Henrywood and Agarwal, Equation (9a)

Average Error: 14.2 → 9.6

Time: 18.5s

Precision: binary64

Cost: 7876

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;D \leq 2.1 \cdot 10^{+155}:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(\frac{h}{\frac{d}{D \cdot M} \cdot \frac{\ell}{0.25}} \cdot \frac{D}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \left(\left(\frac{M}{d} \cdot \frac{D}{-4}\right) \cdot \left(\frac{M}{d} \cdot \frac{h}{\ell}\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (w0 M D h l d)
 :precision binary64
 (if (<= D 2.1e+155)
   (* w0 (sqrt (- 1.0 (* M (* (/ h (* (/ d (* D M)) (/ l 0.25))) (/ D d))))))
   (* w0 (sqrt (+ 1.0 (* D (* (* (/ M d) (/ D -4.0)) (* (/ M d) (/ 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))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2.1e+155) {
		tmp = w0 * sqrt((1.0 - (M * ((h / ((d / (D * M)) * (l / 0.25))) * (D / d)))));
	} else {
		tmp = w0 * sqrt((1.0 + (D * (((M / d) * (D / -4.0)) * ((M / d) * (h / l))))));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 2.1d+155) then
        tmp = w0 * sqrt((1.0d0 - (m * ((h / ((d_1 / (d * m)) * (l / 0.25d0))) * (d / d_1)))))
    else
        tmp = w0 * sqrt((1.0d0 + (d * (((m / d_1) * (d / (-4.0d0))) * ((m / d_1) * (h / l))))))
    end if
    code = tmp
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))));
}

↓

public static double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if (D <= 2.1e+155) {
		tmp = w0 * Math.sqrt((1.0 - (M * ((h / ((d / (D * M)) * (l / 0.25))) * (D / d)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + (D * (((M / d) * (D / -4.0)) * ((M / d) * (h / l))))));
	}
	return tmp;
}

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))))

↓

def code(w0, M, D, h, l, d):
	tmp = 0
	if D <= 2.1e+155:
		tmp = w0 * math.sqrt((1.0 - (M * ((h / ((d / (D * M)) * (l / 0.25))) * (D / d)))))
	else:
		tmp = w0 * math.sqrt((1.0 + (D * (((M / d) * (D / -4.0)) * ((M / d) * (h / l))))))
	return tmp

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

↓

function code(w0, M, D, h, l, d)
	tmp = 0.0
	if (D <= 2.1e+155)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(M * Float64(Float64(h / Float64(Float64(d / Float64(D * M)) * Float64(l / 0.25))) * Float64(D / d))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(D * Float64(Float64(Float64(M / d) * Float64(D / -4.0)) * Float64(Float64(M / d) * Float64(h / l)))))));
	end
	return tmp
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

↓

function tmp_2 = code(w0, M, D, h, l, d)
	tmp = 0.0;
	if (D <= 2.1e+155)
		tmp = w0 * sqrt((1.0 - (M * ((h / ((d / (D * M)) * (l / 0.25))) * (D / d)))));
	else
		tmp = w0 * sqrt((1.0 + (D * (((M / d) * (D / -4.0)) * ((M / d) * (h / l))))));
	end
	tmp_2 = tmp;
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]

↓

code[w0_, M_, D_, h_, l_, d_] := If[LessEqual[D, 2.1e+155], N[(w0 * N[Sqrt[N[(1.0 - N[(M * N[(N[(h / N[(N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision] * N[(l / 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(D * N[(N[(N[(M / d), $MachinePrecision] * N[(D / -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 2.1e155

