Henrywood and Agarwal, Equation (9a)

Percentage Accurate: 81.1% → 89.2%

Time: 17.8s

Alternatives: 14

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: 81.1% 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: 89.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{M\_m}{\frac{d \cdot 2}{D}}\\ \mathbf{if}\;M\_m \leq 5 \cdot 10^{-242}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{\frac{D}{\frac{\frac{d}{M\_m \cdot D}}{M\_m}}}{-4}}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{t\_0}{\ell} \cdot \frac{t\_0}{\frac{-1}{h}}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (/ M_m (/ (* d 2.0) D))))
   (if (<= M_m 5e-242)
     (*
      w0
      (sqrt (+ 1.0 (/ (* h (/ (/ (/ D (/ (/ d (* M_m D)) M_m)) -4.0) l)) d))))
     (* w0 (sqrt (+ 1.0 (* (/ t_0 l) (/ t_0 (/ -1.0 h)))))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = M_m / ((d * 2.0) / D);
	double tmp;
	if (M_m <= 5e-242) {
		tmp = w0 * sqrt((1.0 + ((h * (((D / ((d / (M_m * D)) / M_m)) / -4.0) / l)) / d)));
	} else {
		tmp = w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m / ((d_1 * 2.0d0) / d)
    if (m_m <= 5d-242) then
        tmp = w0 * sqrt((1.0d0 + ((h * (((d / ((d_1 / (m_m * d)) / m_m)) / (-4.0d0)) / l)) / d_1)))
    else
        tmp = w0 * sqrt((1.0d0 + ((t_0 / l) * (t_0 / ((-1.0d0) / h)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = M_m / ((d * 2.0) / D);
	double tmp;
	if (M_m <= 5e-242) {
		tmp = w0 * Math.sqrt((1.0 + ((h * (((D / ((d / (M_m * D)) / M_m)) / -4.0) / l)) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	t_0 = M_m / ((d * 2.0) / D)
	tmp = 0
	if M_m <= 5e-242:
		tmp = w0 * math.sqrt((1.0 + ((h * (((D / ((d / (M_m * D)) / M_m)) / -4.0) / l)) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(M_m / Float64(Float64(d * 2.0) / D))
	tmp = 0.0
	if (M_m <= 5e-242)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * Float64(Float64(Float64(D / Float64(Float64(d / Float64(M_m * D)) / M_m)) / -4.0) / l)) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 / l) * Float64(t_0 / Float64(-1.0 / h))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = M_m / ((d * 2.0) / D);
	tmp = 0.0;
	if (M_m <= 5e-242)
		tmp = w0 * sqrt((1.0 + ((h * (((D / ((d / (M_m * D)) / M_m)) / -4.0) / l)) / d)));
	else
		tmp = w0 * sqrt((1.0 + ((t_0 / l) * (t_0 / (-1.0 / h)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M$95$m / N[(N[(d * 2.0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 5e-242], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[(N[(N[(D / N[(N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 4.9999999999999998e-242

   1. Initial program 87.2%

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

   3. Simplified 82.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 4.9999999999999998e-242 < M

   1. Initial program 79.8%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-/l* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

