cdot \left(M \cdot M\right)\right)}{d}}{\frac{\ell}{\frac{D}{d}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{M}{\frac{d}{D}}\\ \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{\frac{t\_0 \cdot t\_0}{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{d} \cdot \frac{D}{\frac{4 \cdot \left(\ell \cdot \frac{\frac{d}{D}}{M}\right)}{h}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ M (/ d D))))
   (if (<= (/ h l) -5e+84)
     (* w0 (sqrt (- 1.0 (* h (/ (/ (* t_0 t_0) 4.0) l)))))
     (*
      w0
      (sqrt (- 1.0 (* (/ M d) (/ D (/ (* 4.0 (* l (/ (/ d D) M))) h)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M / (d / D);
	double tmp;
	if ((h / l) <= -5e+84) {
		tmp = w0 * sqrt((1.0 - (h * (((t_0 * t_0) / 4.0) / l))));
	} else {
		tmp = w0 * sqrt((1.0 - ((M / d) * (D / ((4.0 * (l * ((d / D) / M))) / h)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m / (d_1 / d)
    if ((h / l) <= (-5d+84)) then
        tmp = w0 * sqrt((1.0d0 - (h * (((t_0 * t_0) / 4.0d0) / l))))
    else
        tmp = w0 * sqrt((1.0d0 - ((m / d_1) * (d / ((4.0d0 * (l * ((d_1 / d) / m))) / h)))))
    end if
    code = tmp
end function

Java

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 ((h / l) <= -5e+84) {
		tmp = w0 * Math.sqrt((1.0 - (h * (((t_0 * t_0) / 4.0) / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 - ((M / d) * (D / ((4.0 * (l * ((d / D) / M))) / h)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	t_0 = M / (d / D)
	tmp = 0
	if (h / l) <= -5e+84:
		tmp = w0 * math.sqrt((1.0 - (h * (((t_0 * t_0) / 4.0) / l))))
	else:
		tmp = w0 * math.sqrt((1.0 - ((M / d) * (D / ((4.0 * (l * ((d / D) / M))) / h)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	t_0 = Float64(M / Float64(d / D))
	tmp = 0.0
	if (Float64(h / l) <= -5e+84)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h * Float64(Float64(Float64(t_0 * t_0) / 4.0) / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M / d) * Float64(D / Float64(Float64(4.0 * Float64(l * Float64(Float64(d / D) / M))) / h))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M / (d / D);
	tmp = 0.0;
	if ((h / l) <= -5e+84)
		tmp = w0 * sqrt((1.0 - (h * (((t_0 * t_0) / 4.0) / l))));
	else
		tmp = w0 * sqrt((1.0 - ((M / d) * (D / ((4.0 * (l * ((d / D) / M))) / h)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(h / l), $MachinePrecision], -5e+84], N[(w0 * N[Sqrt[N[(1.0 - N[(h * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M / d), $MachinePrecision] * N[(D / N[(N[(4.0 * N[(l * N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -5.0000000000000001e84

   1. Initial program 65.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d \cdot 2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{d}}{2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 73.4%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{\frac{\frac{M}{\frac{d}{D}}}{2}}{\ell} \cdot \frac{\frac{M}{\frac{d}{D}}}{2}}{\frac{1}{h}}\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{\frac{\frac{M}{\frac{d}{D}}}{2}}{\ell} \cdot \frac{\frac{M}{\frac{d}{D}}}{2}\right) \cdot \frac{1}{\frac{1}{h}}\right)\right)\right)\right) \]

      3. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\left(\frac{\frac{\frac{M}{\frac{d}{D}}}{2}}{\ell} \cdot \frac{\frac{M}{\frac{d}{D}}}{2}\right) \cdot h\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{\frac{M}{\frac{d}{D}}}{2}}{\ell} \cdot \frac{\frac{M}{\frac{d}{D}}}{2}\right), h\right)\right)\right)\right) \]

   6. Applied egg-rr 72.1%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{\frac{M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}}{4}}{\ell} \cdot h}} \]

   if -5.0000000000000001e84 < (/.f64 h l)

   1. Initial program 85.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d \cdot 2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{d}}{2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 93.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\frac{\frac{M}{\frac{d}{D}}}{2}}{\frac{1}{h}}\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\frac{M}{\frac{d}{D}}}{\frac{1}{h} \cdot 2}\right)\right)\right)\right) \]

      3. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1 \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1 \cdot \frac{1}{\frac{\frac{d}{D}}{M}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{\frac{\frac{d}{D}}{M}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}\right), \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 93.9%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{0.5} \cdot \frac{2}{h}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M}{d} \cdot D}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{M}{d} \cdot \frac{D}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M}{d}\right), \left(\frac{D}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \left(\frac{D}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(D, \left(\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(D, \left(\frac{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot 2}{h}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot 2\right), h\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 92.5%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{M}{d} \cdot \frac{D}{\frac{\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4}{h}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+84}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \frac{\frac{\frac{M}{\frac{d}{D}} \cdot \frac{M}{\frac{d}{D}}}{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{d} \cdot \frac{D}{\frac{4 \cdot \left(\ell \cdot \frac{\frac{d}{D}}{M}\right)}{h}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 3 \cdot 10^{+23}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{d} \cdot \frac{D}{\frac{4 \cdot \left(\ell \cdot \frac{\frac{d}{D}}{M}\right)}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 3e+23)
   (* w0 (sqrt (- 1.0 (* (/ M d) (/ D (/ (* 4.0 (* l (/ (/ d D) M))) h))))))
   (* w0 (sqrt (+ 1.0 (/ (* (* M h) (/ (/ M (/ d D)) -4.0)) (* l (/ d D))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 3e+23) {
		tmp = w0 * sqrt((1.0 - ((M / d) * (D / ((4.0 * (l * ((d / D) / M))) / h)))));
	} else {
		tmp = w0 * sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d_1 <= 3d+23) then
        tmp = w0 * sqrt((1.0d0 - ((m / d_1) * (d / ((4.0d0 * (l * ((d_1 / d) / m))) / h)))))
    else
        tmp = w0 * sqrt((1.0d0 + (((m * h) * ((m / (d_1 / d)) / (-4.0d0))) / (l * (d_1 / d)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 3e+23) {
		tmp = w0 * Math.sqrt((1.0 - ((M / d) * (D / ((4.0 * (l * ((d / D) / M))) / h)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 3e+23:
		tmp = w0 * math.sqrt((1.0 - ((M / d) * (D / ((4.0 * (l * ((d / D) / M))) / h)))))
	else:
		tmp = w0 * math.sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 3e+23)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(M / d) * Float64(D / Float64(Float64(4.0 * Float64(l * Float64(Float64(d / D) / M))) / h))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M * h) * Float64(Float64(M / Float64(d / D)) / -4.0)) / Float64(l * Float64(d / D))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 3e+23)
		tmp = w0 * sqrt((1.0 - ((M / d) * (D / ((4.0 * (l * ((d / D) / M))) / h)))));
	else
		tmp = w0 * sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 3e+23], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(M / d), $MachinePrecision] * N[(D / N[(N[(4.0 * N[(l * N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M * h), $MachinePrecision] * N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 3.0000000000000001e23