   1. Initial program 12.1

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

   2. Applied egg-rr 13.0

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

   3. Simplified 13.8

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

      [Start] 13.0	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(-D\right)}{\frac{\ell}{h} \cdot \left(\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \left(d \cdot -2\right)\right)}} \]
      *-commutative [<=] 13.0	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(-D\right)}{\frac{\ell}{h} \cdot \color{blue}{\left(\left(d \cdot -2\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)\right)}}} \]
      times-frac [=>] 14.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{M}{\frac{\ell}{h}} \cdot \frac{-D}{\left(d \cdot -2\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}}} \]
      associate-/r/ [=>] 13.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M}{\ell} \cdot h\right)} \cdot \frac{-D}{\left(d \cdot -2\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}} \]
      associate-*r* [=>] 13.9	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\color{blue}{\left(\left(d \cdot -2\right) \cdot \frac{2}{D}\right) \cdot \frac{d}{M}}}} \]
      *-commutative [=>] 13.9	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\color{blue}{\frac{d}{M} \cdot \left(\left(d \cdot -2\right) \cdot \frac{2}{D}\right)}}} \]
      associate-*r/ [=>] 13.8	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \color{blue}{\frac{\left(d \cdot -2\right) \cdot 2}{D}}}} \]
      associate-*l* [=>] 13.8	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{\color{blue}{d \cdot \left(-2 \cdot 2\right)}}{D}}} \]
      metadata-eval [=>] 13.8	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot \color{blue}{-4}}{D}}} \]

   4. Applied egg-rr 13.8

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

   5. Simplified 8.8

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

      [Start] 13.8	\[ w0 \cdot \sqrt{1 - \left(\left(\left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}} + 1\right) - 1\right)} \]
      associate--l+ [=>] 13.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}} + \left(1 - 1\right)\right)}} \]
      metadata-eval [=>] 13.8	\[ w0 \cdot \sqrt{1 - \left(\left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}} + \color{blue}{0}\right)} \]
      +-rgt-identity [=>] 13.8	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}} \]
      associate-*l* [=>] 12.1	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{M}{\ell} \cdot \left(h \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}\right)}} \]
      associate-*l/ [=>] 10.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{M \cdot \left(h \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}\right)}{\ell}}} \]
      *-commutative [=>] 10.0	\[ w0 \cdot \sqrt{1 - \frac{\color{blue}{\left(h \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}\right) \cdot M}}{\ell}} \]
      associate-*l/ [<=] 10.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{h \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}{\ell} \cdot M}} \]
      *-commutative [=>] 10.0	\[ w0 \cdot \sqrt{1 - \color{blue}{M \cdot \frac{h \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}{\ell}}} \]
      associate-/l* [=>] 9.9	\[ w0 \cdot \sqrt{1 - M \cdot \color{blue}{\frac{h}{\frac{\ell}{\frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}}}} \]
      neg-mul-1 [=>] 9.9	\[ w0 \cdot \sqrt{1 - M \cdot \frac{h}{\frac{\ell}{\frac{\color{blue}{-1 \cdot D}}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}}} \]
      *-commutative [=>] 9.9	\[ w0 \cdot \sqrt{1 - M \cdot \frac{h}{\frac{\ell}{\frac{-1 \cdot D}{\color{blue}{\frac{d \cdot -4}{D} \cdot \frac{d}{M}}}}}} \]
      associate-*r/ [=>] 11.9	\[ w0 \cdot \sqrt{1 - M \cdot \frac{h}{\frac{\ell}{\frac{-1 \cdot D}{\color{blue}{\frac{\frac{d \cdot -4}{D} \cdot d}{M}}}}}} \]
      associate-*l/ [<=] 9.6	\[ w0 \cdot \sqrt{1 - M \cdot \frac{h}{\frac{\ell}{\frac{-1 \cdot D}{\color{blue}{\frac{\frac{d \cdot -4}{D}}{M} \cdot d}}}}} \]
      times-frac [=>] 8.8	\[ w0 \cdot \sqrt{1 - M \cdot \frac{h}{\frac{\ell}{\color{blue}{\frac{-1}{\frac{\frac{d \cdot -4}{D}}{M}} \cdot \frac{D}{d}}}}} \]