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

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

   4. Applied egg-rr 92.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-242}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{\frac{D}{\frac{\frac{d}{M \cdot D}}{M}}}{-4}}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M}{\frac{d \cdot 2}{D}}}{\ell} \cdot \frac{\frac{M}{\frac{d \cdot 2}{D}}}{\frac{-1}{h}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2 \cdot 10^{-133}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{\frac{D}{\frac{\frac{d}{M\_m \cdot D}}{M\_m}}}{-4}}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M\_m}{\frac{d \cdot 2}{D}}}{\ell} \cdot \left(\frac{M\_m}{d \cdot 2} \cdot \left(h \cdot D\right)\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 2e-133)
   (*
    w0
    (sqrt (+ 1.0 (/ (* h (/ (/ (/ D (/ (/ d (* M_m D)) M_m)) -4.0) l)) d))))
   (*
    w0
    (sqrt
     (-
      1.0
      (* (/ (/ M_m (/ (* d 2.0) D)) l) (* (/ M_m (* d 2.0)) (* h D))))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2e-133) {
		tmp = w0 * sqrt((1.0 + ((h * (((D / ((d / (M_m * D)) / M_m)) / -4.0) / l)) / d)));
	} else {
		tmp = w0 * sqrt((1.0 - (((M_m / ((d * 2.0) / D)) / l) * ((M_m / (d * 2.0)) * (h * D)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 2d-133) then
        tmp = w0 * sqrt((1.0d0 + ((h * (((d / ((d_1 / (m_m * d)) / m_m)) / (-4.0d0)) / l)) / d_1)))
    else
        tmp = w0 * sqrt((1.0d0 - (((m_m / ((d_1 * 2.0d0) / d)) / l) * ((m_m / (d_1 * 2.0d0)) * (h * d)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2e-133) {
		tmp = w0 * Math.sqrt((1.0 + ((h * (((D / ((d / (M_m * D)) / M_m)) / -4.0) / l)) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((M_m / ((d * 2.0) / D)) / l) * ((M_m / (d * 2.0)) * (h * D)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 2e-133:
		tmp = w0 * math.sqrt((1.0 + ((h * (((D / ((d / (M_m * D)) / M_m)) / -4.0) / l)) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 - (((M_m / ((d * 2.0) / D)) / l) * ((M_m / (d * 2.0)) * (h * D)))))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 2e-133)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * Float64(Float64(Float64(D / Float64(Float64(d / Float64(M_m * D)) / M_m)) / -4.0) / l)) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(M_m / Float64(Float64(d * 2.0) / D)) / l) * Float64(Float64(M_m / Float64(d * 2.0)) * Float64(h * D))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 2e-133)
		tmp = w0 * sqrt((1.0 + ((h * (((D / ((d / (M_m * D)) / M_m)) / -4.0) / l)) / d)));
	else
		tmp = w0 * sqrt((1.0 - (((M_m / ((d * 2.0) / D)) / l) * ((M_m / (d * 2.0)) * (h * D)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 2e-133], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[(N[(N[(D / N[(N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision] / M$95$m), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(M$95$m / N[(N[(d * 2.0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(h * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.0000000000000001e-133

   1. Initial program 87.6%

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

   3. Simplified 83.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 88.3%

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

   6. Applied egg-rr 88.3%

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

   if 2.0000000000000001e-133 < M

   1. Initial program 77.1%

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   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. associate-/l* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

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

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

   4. Applied egg-rr 91.0%

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

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 87.3%

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

   6. Applied egg-rr 87.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2 \cdot 10^{-133}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{\frac{D}{\frac{\frac{d}{M \cdot D}}{M}}}{-4}}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{d \cdot 2}{D}}}{\ell} \cdot \left(\frac{M}{d \cdot 2} \cdot \left(h \cdot D\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{d}{M\_m \cdot D}\\ \mathbf{if}\;M\_m \leq 5 \cdot 10^{-242}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{\frac{D}{\frac{t\_0}{M\_m}}}{-4}}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M\_m}{t\_0}}{-4}}{\ell} \cdot \frac{D}{d}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (let* ((t_0 (/ d (* M_m D))))
   (if (<= M_m 5e-242)
     (* w0 (sqrt (+ 1.0 (/ (* h (/ (/ (/ D (/ t_0 M_m)) -4.0) l)) d))))
     (* w0 (sqrt (+ 1.0 (* (/ (* h (/ (/ M_m t_0) -4.0)) l) (/ D d))))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = d / (M_m * D);
	double tmp;
	if (M_m <= 5e-242) {
		tmp = w0 * sqrt((1.0 + ((h * (((D / (t_0 / M_m)) / -4.0) / l)) / d)));
	} else {
		tmp = w0 * sqrt((1.0 + (((h * ((M_m / t_0) / -4.0)) / l) * (D / d))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d_1 / (m_m * d)
    if (m_m <= 5d-242) then
        tmp = w0 * sqrt((1.0d0 + ((h * (((d / (t_0 / m_m)) / (-4.0d0)) / l)) / d_1)))
    else
        tmp = w0 * sqrt((1.0d0 + (((h * ((m_m / t_0) / (-4.0d0))) / l) * (d / d_1))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double t_0 = d / (M_m * D);
	double tmp;
	if (M_m <= 5e-242) {
		tmp = w0 * Math.sqrt((1.0 + ((h * (((D / (t_0 / M_m)) / -4.0) / l)) / d)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (((h * ((M_m / t_0) / -4.0)) / l) * (D / d))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	t_0 = d / (M_m * D)
	tmp = 0
	if M_m <= 5e-242:
		tmp = w0 * math.sqrt((1.0 + ((h * (((D / (t_0 / M_m)) / -4.0) / l)) / d)))
	else:
		tmp = w0 * math.sqrt((1.0 + (((h * ((M_m / t_0) / -4.0)) / l) * (D / d))))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	t_0 = Float64(d / Float64(M_m * D))
	tmp = 0.0
	if (M_m <= 5e-242)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(h * Float64(Float64(Float64(D / Float64(t_0 / M_m)) / -4.0) / l)) / d))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(h * Float64(Float64(M_m / t_0) / -4.0)) / l) * Float64(D / d)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	t_0 = d / (M_m * D);
	tmp = 0.0;
	if (M_m <= 5e-242)
		tmp = w0 * sqrt((1.0 + ((h * (((D / (t_0 / M_m)) / -4.0) / l)) / d)));
	else
		tmp = w0 * sqrt((1.0 + (((h * ((M_m / t_0) / -4.0)) / l) * (D / d))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := Block[{t$95$0 = N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 5e-242], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(h * N[(N[(N[(D / N[(t$95$0 / M$95$m), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(N[(M$95$m / t$95$0), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 4.9999999999999998e-242