   1. Initial program 80.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d \cdot 2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{d}}{2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 87.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\frac{\frac{M}{\frac{d}{D}}}{2}}{\frac{1}{h}}\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\frac{M}{\frac{d}{D}}}{\frac{1}{h} \cdot 2}\right)\right)\right)\right) \]

      3. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1 \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1 \cdot \frac{1}{\frac{\frac{d}{D}}{M}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{\frac{\frac{d}{D}}{M}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}} \cdot \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{2}}\right), \left(\frac{1}{h} \cdot 2\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 88.2%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{0.5} \cdot \frac{2}{h}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M}{d} \cdot D}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}}\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{M}{d} \cdot \frac{D}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{M}{d}\right), \left(\frac{D}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \left(\frac{D}{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}}\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(D, \left(\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot \frac{2}{h}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(D, \left(\frac{\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot 2}{h}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(M, d\right), \mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{M}{\frac{d}{D}}}}{\frac{1}{2}} \cdot 2\right), h\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 85.6%

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{M}{d} \cdot \frac{D}{\frac{\left(\ell \cdot \frac{\frac{d}{D}}{M}\right) \cdot 4}{h}}}} \]

   if 3.0000000000000001e23 < d

   1. Initial program 80.7%

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

   3. Simplified 75.1%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right), \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right), \ell\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{1}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{M \cdot \left(M \cdot D\right)}{-4 \cdot d}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(D \cdot M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(d \cdot -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      17. /-lowering-/.f64 82.2%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \mathsf{/.f64}\left(d, D\right)\right)\right), \ell\right)\right)\right)\right) \]

   6. Applied egg-rr 82.2%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\ell \cdot \frac{d}{D}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \left(M \cdot \frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(h \cdot M\right) \cdot \frac{D \cdot M}{d \cdot -4}\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot M\right), \left(\frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{D \cdot M}{d}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M \cdot D}{d}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M}{d} \cdot D}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M}{\frac{d}{D}}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \mathsf{*.f64}\left(\ell, \left(\frac{d}{D}\right)\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 83.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(d, D\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 83.4%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\left(h \cdot M\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 3 \cdot 10^{+23}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{d} \cdot \frac{D}{\frac{4 \cdot \left(\ell \cdot \frac{\frac{d}{D}}{M}\right)}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 9.4 \cdot 10^{-33}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\frac{D}{\frac{\ell}{h}}}{-4} \cdot \frac{M}{\frac{d}{M \cdot D}}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 9.4e-33)
   (* w0 (sqrt (+ 1.0 (/ (* (/ (/ D (/ l h)) -4.0) (/ M (/ d (* M D)))) d))))
   (* w0 (sqrt (+ 1.0 (/ (* (* M h) (/ (/ M (/ d D)) -4.0)) (* l (/ d D))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 9.4e-33) {
		tmp = w0 * sqrt((1.0 + ((((D / (l / h)) / -4.0) * (M / (d / (M * D)))) / d)));
	} else {
		tmp = w0 * sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d_1 <= 9.4d-33) then
        tmp = w0 * sqrt((1.0d0 + ((((d / (l / h)) / (-4.0d0)) * (m / (d_1 / (m * d)))) / d_1)))
    else
        tmp = w0 * sqrt((1.0d0 + (((m * h) * ((m / (d_1 / d)) / (-4.0d0))) / (l * (d_1 / d)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 9.4e-33) {
		tmp = w0 * Math.sqrt((1.0 + ((((D / (l / h)) / -4.0) * (M / (d / (M * D)))) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 9.4e-33:
		tmp = w0 * math.sqrt((1.0 + ((((D / (l / h)) / -4.0) * (M / (d / (M * D)))) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 9.4e-33)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(D / Float64(l / h)) / -4.0) * Float64(M / Float64(d / Float64(M * D)))) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M * h) * Float64(Float64(M / Float64(d / D)) / -4.0)) / Float64(l * Float64(d / D))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 9.4e-33)
		tmp = w0 * sqrt((1.0 + ((((D / (l / h)) / -4.0) * (M / (d / (M * D)))) / d)));
	else
		tmp = w0 * sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 9.4e-33], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(D / N[(l / h), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision] * N[(M / N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M * h), $MachinePrecision] * N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.4000000000000004e-33

   1. Initial program 79.9%

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

   3. Simplified 76.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right), d\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\frac{h}{\ell} \cdot D\right) \cdot \frac{M \cdot \left(M \cdot D\right)}{-4 \cdot d}\right), d\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\left(\frac{h}{\ell} \cdot D\right) \cdot \left(M \cdot \left(M \cdot D\right)\right)}{-4 \cdot d}\right), d\right)\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{h}{\ell} \cdot D}{-4} \cdot \frac{M \cdot \left(M \cdot D\right)}{d}\right), d\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{h}{\ell} \cdot D}{-4}\right), \left(\frac{M \cdot \left(M \cdot D\right)}{d}\right)\right), d\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{h}{\ell} \cdot D\right), -4\right), \left(\frac{M \cdot \left(M \cdot D\right)}{d}\right)\right), d\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \frac{h}{\ell}\right), -4\right), \left(\frac{M \cdot \left(M \cdot D\right)}{d}\right)\right), d\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(D \cdot \frac{1}{\frac{\ell}{h}}\right), -4\right), \left(\frac{M \cdot \left(M \cdot D\right)}{d}\right)\right), d\right)\right)\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{D}{\frac{\ell}{h}}\right), -4\right), \left(\frac{M \cdot \left(M \cdot D\right)}{d}\right)\right), d\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \left(\frac{\ell}{h}\right)\right), -4\right), \left(\frac{M \cdot \left(M \cdot D\right)}{d}\right)\right), d\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\ell, h\right)\right), -4\right), \left(\frac{M \cdot \left(M \cdot D\right)}{d}\right)\right), d\right)\right)\right)\right) \]