   1. Initial program 87.2%

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

   3. Simplified 82.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 4.9999999999999998e-242 < M

   1. Initial program 79.8%

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

   3. Simplified 71.2%

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

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

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

   6. Applied egg-rr 82.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 5 \cdot 10^{-242}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{\frac{D}{\frac{\frac{d}{M \cdot D}}{M}}}{-4}}{\ell}}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M}{\frac{d}{M \cdot D}}}{-4}}{\ell} \cdot \frac{D}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.95 \cdot 10^{-181}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{d} \cdot \frac{-0.25 \cdot \left(D \cdot \frac{h \cdot \left(M\_m \cdot M\_m\right)}{d}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 3.95e-181)
   w0
   (*
    w0
    (sqrt (+ 1.0 (* (/ D d) (/ (* -0.25 (* D (/ (* h (* M_m M_m)) d))) l)))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 3.95e-181) {
		tmp = w0;
	} else {
		tmp = w0 * sqrt((1.0 + ((D / d) * ((-0.25 * (D * ((h * (M_m * M_m)) / d))) / l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 3.95d-181) then
        tmp = w0
    else
        tmp = w0 * sqrt((1.0d0 + ((d / d_1) * (((-0.25d0) * (d * ((h * (m_m * m_m)) / d_1))) / l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 3.95e-181) {
		tmp = w0;
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((D / d) * ((-0.25 * (D * ((h * (M_m * M_m)) / d))) / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 3.95e-181:
		tmp = w0
	else:
		tmp = w0 * math.sqrt((1.0 + ((D / d) * ((-0.25 * (D * ((h * (M_m * M_m)) / d))) / l))))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 3.95e-181)
		tmp = w0;
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(D / d) * Float64(Float64(-0.25 * Float64(D * Float64(Float64(h * Float64(M_m * M_m)) / d))) / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 3.95e-181)
		tmp = w0;
	else
		tmp = w0 * sqrt((1.0 + ((D / d) * ((-0.25 * (D * ((h * (M_m * M_m)) / d))) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 3.95e-181], w0, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(D / d), $MachinePrecision] * N[(N[(-0.25 * N[(D * N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.95e-181

   1. Initial program 86.9%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 74.6%

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

   if 3.95e-181 < M

   1. Initial program 79.5%

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

   3. Simplified 70.0%

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

      1. associate-/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. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

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

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in h around 0

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

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

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 70.0%

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

   9. Simplified 70.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.95 \cdot 10^{-181}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{D}{d} \cdot \frac{-0.25 \cdot \left(D \cdot \frac{h \cdot \left(M \cdot M\right)}{d}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ w0 \cdot \sqrt{1 + \frac{h \cdot \frac{\frac{M\_m}{\frac{d}{M\_m \cdot D}}}{-4}}{\ell} \cdot \frac{D}{d}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (*
  w0
  (sqrt (+ 1.0 (* (/ (* h (/ (/ M_m (/ d (* M_m D))) -4.0)) l) (/ D d))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 + (((h * ((M_m / (d / (M_m * D))) / -4.0)) / l) * (D / d))));
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_1 / (m_m * d))) / (-4.0d0))) / l) * (d / d_1))))
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 + (((h * ((M_m / (d / (M_m * D))) / -4.0)) / l) * (D / d))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(h * N[(N[(M$95$m / N[(d / N[(M$95$m * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 77.7%

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

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. times-frac N/A

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

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

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

6. Applied egg-rr 84.4%

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

7. Final simplification 84.4%

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

Alternative 6: 65.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;D \leq 3.7 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{h \cdot \left(M\_m \cdot M\_m\right)}{d} \cdot \frac{D \cdot D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D \cdot -0.125\right) \cdot \frac{D \cdot \left(M\_m \cdot \left(M\_m \cdot \left(w0 \cdot h\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= D 3.7e+115)
   (* w0 (+ 1.0 (* (/ -0.125 l) (* (/ (* h (* M_m M_m)) d) (/ (* D D) d)))))
   (+ w0 (* (* D -0.125) (/ (* D (* M_m (* M_m (* w0 h)))) (* d (* d l)))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3.7e+115) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((h * (M_m * M_m)) / d) * ((D * D) / d))));
	} else {
		tmp = w0 + ((D * -0.125) * ((D * (M_m * (M_m * (w0 * h)))) / (d * (d * l))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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 <= 3.7d+115) then
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (((h * (m_m * m_m)) / d_1) * ((d * d) / d_1))))
    else
        tmp = w0 + ((d * (-0.125d0)) * ((d * (m_m * (m_m * (w0 * h)))) / (d_1 * (d_1 * l))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (D <= 3.7e+115) {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((h * (M_m * M_m)) / d) * ((D * D) / d))));
	} else {
		tmp = w0 + ((D * -0.125) * ((D * (M_m * (M_m * (w0 * h)))) / (d * (d * l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if D <= 3.7e+115:
		tmp = w0 * (1.0 + ((-0.125 / l) * (((h * (M_m * M_m)) / d) * ((D * D) / d))))
	else:
		tmp = w0 + ((D * -0.125) * ((D * (M_m * (M_m * (w0 * h)))) / (d * (d * l))))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (D <= 3.7e+115)
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(Float64(Float64(h * Float64(M_m * M_m)) / d) * Float64(Float64(D * D) / d)))));
	else
		tmp = Float64(w0 + Float64(Float64(D * -0.125) * Float64(Float64(D * Float64(M_m * Float64(M_m * Float64(w0 * h)))) / Float64(d * Float64(d * l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (D <= 3.7e+115)
		tmp = w0 * (1.0 + ((-0.125 / l) * (((h * (M_m * M_m)) / d) * ((D * D) / d))));
	else
		tmp = w0 + ((D * -0.125) * ((D * (M_m * (M_m * (w0 * h)))) / (d * (d * l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[D, 3.7e+115], N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w0 + N[(N[(D * -0.125), $MachinePrecision] * N[(N[(D * N[(M$95$m * N[(M$95$m * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 3.70000000000000006e115