      12. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\ell, h\right)\right), -4\right), \left(M \cdot \frac{M \cdot D}{d}\right)\right), d\right)\right)\right)\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\ell, h\right)\right), -4\right), \left(M \cdot \frac{1}{\frac{d}{M \cdot D}}\right)\right), d\right)\right)\right)\right) \]

      14. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\ell, h\right)\right), -4\right), \left(\frac{M}{\frac{d}{M \cdot D}}\right)\right), d\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\ell, h\right)\right), -4\right), \mathsf{/.f64}\left(M, \left(\frac{d}{M \cdot D}\right)\right)\right), d\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\ell, h\right)\right), -4\right), \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, \left(M \cdot D\right)\right)\right)\right), d\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\ell, h\right)\right), -4\right), \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, \left(D \cdot M\right)\right)\right)\right), d\right)\right)\right)\right) \]

      18. *-lowering-*.f64 79.5%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(D, \mathsf{/.f64}\left(\ell, h\right)\right), -4\right), \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, \mathsf{*.f64}\left(D, M\right)\right)\right)\right), d\right)\right)\right)\right) \]

   6. Applied egg-rr 79.5%

      \[\leadsto w0 \cdot \sqrt{1 + \frac{\color{blue}{\frac{\frac{D}{\frac{\ell}{h}}}{-4} \cdot \frac{M}{\frac{d}{D \cdot M}}}}{d}} \]

   if 9.4000000000000004e-33 < d

   1. Initial program 81.2%

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

   3. Simplified 74.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right), \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right), \ell\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{1}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{M \cdot \left(M \cdot D\right)}{-4 \cdot d}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(D \cdot M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(d \cdot -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      17. /-lowering-/.f64 81.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \mathsf{/.f64}\left(d, D\right)\right)\right), \ell\right)\right)\right)\right) \]

   6. Applied egg-rr 81.4%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\ell \cdot \frac{d}{D}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \left(M \cdot \frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(h \cdot M\right) \cdot \frac{D \cdot M}{d \cdot -4}\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot M\right), \left(\frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{D \cdot M}{d}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M \cdot D}{d}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M}{d} \cdot D}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M}{\frac{d}{D}}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \mathsf{*.f64}\left(\ell, \left(\frac{d}{D}\right)\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 83.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(d, D\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 83.6%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\left(h \cdot M\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9.4 \cdot 10^{-33}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{\frac{D}{\frac{\ell}{h}}}{-4} \cdot \frac{M}{\frac{d}{M \cdot D}}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(M \cdot D\right)}{-4}\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 5.6e-33)
   (* w0 (sqrt (+ 1.0 (/ (* (/ h l) (* D (/ (* (/ M d) (* M D)) -4.0))) d))))
   (* w0 (sqrt (+ 1.0 (/ (* (* M h) (/ (/ M (/ d D)) -4.0)) (* l (/ d D))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.6e-33) {
		tmp = w0 * sqrt((1.0 + (((h / l) * (D * (((M / d) * (M * D)) / -4.0))) / d)));
	} else {
		tmp = w0 * sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d_1 <= 5.6d-33) then
        tmp = w0 * sqrt((1.0d0 + (((h / l) * (d * (((m / d_1) * (m * d)) / (-4.0d0)))) / d_1)))
    else
        tmp = w0 * sqrt((1.0d0 + (((m * h) * ((m / (d_1 / d)) / (-4.0d0))) / (l * (d_1 / d)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 5.6e-33) {
		tmp = w0 * Math.sqrt((1.0 + (((h / l) * (D * (((M / d) * (M * D)) / -4.0))) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 5.6e-33:
		tmp = w0 * math.sqrt((1.0 + (((h / l) * (D * (((M / d) * (M * D)) / -4.0))) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 5.6e-33)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h / l) * Float64(D * Float64(Float64(Float64(M / d) * Float64(M * D)) / -4.0))) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M * h) * Float64(Float64(M / Float64(d / D)) / -4.0)) / Float64(l * Float64(d / D))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 5.6e-33)
		tmp = w0 * sqrt((1.0 + (((h / l) * (D * (((M / d) * (M * D)) / -4.0))) / d)));
	else
		tmp = w0 * sqrt((1.0 + (((M * h) * ((M / (d / D)) / -4.0)) / (l * (d / D)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 5.6e-33], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h / l), $MachinePrecision] * N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M * h), $MachinePrecision] * N[(N[(M / N[(d / D), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.6e-33

   1. Initial program 79.9%

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

   3. Simplified 76.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\frac{\left(M \cdot D\right) \cdot M}{d}\right), -4\right)\right)\right), d\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\left(\left(M \cdot D\right) \cdot \frac{M}{d}\right), -4\right)\right)\right), d\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(M \cdot D\right), \left(\frac{M}{d}\right)\right), -4\right)\right)\right), d\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(D \cdot M\right), \left(\frac{M}{d}\right)\right), -4\right)\right)\right), d\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, M\right), \left(\frac{M}{d}\right)\right), -4\right)\right)\right), d\right)\right)\right)\right) \]

      6. /-lowering-/.f64 78.8%

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(h, \ell\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(D, M\right), \mathsf{/.f64}\left(M, d\right)\right), -4\right)\right)\right), d\right)\right)\right)\right) \]