   1. Initial program 85.0%

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

   3. Simplified 78.2%

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

      1. *-commutative N/A

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

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

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in D around 0

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

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

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 66.1%

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

   9. Simplified 66.1%

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

   if 3.70000000000000006e115 < D

   1. Initial program 76.5%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 46.7%

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

   7. Simplified 46.7%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. *-lowering-*.f64 72.1%

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

   9. Applied egg-rr 72.1%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 3.7 \cdot 10^{+115}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{h \cdot \left(M \cdot M\right)}{d} \cdot \frac{D \cdot D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 + \left(D \cdot -0.125\right) \cdot \frac{D \cdot \left(M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)\right)}{d \cdot \left(d \cdot \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;d \leq 1.3 \cdot 10^{-79}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{\frac{\left(M\_m \cdot \left(w0 \cdot h\right)\right) \cdot \frac{M\_m}{d}}{\ell}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= d 1.3e-79)
   (+ w0 (* -0.125 (* (* D D) (/ (/ (* (* M_m (* w0 h)) (/ M_m d)) l) d))))
   w0))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.3e-79) {
		tmp = w0 + (-0.125 * ((D * D) * ((((M_m * (w0 * h)) * (M_m / d)) / l) / d)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d_1 <= 1.3d-79) then
        tmp = w0 + ((-0.125d0) * ((d * d) * ((((m_m * (w0 * h)) * (m_m / d_1)) / l) / d_1)))
    else
        tmp = w0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (d <= 1.3e-79) {
		tmp = w0 + (-0.125 * ((D * D) * ((((M_m * (w0 * h)) * (M_m / d)) / l) / d)));
	} else {
		tmp = w0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if d <= 1.3e-79:
		tmp = w0 + (-0.125 * ((D * D) * ((((M_m * (w0 * h)) * (M_m / d)) / l) / d)))
	else:
		tmp = w0
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (d <= 1.3e-79)
		tmp = Float64(w0 + Float64(-0.125 * Float64(Float64(D * D) * Float64(Float64(Float64(Float64(M_m * Float64(w0 * h)) * Float64(M_m / d)) / l) / d))));
	else
		tmp = w0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (d <= 1.3e-79)
		tmp = w0 + (-0.125 * ((D * D) * ((((M_m * (w0 * h)) * (M_m / d)) / l) / d)));
	else
		tmp = w0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[d, 1.3e-79], N[(w0 + N[(-0.125 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(N[(M$95$m * N[(w0 * h), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if d < 1.29999999999999997e-79

   1. Initial program 83.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 58.1%

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

   7. Simplified 58.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. times-frac N/A

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

       2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f64 65.5%

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

   11. Applied egg-rr 65.5%

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

   if 1.29999999999999997e-79 < d

   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 76.8%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 81.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.3 \cdot 10^{-79}:\\ \;\;\;\;w0 + -0.125 \cdot \left(\left(D \cdot D\right) \cdot \frac{\frac{\left(M \cdot \left(w0 \cdot h\right)\right) \cdot \frac{M}{d}}{\ell}}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3.3 \cdot 10^{-95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{h \cdot \left(M\_m \cdot M\_m\right)}{d} \cdot \frac{D \cdot D}{d}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 3.3e-95)
   w0
   (* w0 (+ 1.0 (* (/ -0.125 l) (* (/ (* h (* M_m M_m)) d) (/ (* D D) d)))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 3.3e-95) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((h * (M_m * M_m)) / d) * ((D * D) / d))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 3.3d-95) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + (((-0.125d0) / l) * (((h * (m_m * m_m)) / d_1) * ((d * d) / d_1))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 3.3e-95) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((-0.125 / l) * (((h * (M_m * M_m)) / d) * ((D * D) / d))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 3.3e-95:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((-0.125 / l) * (((h * (M_m * M_m)) / d) * ((D * D) / d))))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 3.3e-95)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(-0.125 / l) * Float64(Float64(Float64(h * Float64(M_m * M_m)) / d) * Float64(Float64(D * D) / d)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 3.3e-95)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((-0.125 / l) * (((h * (M_m * M_m)) / d) * ((D * D) / d))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 3.3e-95], w0, N[(w0 * N[(1.0 + N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3.3e-95

   1. Initial program 87.9%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 76.6%

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

   if 3.3e-95 < M

   1. Initial program 74.6%

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

   3. Simplified 62.1%

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

      1. *-commutative N/A

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

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

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

   6. Applied egg-rr 72.6%

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

   7. Taylor expanded in D around 0

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

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

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

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

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 47.0%

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

   9. Simplified 47.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3.3 \cdot 10^{-95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \frac{-0.125}{\ell} \cdot \left(\frac{h \cdot \left(M \cdot M\right)}{d} \cdot \frac{D \cdot D}{d}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 3 \cdot 10^{-95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot D\right) \cdot \frac{\left(h \cdot \left(M\_m \cdot M\_m\right)\right) \cdot -0.125}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 3e-95)
   w0
   (* w0 (+ 1.0 (* (* D D) (/ (* (* h (* M_m M_m)) -0.125) (* d (* d l))))))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 3e-95) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D * D) * (((h * (M_m * M_m)) * -0.125) / (d * (d * l)))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 3d-95) then
        tmp = w0
    else
        tmp = w0 * (1.0d0 + ((d * d) * (((h * (m_m * m_m)) * (-0.125d0)) / (d_1 * (d_1 * l)))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 3e-95) {
		tmp = w0;
	} else {
		tmp = w0 * (1.0 + ((D * D) * (((h * (M_m * M_m)) * -0.125) / (d * (d * l)))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 3e-95:
		tmp = w0
	else:
		tmp = w0 * (1.0 + ((D * D) * (((h * (M_m * M_m)) * -0.125) / (d * (d * l)))))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 3e-95)
		tmp = w0;
	else
		tmp = Float64(w0 * Float64(1.0 + Float64(Float64(D * D) * Float64(Float64(Float64(h * Float64(M_m * M_m)) * -0.125) / Float64(d * Float64(d * l))))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 3e-95)
		tmp = w0;
	else
		tmp = w0 * (1.0 + ((D * D) * (((h * (M_m * M_m)) * -0.125) / (d * (d * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 3e-95], w0, N[(w0 * N[(1.0 + N[(N[(D * D), $MachinePrecision] * N[(N[(N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 3e-95

   1. Initial program 87.9%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 76.6%

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

   if 3e-95 < M

   1. Initial program 74.6%

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

   3. Simplified 62.1%

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 + \frac{\frac{h}{\ell} \cdot \left(D \cdot \frac{\frac{M \cdot \left(M \cdot D\right)}{d}}{-4}\right)}{d}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 45.3%

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

   7. Simplified 45.3%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 3 \cdot 10^{-95}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(D \cdot D\right) \cdot \frac{\left(h \cdot \left(M \cdot M\right)\right) \cdot -0.125}{d \cdot \left(d \cdot \ell\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{h \cdot \left(-0.125 \cdot \left(D \cdot D\right)\right)}{d} \cdot \left(M\_m \cdot \frac{M\_m \cdot w0}{d}\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 2.1e+33)
   w0
   (/ (* (/ (* h (* -0.125 (* D D))) d) (* M_m (/ (* M_m w0) d))) l)))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.1e+33) {
		tmp = w0;
	} else {
		tmp = (((h * (-0.125 * (D * D))) / d) * (M_m * ((M_m * w0) / d))) / l;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 2.1d+33) then
        tmp = w0
    else
        tmp = (((h * ((-0.125d0) * (d * d))) / d_1) * (m_m * ((m_m * w0) / d_1))) / l
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.1e+33) {
		tmp = w0;
	} else {
		tmp = (((h * (-0.125 * (D * D))) / d) * (M_m * ((M_m * w0) / d))) / l;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 2.1e+33:
		tmp = w0
	else:
		tmp = (((h * (-0.125 * (D * D))) / d) * (M_m * ((M_m * w0) / d))) / l
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 2.1e+33)
		tmp = w0;
	else
		tmp = Float64(Float64(Float64(Float64(h * Float64(-0.125 * Float64(D * D))) / d) * Float64(M_m * Float64(Float64(M_m * w0) / d))) / l);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.1e+33)
		tmp = w0;
	else
		tmp = (((h * (-0.125 * (D * D))) / d) * (M_m * ((M_m * w0) / d))) / l;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 2.1e+33], w0, N[(N[(N[(N[(h * N[(-0.125 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(M$95$m * N[(N[(M$95$m * w0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.1000000000000001e33

   1. Initial program 87.0%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 74.8%

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

   if 2.1000000000000001e33 < M

   1. Initial program 72.0%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 34.8%

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

   7. Simplified 34.8%

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

   8. Taylor expanded in D around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 21.0%

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

   10. Simplified 21.0%

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

       1. associate-*l* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       7. associate-*l* N/A