   6. Applied egg-rr 78.8%

      \[\leadsto w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \frac{M}{d}}}{-4}\right)}{d}} \]

   if 5.6e-33 < d

   1. Initial program 81.2%

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

   3. Simplified 74.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right), \ell\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right), \ell\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{D}{d}\right)\right), \ell\right)\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{1}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      10. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{M \cdot \left(M \cdot D\right)}{-4 \cdot d}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(D \cdot M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(d \cdot -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

      17. /-lowering-/.f64 81.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \mathsf{/.f64}\left(d, D\right)\right)\right), \ell\right)\right)\right)\right) \]

   6. Applied egg-rr 81.4%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}\right)\right)\right)\right) \]

      2. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\ell \cdot \frac{d}{D}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \left(M \cdot \frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(h \cdot M\right) \cdot \frac{D \cdot M}{d \cdot -4}\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(h \cdot M\right), \left(\frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{D \cdot M}{d \cdot -4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{D \cdot M}{d}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M \cdot D}{d}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M}{d} \cdot D}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      11. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \left(\frac{\frac{M}{\frac{d}{D}}}{-4}\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \left(\ell \cdot \frac{d}{D}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \mathsf{*.f64}\left(\ell, \left(\frac{d}{D}\right)\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 83.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(h, M\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), -4\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(d, D\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 83.6%

      \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{\left(h \cdot M\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.6 \cdot 10^{-33}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M}{d} \cdot \left(M \cdot D\right)}{-4}\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(M \cdot h\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{-4}}{\ell \cdot \frac{d}{D}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (+ 1.0 (/ (* h (/ (/ (* M (* M D)) (* d -4.0)) (/ d D))) l)))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 + ((h * (((M * (M * D)) / (d * -4.0)) / (d / D))) / 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 + ((h * (((m * (m * d)) / (d_1 * (-4.0d0))) / (d_1 / d))) / 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 + ((h * (((M * (M * D)) / (d * -4.0)) / (d / D))) / l)));
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 + ((h * (((M * (M * D)) / (d * -4.0)) / (d / D))) / l)))

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * Float64(Float64(Float64(M * Float64(M * D)) / Float64(d * -4.0)) / Float64(d / D))) / l))))
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + ((h * (((M * (M * D)) / (d * -4.0)) / (d / D))) / l)));
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[(N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * -4.0), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

3. Simplified 75.7%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h}{\ell} \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right)\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}}{\ell}\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(h \cdot \frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right), \ell\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{d}\right)\right), \ell\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot D}{d}\right)\right), \ell\right)\right)\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{D}{d}\right)\right), \ell\right)\right)\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4} \cdot \frac{1}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

   8. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \left(\frac{\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}}{\frac{d}{D}}\right)\right), \ell\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

   10. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\left(\frac{M \cdot \left(M \cdot D\right)}{-4 \cdot d}\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot D\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(D \cdot M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(-4 \cdot d\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \left(d \cdot -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \left(\frac{d}{D}\right)\right)\right), \ell\right)\right)\right)\right) \]

   17. /-lowering-/.f64 80.0%

       \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(h, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(D, M\right)\right), \mathsf{*.f64}\left(d, -4\right)\right), \mathsf{/.f64}\left(d, D\right)\right)\right), \ell\right)\right)\right)\right) \]

6. Applied egg-rr 80.0%

   \[\leadsto w0 \cdot \sqrt{1 + \color{blue}{\frac{h \cdot \frac{\frac{M \cdot \left(D \cdot M\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}}} \]

7. Final simplification 80.0%

   \[\leadsto w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d \cdot -4}}{\frac{d}{D}}}{\ell}} \]
8. Add Preprocessing

Alternative 10: 69.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-155}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D \cdot D\right) \cdot \left(\left(h \cdot \frac{M}{\frac{d}{M}}\right) \cdot \left(\frac{-0.125}{\ell} \cdot \frac{w0}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.7e-155)
   w0
   (if (<= M 2.15e+99)
     (* w0 (+ 1.0 (* D (/ D (/ (/ (* d (* d l)) h) (* M (* M -0.125)))))))
     (+ w0 (* (* D D) (* (* h (/ M (/ d M))) (* (/ -0.125 l) (/ w0 d))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.7e-155) {
		tmp = w0;
	} else if (M <= 2.15e+99) {
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	} else {
		tmp = w0 + ((D * D) * ((h * (M / (d / M))) * ((-0.125 / l) * (w0 / d))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 1.7d-155) then
        tmp = w0
    else if (m <= 2.15d+99) then
        tmp = w0 * (1.0d0 + (d * (d / (((d_1 * (d_1 * l)) / h) / (m * (m * (-0.125d0)))))))
    else
        tmp = w0 + ((d * d) * ((h * (m / (d_1 / m))) * (((-0.125d0) / l) * (w0 / d_1))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.7e-155) {
		tmp = w0;
	} else if (M <= 2.15e+99) {
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	} else {
		tmp = w0 + ((D * D) * ((h * (M / (d / M))) * ((-0.125 / l) * (w0 / d))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.7e-155:
		tmp = w0
	elif M <= 2.15e+99:
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))))
	else:
		tmp = w0 + ((D * D) * ((h * (M / (d / M))) * ((-0.125 / l) * (w0 / d))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.7e-155)
		tmp = w0;
	elseif (M <= 2.15e+99)
		tmp = Float64(w0 * Float64(1.0 + Float64(D * Float64(D / Float64(Float64(Float64(d * Float64(d * l)) / h) / Float64(M * Float64(M * -0.125)))))));
	else
		tmp = Float64(w0 + Float64(Float64(D * D) * Float64(Float64(h * Float64(M / Float64(d / M))) * Float64(Float64(-0.125 / l) * Float64(w0 / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.7e-155)
		tmp = w0;
	elseif (M <= 2.15e+99)
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	else
		tmp = w0 + ((D * D) * ((h * (M / (d / M))) * ((-0.125 / l) * (w0 / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.7e-155], w0, If[LessEqual[M, 2.15e+99], N[(w0 * N[(1.0 + N[(D * N[(D / N[(N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(M * N[(M * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 + N[(N[(D * D), $MachinePrecision] * N[(N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 / l), $MachinePrecision] * N[(w0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < 1.7e-155

   1. Initial program 84.8%

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

   3. Simplified 79.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 70.6%

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

   if 1.7e-155 < M < 2.1500000000000001e99

   1. Initial program 78.1%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 63.1%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right)\right) \]

   7. Simplified 63.1%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(D \cdot \color{blue}{\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \color{blue}{D}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right), \color{blue}{D}\right)\right)\right) \]

   9. Applied egg-rr 71.4%

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

   if 2.1500000000000001e99 < M

   1. Initial program 67.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d \cdot 2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{d}}{2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 81.9%

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

   5. Taylor expanded in M around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{M}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right)\right)\right) \]

   7. Simplified 43.3%

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

      1. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8}}{\ell} \cdot \left(\frac{w0}{d} \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d}}\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\left(\frac{\frac{-1}{8}}{\ell} \cdot \frac{w0}{d}\right) \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell} \cdot \frac{w0}{d}\right), \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot h}{d}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \left(\frac{w0}{d}\right)\right), \left(\frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{d}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{w0}{d}\right)\right), \left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot h}{d}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \left(\frac{\left(M \cdot M\right) \cdot \color{blue}{h}}{d}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \left(\frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \left(h \cdot \color{blue}{\frac{M \cdot M}{d}}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \left(h \cdot \left(M \cdot \color{blue}{\frac{M}{d}}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \mathsf{*.f64}\left(h, \color{blue}{\left(M \cdot \frac{M}{d}\right)}\right)\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \mathsf{*.f64}\left(h, \left(M \cdot \frac{1}{\color{blue}{\frac{d}{M}}}\right)\right)\right)\right)\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \mathsf{*.f64}\left(h, \left(\frac{M}{\color{blue}{\frac{d}{M}}}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(M, \color{blue}{\left(\frac{d}{M}\right)}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 43.9%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(w0, d\right)\right), \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, \color{blue}{M}\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 43.9%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.7 \cdot 10^{-155}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq 2.15 \cdot 10^{+99}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D \cdot D\right) \cdot \left(\left(h \cdot \frac{M}{\frac{d}{M}}\right) \cdot \left(\frac{-0.125}{\ell} \cdot \frac{w0}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-77}:\\ \;\;\;\;w0 + \frac{\frac{D \cdot D}{\frac{\ell}{-0.125}} \cdot \left(w0 \cdot \left(h \cdot \frac{M}{\frac{d}{M}}\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= d 2e-77)
   (+ w0 (/ (* (/ (* D D) (/ l -0.125)) (* w0 (* h (/ M (/ d M))))) d))
   (* w0 (+ 1.0 (* D (/ D (/ (/ (* d (* d l)) h) (* M (* M -0.125)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2e-77) {
		tmp = w0 + ((((D * D) / (l / -0.125)) * (w0 * (h * (M / (d / M))))) / d);
	} else {
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d_1 <= 2d-77) then
        tmp = w0 + ((((d * d) / (l / (-0.125d0))) * (w0 * (h * (m / (d_1 / m))))) / d_1)
    else
        tmp = w0 * (1.0d0 + (d * (d / (((d_1 * (d_1 * l)) / h) / (m * (m * (-0.125d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (d <= 2e-77) {
		tmp = w0 + ((((D * D) / (l / -0.125)) * (w0 * (h * (M / (d / M))))) / d);
	} else {
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if d <= 2e-77:
		tmp = w0 + ((((D * D) / (l / -0.125)) * (w0 * (h * (M / (d / M))))) / d)
	else:
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (d <= 2e-77)
		tmp = Float64(w0 + Float64(Float64(Float64(Float64(D * D) / Float64(l / -0.125)) * Float64(w0 * Float64(h * Float64(M / Float64(d / M))))) / d));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(D * Float64(D / Float64(Float64(Float64(d * Float64(d * l)) / h) / Float64(M * Float64(M * -0.125)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (d <= 2e-77)
		tmp = w0 + ((((D * D) / (l / -0.125)) * (w0 * (h * (M / (d / M))))) / d);
	else
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[d, 2e-77], N[(w0 + N[(N[(N[(N[(D * D), $MachinePrecision] / N[(l / -0.125), $MachinePrecision]), $MachinePrecision] * N[(w0 * N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(D * N[(D / N[(N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(M * N[(M * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.9999999999999999e-77

   1. Initial program 79.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d \cdot 2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{d}}{2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 87.4%

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

   5. Taylor expanded in M around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{M}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right)\right)\right) \]

   7. Simplified 56.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left(\left(D \cdot D\right) \cdot \frac{\frac{-1}{8}}{\ell}\right) \cdot \color{blue}{\frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left(\left(D \cdot D\right) \cdot \frac{\frac{-1}{8}}{\ell}\right) \cdot \frac{\frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}}{\color{blue}{d}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{\left(\left(D \cdot D\right) \cdot \frac{\frac{-1}{8}}{\ell}\right) \cdot \frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}}{\color{blue}{d}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\left(D \cdot D\right) \cdot \frac{\frac{-1}{8}}{\ell}\right) \cdot \frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}\right), \color{blue}{d}\right)\right) \]

   9. Applied egg-rr 67.0%

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

   if 1.9999999999999999e-77 < d

   1. Initial program 81.5%

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

   3. Simplified 76.1%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 59.0%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right)\right) \]

   7. Simplified 59.0%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(D \cdot \color{blue}{\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \color{blue}{D}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right), \color{blue}{D}\right)\right)\right) \]