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

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

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

       9. *-lowering-*.f64 21.7%

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

   12. Applied egg-rr 21.7%

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

       1. times-frac N/A

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

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

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

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

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. associate-/l* N/A

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

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

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

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

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

       11. *-lowering-*.f64 22.3%

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

   14. Applied egg-rr 22.3%

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

Alternative 11: 61.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{D}{d} \cdot \left(\frac{-0.125}{\ell} \cdot \frac{D \cdot \left(M\_m \cdot \left(w0 \cdot \left(M\_m \cdot h\right)\right)\right)}{d}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 2.1e+33)
   w0
   (* (/ D d) (* (/ -0.125 l) (/ (* D (* M_m (* w0 (* M_m h)))) d)))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.1e+33) {
		tmp = w0;
	} else {
		tmp = (D / d) * ((-0.125 / l) * ((D * (M_m * (w0 * (M_m * h)))) / d));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 2.1d+33) then
        tmp = w0
    else
        tmp = (d / d_1) * (((-0.125d0) / l) * ((d * (m_m * (w0 * (m_m * h)))) / d_1))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.1e+33) {
		tmp = w0;
	} else {
		tmp = (D / d) * ((-0.125 / l) * ((D * (M_m * (w0 * (M_m * h)))) / d));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 2.1e+33:
		tmp = w0
	else:
		tmp = (D / d) * ((-0.125 / l) * ((D * (M_m * (w0 * (M_m * h)))) / d))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 2.1e+33)
		tmp = w0;
	else
		tmp = Float64(Float64(D / d) * Float64(Float64(-0.125 / l) * Float64(Float64(D * Float64(M_m * Float64(w0 * Float64(M_m * h)))) / d)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.1e+33)
		tmp = w0;
	else
		tmp = (D / d) * ((-0.125 / l) * ((D * (M_m * (w0 * (M_m * h)))) / d));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 2.1e+33], w0, N[(N[(D / d), $MachinePrecision] * N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(D * N[(M$95$m * N[(w0 * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.1000000000000001e33

   1. Initial program 87.0%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 74.8%

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

   if 2.1000000000000001e33 < M

   1. Initial program 72.0%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 34.8%

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

   7. Simplified 34.8%

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

   8. Taylor expanded in D around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 21.0%

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

   10. Simplified 21.0%

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

       1. associate-/l/ N/A

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

       2. associate-*l* N/A

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

       3. times-frac N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       16. *-lowering-*.f64 28.6%

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

   12. Applied egg-rr 28.6%

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

       1. *-commutative N/A

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

       2. times-frac N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

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

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

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

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

       14. /-lowering-/.f64 28.8%

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

   14. Applied egg-rr 28.8%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{D}{d} \cdot \left(\frac{-0.125}{\ell} \cdot \frac{D \cdot \left(M \cdot \left(w0 \cdot \left(M \cdot h\right)\right)\right)}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{\ell} \cdot \frac{\left(M\_m \cdot D\right) \cdot \left(D \cdot \left(w0 \cdot \left(M\_m \cdot h\right)\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 2.1e+33)
   w0
   (* (/ -0.125 l) (/ (* (* M_m D) (* D (* w0 (* M_m h)))) (* d d)))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.1e+33) {
		tmp = w0;
	} else {
		tmp = (-0.125 / l) * (((M_m * D) * (D * (w0 * (M_m * h)))) / (d * d));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 2.1d+33) then
        tmp = w0
    else
        tmp = ((-0.125d0) / l) * (((m_m * d) * (d * (w0 * (m_m * h)))) / (d_1 * d_1))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 2.1e+33) {
		tmp = w0;
	} else {
		tmp = (-0.125 / l) * (((M_m * D) * (D * (w0 * (M_m * h)))) / (d * d));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 2.1e+33:
		tmp = w0
	else:
		tmp = (-0.125 / l) * (((M_m * D) * (D * (w0 * (M_m * h)))) / (d * d))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 2.1e+33)
		tmp = w0;
	else
		tmp = Float64(Float64(-0.125 / l) * Float64(Float64(Float64(M_m * D) * Float64(D * Float64(w0 * Float64(M_m * h)))) / Float64(d * d)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 2.1e+33)
		tmp = w0;
	else
		tmp = (-0.125 / l) * (((M_m * D) * (D * (w0 * (M_m * h)))) / (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 2.1e+33], w0, N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(N[(M$95$m * D), $MachinePrecision] * N[(D * N[(w0 * N[(M$95$m * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 2.1000000000000001e33

   1. Initial program 87.0%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 74.8%

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

   if 2.1000000000000001e33 < M

   1. Initial program 72.0%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 34.8%

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

   7. Simplified 34.8%

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

   8. Taylor expanded in D around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 21.0%

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

   10. Simplified 21.0%

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

       1. associate-/l/ N/A

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

       2. associate-*l* N/A

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

       3. times-frac N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       16. *-lowering-*.f64 28.6%