   9. Applied egg-rr 70.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{-77}:\\ \;\;\;\;w0 + \frac{\frac{D \cdot D}{\frac{\ell}{-0.125}} \cdot \left(w0 \cdot \left(h \cdot \frac{M}{\frac{d}{M}}\right)\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 9.8 \cdot 10^{+153}:\\ \;\;\;\;w0 + \left(D \cdot D\right) \cdot \frac{\left(w0 \cdot \left(h \cdot \frac{M}{\frac{d}{M}}\right)\right) \cdot \frac{-0.125}{\ell}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 9.8e+153)
   (+ w0 (* (* D D) (/ (* (* w0 (* h (/ M (/ d M)))) (/ -0.125 l)) d)))
   (* w0 (+ 1.0 (/ (* -0.125 (* D (* D (* h (* M M))))) (* d (* d l)))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 9.8e+153) {
		tmp = w0 + ((D * D) * (((w0 * (h * (M / (d / M)))) * (-0.125 / l)) / d));
	} else {
		tmp = w0 * (1.0 + ((-0.125 * (D * (D * (h * (M * M))))) / (d * (d * l))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (d <= 9.8d+153) then
        tmp = w0 + ((d * d) * (((w0 * (h * (m / (d_1 / m)))) * ((-0.125d0) / l)) / d_1))
    else
        tmp = w0 * (1.0d0 + (((-0.125d0) * (d * (d * (h * (m * m))))) / (d_1 * (d_1 * l))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 9.8e+153) {
		tmp = w0 + ((D * D) * (((w0 * (h * (M / (d / M)))) * (-0.125 / l)) / d));
	} else {
		tmp = w0 * (1.0 + ((-0.125 * (D * (D * (h * (M * M))))) / (d * (d * l))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 9.8e+153:
		tmp = w0 + ((D * D) * (((w0 * (h * (M / (d / M)))) * (-0.125 / l)) / d))
	else:
		tmp = w0 * (1.0 + ((-0.125 * (D * (D * (h * (M * M))))) / (d * (d * l))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 9.8e+153)
		tmp = Float64(w0 + Float64(Float64(D * D) * Float64(Float64(Float64(w0 * Float64(h * Float64(M / Float64(d / M)))) * Float64(-0.125 / l)) / d)));
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 * Float64(D * Float64(D * Float64(h * Float64(M * M))))) / Float64(d * Float64(d * l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 9.8e+153)
		tmp = w0 + ((D * D) * (((w0 * (h * (M / (d / M)))) * (-0.125 / l)) / d));
	else
		tmp = w0 * (1.0 + ((-0.125 * (D * (D * (h * (M * M))))) / (d * (d * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 9.8e+153], N[(w0 + N[(N[(D * D), $MachinePrecision] * N[(N[(N[(w0 * N[(h * N[(M / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 * N[(1.0 + N[(N[(-0.125 * N[(D * N[(D * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 9.80000000000000003e153

   1. Initial program 82.3%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d \cdot 2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{d}}{2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in M around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2} \cdot \ell}\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot \left(h \cdot w0\right)\right)}{\ell \cdot \color{blue}{{d}^{2}}}\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8}}{\ell} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \color{blue}{\left(\frac{{M}^{2} \cdot \left(h \cdot w0\right)}{{d}^{2}}\right)}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{\color{blue}{{M}^{2} \cdot \left(h \cdot w0\right)}}{{d}^{2}}\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{/.f64}\left(\left({M}^{2} \cdot \left(h \cdot w0\right)\right), \color{blue}{\left({d}^{2}\right)}\right)\right)\right)\right) \]

   7. Simplified 57.9%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8}}{\ell} \cdot \frac{\frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}}{\color{blue}{d}}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{\frac{-1}{8}}{\ell} \cdot \frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}}{\color{blue}{d}}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{\frac{-1}{8}}{\ell} \cdot \frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}\right), \color{blue}{d}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{\ell}\right), \left(\frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}\right)\right), d\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(\frac{w0 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}\right)\right), d\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \left(w0 \cdot \frac{\left(M \cdot M\right) \cdot h}{d}\right)\right), d\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \left(\frac{\left(M \cdot M\right) \cdot h}{d}\right)\right)\right), d\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \left(\frac{h \cdot \left(M \cdot M\right)}{d}\right)\right)\right), d\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \left(h \cdot \frac{M \cdot M}{d}\right)\right)\right), d\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \left(h \cdot \left(M \cdot \frac{M}{d}\right)\right)\right)\right), d\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(h, \left(M \cdot \frac{M}{d}\right)\right)\right)\right), d\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(h, \left(M \cdot \frac{1}{\frac{d}{M}}\right)\right)\right)\right), d\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(h, \left(\frac{M}{\frac{d}{M}}\right)\right)\right)\right), d\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(M, \left(\frac{d}{M}\right)\right)\right)\right)\right), d\right)\right)\right) \]

      15. /-lowering-/.f64 68.5%

          \[\leadsto \mathsf{+.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, \ell\right), \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(h, \mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, M\right)\right)\right)\right)\right), d\right)\right)\right) \]

   9. Applied egg-rr 68.5%

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

   if 9.80000000000000003e153 < D

   1. Initial program 61.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell \cdot \frac{1}{h}}\right)\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d \cdot 2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{M \cdot D}{d}}{2}\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M \cdot D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{D}{d}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \frac{1}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{M}{\frac{d}{D}}\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \left(\frac{d}{D}\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \left(\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{1}{h}}\right)\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(M, \mathsf{/.f64}\left(d, D\right)\right), 2\right), \ell\right), \mathsf{/.f64}\left(\left(\frac{M \cdot D}{2 \cdot d}\right), \left(\frac{1}{h}\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 81.0%

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

   5. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(D \cdot D\right) \cdot \left({M}^{2} \cdot h\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \left({M}^{2} \cdot h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left({M}^{2}\right), h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\left(M \cdot M\right), h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \left(\left(d \cdot d\right) \cdot \ell\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \left(d \cdot \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 64.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(D, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right)\right)\right)\right), \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right)\right) \]

   7. Simplified 64.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 9.8 \cdot 10^{+153}:\\ \;\;\;\;w0 + \left(D \cdot D\right) \cdot \frac{\left(w0 \cdot \left(h \cdot \frac{M}{\frac{d}{M}}\right)\right) \cdot \frac{-0.125}{\ell}}{d}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125 \cdot \left(D \cdot \left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.72 \cdot 10^{-155}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.72e-155)
   w0
   (* w0 (+ 1.0 (* D (/ D (/ (/ (* d (* d l)) h) (* M (* M -0.125)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.72e-155) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 1.72d-155) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (d * (d / (((d_1 * (d_1 * l)) / h) / (m * (m * (-0.125d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.72e-155) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.72e-155:
		tmp = w0
	else:
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.72e-155)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(D * Float64(D / Float64(Float64(Float64(d * Float64(d * l)) / h) / Float64(M * Float64(M * -0.125)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.72e-155)
		tmp = w0;
	else
		tmp = w0 * (1.0 + (D * (D / (((d * (d * l)) / h) / (M * (M * -0.125))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.72e-155], w0, N[(w0 * N[(1.0 + N[(D * N[(D / N[(N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(M * N[(M * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.71999999999999991e-155