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

   12. Applied egg-rr 28.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

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

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

       5. *-commutative N/A

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

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

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

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

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

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

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

       11. *-lowering-*.f64 28.4%

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

   14. Applied egg-rr 28.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{\ell} \cdot \frac{\left(M \cdot D\right) \cdot \left(D \cdot \left(w0 \cdot \left(M \cdot h\right)\right)\right)}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.4 \cdot 10^{+30}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{\ell} \cdot \frac{D \cdot \left(D \cdot \left(M\_m \cdot \left(M\_m \cdot \left(w0 \cdot h\right)\right)\right)\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d)
 :precision binary64
 (if (<= M_m 4.4e+30)
   w0
   (* (/ -0.125 l) (/ (* D (* D (* M_m (* M_m (* w0 h))))) (* d d)))))

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.4e+30) {
		tmp = w0;
	} else {
		tmp = (-0.125 / l) * ((D * (D * (M_m * (M_m * (w0 * h))))) / (d * d));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_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_m <= 4.4d+30) then
        tmp = w0
    else
        tmp = ((-0.125d0) / l) * ((d * (d * (m_m * (m_m * (w0 * h))))) / (d_1 * d_1))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double w0, double M_m, double D, double h, double l, double d) {
	double tmp;
	if (M_m <= 4.4e+30) {
		tmp = w0;
	} else {
		tmp = (-0.125 / l) * ((D * (D * (M_m * (M_m * (w0 * h))))) / (d * d));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	tmp = 0
	if M_m <= 4.4e+30:
		tmp = w0
	else:
		tmp = (-0.125 / l) * ((D * (D * (M_m * (M_m * (w0 * h))))) / (d * d))
	return tmp

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	tmp = 0.0
	if (M_m <= 4.4e+30)
		tmp = w0;
	else
		tmp = Float64(Float64(-0.125 / l) * Float64(Float64(D * Float64(D * Float64(M_m * Float64(M_m * Float64(w0 * h))))) / Float64(d * d)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(w0, M_m, D, h, l, d)
	tmp = 0.0;
	if (M_m <= 4.4e+30)
		tmp = w0;
	else
		tmp = (-0.125 / l) * ((D * (D * (M_m * (M_m * (w0 * h))))) / (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := If[LessEqual[M$95$m, 4.4e+30], w0, N[(N[(-0.125 / l), $MachinePrecision] * N[(N[(D * N[(D * N[(M$95$m * N[(M$95$m * N[(w0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.4e30

   1. Initial program 87.0%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in h around 0

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

   7. Simplified 74.8%

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

   if 4.4e30 < M

   1. Initial program 72.0%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in h around 0

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 34.8%

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

   7. Simplified 34.8%

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

   8. Taylor expanded in D around inf

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. associate-*r* N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 21.0%

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

   10. Simplified 21.0%

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

       1. associate-/l/ N/A

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

       2. associate-*l* N/A

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

       3. times-frac N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       16. *-lowering-*.f64 28.6%

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

   12. Applied egg-rr 28.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.4 \cdot 10^{+30}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125}{\ell} \cdot \frac{D \cdot \left(D \cdot \left(M \cdot \left(M \cdot \left(w0 \cdot h\right)\right)\right)\right)}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.3% accurate, 216.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ w0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (w0 M_m D h l d) :precision binary64 w0)

C

M_m = fabs(M);
double code(double w0, double M_m, double D, double h, double l, double d) {
	return w0;
}

Fortran

M_m = abs(m)
real(8) function code(w0, m_m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0
end function

Java

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

Python

M_m = math.fabs(M)
def code(w0, M_m, D, h, l, d):
	return w0

Julia

M_m = abs(M)
function code(w0, M_m, D, h, l, d)
	return w0
end

MATLAB

M_m = abs(M);
function tmp = code(w0, M_m, D, h, l, d)
	tmp = w0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[w0_, M$95$m_, D_, h_, l_, d_] := w0

Derivation

1. Initial program 84.2%

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

3. Simplified 77.7%

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

5. Taylor expanded in h around 0

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

7. Simplified 70.3%

   \[\leadsto \color{blue}{w0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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))))))