   1. Initial program 84.8%

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

   3. Simplified 79.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 70.6%

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

   if 1.71999999999999991e-155 < M

   1. Initial program 74.1%

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

   3. Simplified 70.3%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 55.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right)\right) \]

   7. Simplified 55.4%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(D \cdot \color{blue}{\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \color{blue}{D}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot \frac{\left(\frac{-1}{8} \cdot \left(M \cdot M\right)\right) \cdot h}{\left(d \cdot d\right) \cdot \ell}\right), \color{blue}{D}\right)\right)\right) \]

   9. Applied egg-rr 60.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 1.72 \cdot 10^{-155}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + D \cdot \frac{D}{\frac{\frac{d \cdot \left(d \cdot \ell\right)}{h}}{M \cdot \left(M \cdot -0.125\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 1.65 \cdot 10^{-165}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot D\right) \cdot \left(\frac{-0.125}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot \ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 1.65e-165)
   w0
   (* w0 (+ 1.0 (* (* D D) (* (/ -0.125 d) (/ (* M (* M h)) (* d l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.65e-165) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D * D) * ((-0.125 / d) * ((M * (M * h)) / (d * l)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 1.65d-165) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((d * d) * (((-0.125d0) / d_1) * ((m * (m * h)) / (d_1 * l)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 1.65e-165) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D * D) * ((-0.125 / d) * ((M * (M * h)) / (d * l)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 1.65e-165:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((D * D) * ((-0.125 / d) * ((M * (M * h)) / (d * l)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 1.65e-165)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * D) * Float64(Float64(-0.125 / d) * Float64(Float64(M * Float64(M * h)) / Float64(d * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 1.65e-165)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((D * D) * ((-0.125 / d) * ((M * (M * h)) / (d * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 1.65e-165], w0, N[(w0 * N[(1.0 + N[(N[(D * D), $MachinePrecision] * N[(N[(-0.125 / d), $MachinePrecision] * N[(N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.6499999999999999e-165

   1. Initial program 84.8%

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

   3. Simplified 79.6%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 70.6%

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

   if 1.6499999999999999e-165 < M

   1. Initial program 74.1%

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

   3. Simplified 70.3%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 55.4%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right)\right) \]

   7. Simplified 55.4%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}\right)\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8}}{d} \cdot \color{blue}{\frac{\left(M \cdot M\right) \cdot h}{d \cdot \ell}}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\left(\frac{\frac{-1}{8}}{d}\right), \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot h}{d \cdot \ell}\right)}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, d\right), \left(\frac{\color{blue}{\left(M \cdot M\right) \cdot h}}{d \cdot \ell}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, d\right), \mathsf{/.f64}\left(\left(\left(M \cdot M\right) \cdot h\right), \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, d\right), \mathsf{/.f64}\left(\left(M \cdot \left(M \cdot h\right)\right), \left(\color{blue}{d} \cdot \ell\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot h\right)\right), \left(\color{blue}{d} \cdot \ell\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \left(d \cdot \ell\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 59.3%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{8}, d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, h\right)\right), \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 59.3%

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

Alternative 15: 64.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 4 \cdot 10^{+43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\frac{M \cdot \left(M \cdot \left(h \cdot -0.125\right)\right)}{d} \cdot \frac{D \cdot D}{d \cdot \ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 4e+43)
   w0
   (* w0 (* (/ (* M (* M (* h -0.125))) d) (/ (* D D) (* d l))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+43) {
		tmp = w0;
	} else {
		tmp = w0 * (((M * (M * (h * -0.125))) / d) * ((D * D) / (d * l)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 4d+43) then
        tmp = w0
    else
        tmp = w0 * (((m * (m * (h * (-0.125d0)))) / d_1) * ((d * d) / (d_1 * l)))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 4e+43) {
		tmp = w0;
	} else {
		tmp = w0 * (((M * (M * (h * -0.125))) / d) * ((D * D) / (d * l)));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 4e+43:
		tmp = w0
	else:
		tmp = w0 * (((M * (M * (h * -0.125))) / d) * ((D * D) / (d * l)))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 4e+43)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(Float64(M * Float64(M * Float64(h * -0.125))) / d) * Float64(Float64(D * D) / Float64(d * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 4e+43)
		tmp = w0;
	else
		tmp = w0 * (((M * (M * (h * -0.125))) / d) * ((D * D) / (d * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 4e+43], w0, N[(w0 * N[(N[(N[(M * N[(M * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.00000000000000006e43

   1. Initial program 83.4%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 71.5%

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

   if 4.00000000000000006e43 < M

   1. Initial program 69.7%

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

   3. Simplified 62.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 49.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right)\right) \]

   7. Simplified 49.6%

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

   8. Taylor expanded in D around inf

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 27.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   10. Simplified 27.6%

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

       1. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \left(D \cdot D\right)}{d \cdot \color{blue}{\left(d \cdot \ell\right)}}\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{d} \cdot \color{blue}{\frac{D \cdot D}{d \cdot \ell}}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}}{d}\right), \color{blue}{\left(\frac{D \cdot D}{d \cdot \ell}\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right), d\right), \left(\frac{\color{blue}{D \cdot D}}{d \cdot \ell}\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{-1}{8}\right)\right), d\right), \left(\frac{\color{blue}{D} \cdot D}{d \cdot \ell}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(M \cdot \left(M \cdot \left(h \cdot \frac{-1}{8}\right)\right)\right), d\right), \left(\frac{\color{blue}{D} \cdot D}{d \cdot \ell}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot \left(h \cdot \frac{-1}{8}\right)\right)\right), d\right), \left(\frac{\color{blue}{D} \cdot D}{d \cdot \ell}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \left(h \cdot \frac{-1}{8}\right)\right)\right), d\right), \left(\frac{D \cdot D}{d \cdot \ell}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), d\right), \left(\frac{D \cdot D}{d \cdot \ell}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), d\right), \mathsf{/.f64}\left(\left(D \cdot D\right), \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{d} \cdot \ell\right)\right)\right)\right) \]

       12. *-lowering-*.f64 26.4%

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), d\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right) \]

   12. Applied egg-rr 26.4%

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

Alternative 16: 64.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 9 \cdot 10^{+43}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\left(M \cdot \left(M \cdot \left(h \cdot -0.125\right)\right)\right) \cdot \left(D \cdot \frac{D}{d \cdot \left(d \cdot \ell\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= M 9e+43)
   w0
   (* w0 (* (* M (* M (* h -0.125))) (* D (/ D (* d (* d l))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e+43) {
		tmp = w0;
	} else {
		tmp = w0 * ((M * (M * (h * -0.125))) * (D * (D / (d * (d * l)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 9d+43) then
        tmp = w0
    else
        tmp = w0 * ((m * (m * (h * (-0.125d0)))) * (d * (d / (d_1 * (d_1 * l)))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (M <= 9e+43) {
		tmp = w0;
	} else {
		tmp = w0 * ((M * (M * (h * -0.125))) * (D * (D / (d * (d * l)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	tmp = 0
	if M <= 9e+43:
		tmp = w0
	else:
		tmp = w0 * ((M * (M * (h * -0.125))) * (D * (D / (d * (d * l)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (M <= 9e+43)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(Float64(M * Float64(M * Float64(h * -0.125))) * Float64(D * Float64(D / Float64(d * Float64(d * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (M <= 9e+43)
		tmp = w0;
	else
		tmp = w0 * ((M * (M * (h * -0.125))) * (D * (D / (d * (d * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[M, 9e+43], w0, N[(w0 * N[(N[(M * N[(M * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D * N[(D / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9e43

   1. Initial program 83.4%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \color{blue}{w0} \]
   6. Step-by-step derivation

   7. Simplified 71.5%

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

   if 9e43 < M

   1. Initial program 69.7%

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

   3. Simplified 62.7%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   4. Add Preprocessing

   5. Taylor expanded in h around 0

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left(\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot \frac{-1}{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \color{blue}{\left(\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \color{blue}{\frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({D}^{2}\right), \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(D \cdot D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\color{blue}{\frac{-1}{8}} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \left(\frac{\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot {M}^{2}\right), h\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({M}^{2}\right)\right), h\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(M \cdot M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \left({d}^{2} \cdot \ell\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 49.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(D, D\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(M, M\right)\right), h\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right)\right)\right) \]

   7. Simplified 49.6%

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

   8. Taylor expanded in D around inf

      \[\leadsto \mathsf{*.f64}\left(w0, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \left(\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{\color{blue}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)\right), \color{blue}{\left({d}^{2} \cdot \ell\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(\left({M}^{2} \cdot h\right) \cdot {D}^{2}\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right) \cdot {D}^{2}\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{8} \cdot \left({M}^{2} \cdot h\right)\right), \left({D}^{2}\right)\right), \left(\color{blue}{{d}^{2}} \cdot \ell\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left({M}^{2} \cdot h\right) \cdot \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2} \cdot h\right), \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({\color{blue}{d}}^{2} \cdot \ell\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), h\right), \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), h\right), \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \left({D}^{2}\right)\right), \left({d}^{2} \cdot \ell\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \left(D \cdot D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(D, D\right)\right), \left({d}^{\color{blue}{2}} \cdot \ell\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left({d}^{2}\right), \color{blue}{\ell}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\left(d \cdot d\right), \ell\right)\right)\right) \]

      15. *-lowering-*.f64 27.6%

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), h\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(D, D\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d, d\right), \ell\right)\right)\right) \]

   10. Simplified 27.6%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right) \cdot \color{blue}{\frac{D \cdot D}{\left(d \cdot d\right) \cdot \ell}}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \frac{-1}{8}\right), \color{blue}{\left(\frac{D \cdot D}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{-1}{8}\right)\right), \left(\frac{\color{blue}{D \cdot D}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\left(M \cdot \left(M \cdot \left(h \cdot \frac{-1}{8}\right)\right)\right), \left(\frac{\color{blue}{D \cdot D}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \left(M \cdot \left(h \cdot \frac{-1}{8}\right)\right)\right), \left(\frac{\color{blue}{D \cdot D}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \left(h \cdot \frac{-1}{8}\right)\right)\right), \left(\frac{D \cdot \color{blue}{D}}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), \left(\frac{D \cdot D}{\left(d \cdot d\right) \cdot \ell}\right)\right)\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), \left(D \cdot \color{blue}{\frac{D}{\left(d \cdot d\right) \cdot \ell}}\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(D, \color{blue}{\left(\frac{D}{\left(d \cdot d\right) \cdot \ell}\right)}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(D, \color{blue}{\left(\left(d \cdot d\right) \cdot \ell\right)}\right)\right)\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(D, \left(d \cdot \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \color{blue}{\left(d \cdot \ell\right)}\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f64 28.2%

           \[\leadsto \mathsf{*.f64}\left(w0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, \mathsf{*.f64}\left(M, \mathsf{*.f64}\left(h, \frac{-1}{8}\right)\right)\right), \mathsf{*.f64}\left(D, \mathsf{/.f64}\left(D, \mathsf{*.f64}\left(d, \mathsf{*.f64}\left(d, \color{blue}{\ell}\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 28.2%

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

Alternative 17: 67.8% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ w0 \end{array} \]

FPCore

(FPCore (w0 M D h l d) :precision binary64 w0)

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

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
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0;
}

Python

def code(w0, M, D, h, l, d):
	return w0

Julia

function code(w0, M, D, h, l, d)
	return w0
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 80.3%

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

3. Simplified 75.7%

   \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
4. Add Preprocessing

5. Taylor expanded in h around 0

   \[\leadsto \color{blue}{w0} \]
6. Step-by-step derivation

7. Simplified 66.0%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(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))))))