   6. Applied egg-rr 7.6

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

   if 2.1e155 < D

   1. Initial program 25.4

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

   2. Applied egg-rr 24.9

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

   3. Simplified 27.2

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

      [Start] 24.9	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(-D\right)}{\frac{\ell}{h} \cdot \left(\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \left(d \cdot -2\right)\right)}} \]
      *-commutative [<=] 24.9	\[ w0 \cdot \sqrt{1 - \frac{M \cdot \left(-D\right)}{\frac{\ell}{h} \cdot \color{blue}{\left(\left(d \cdot -2\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)\right)}}} \]
      times-frac [=>] 28.0	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{M}{\frac{\ell}{h}} \cdot \frac{-D}{\left(d \cdot -2\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}}} \]
      associate-/r/ [=>] 27.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M}{\ell} \cdot h\right)} \cdot \frac{-D}{\left(d \cdot -2\right) \cdot \left(\frac{2}{D} \cdot \frac{d}{M}\right)}} \]
      associate-*r* [=>] 27.2	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\color{blue}{\left(\left(d \cdot -2\right) \cdot \frac{2}{D}\right) \cdot \frac{d}{M}}}} \]
      *-commutative [=>] 27.2	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\color{blue}{\frac{d}{M} \cdot \left(\left(d \cdot -2\right) \cdot \frac{2}{D}\right)}}} \]
      associate-*r/ [=>] 27.2	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \color{blue}{\frac{\left(d \cdot -2\right) \cdot 2}{D}}}} \]
      associate-*l* [=>] 27.2	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{\color{blue}{d \cdot \left(-2 \cdot 2\right)}}{D}}} \]
      metadata-eval [=>] 27.2	\[ w0 \cdot \sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot \color{blue}{-4}}{D}}} \]

   4. Applied egg-rr 27.5

      \[\leadsto w0 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}\right)} - 1\right)} \]

   5. Simplified 21.9

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

      [Start] 27.5	\[ w0 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}\right)} - 1\right) \]
      expm1-def [=>] 27.5	\[ w0 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}\right)\right)} \]
      expm1-log1p [=>] 27.2	\[ w0 \cdot \color{blue}{\sqrt{1 - \left(\frac{M}{\ell} \cdot h\right) \cdot \frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}} \]
      *-commutative [=>] 27.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{-D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}} \cdot \left(\frac{M}{\ell} \cdot h\right)}} \]
      distribute-frac-neg [=>] 27.2	\[ w0 \cdot \sqrt{1 - \color{blue}{\left(-\frac{D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}\right)} \cdot \left(\frac{M}{\ell} \cdot h\right)} \]
      cancel-sign-sub [=>] 27.2	\[ w0 \cdot \sqrt{\color{blue}{1 + \frac{D}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}} \cdot \left(\frac{M}{\ell} \cdot h\right)}} \]
      associate-*l/ [=>] 25.8	\[ w0 \cdot \sqrt{1 + \color{blue}{\frac{D \cdot \left(\frac{M}{\ell} \cdot h\right)}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}}} \]
      *-commutative [<=] 25.8	\[ w0 \cdot \sqrt{1 + \frac{\color{blue}{\left(\frac{M}{\ell} \cdot h\right) \cdot D}}{\frac{d}{M} \cdot \frac{d \cdot -4}{D}}} \]
      *-commutative [=>] 25.8	\[ w0 \cdot \sqrt{1 + \frac{\left(\frac{M}{\ell} \cdot h\right) \cdot D}{\color{blue}{\frac{d \cdot -4}{D} \cdot \frac{d}{M}}}} \]
      times-frac [=>] 23.7	\[ w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{M}{\ell} \cdot h}{\frac{d \cdot -4}{D}} \cdot \frac{D}{\frac{d}{M}}}} \]
      associate-/l* [<=] 26.0	\[ w0 \cdot \sqrt{1 + \frac{\frac{M}{\ell} \cdot h}{\frac{d \cdot -4}{D}} \cdot \color{blue}{\frac{D \cdot M}{d}}} \]
      associate-*r/ [<=] 23.5	\[ w0 \cdot \sqrt{1 + \frac{\frac{M}{\ell} \cdot h}{\frac{d \cdot -4}{D}} \cdot \color{blue}{\left(D \cdot \frac{M}{d}\right)}} \]
      associate-/r/ [<=] 26.7	\[ w0 \cdot \sqrt{1 + \color{blue}{\frac{\frac{M}{\ell} \cdot h}{\frac{\frac{d \cdot -4}{D}}{D \cdot \frac{M}{d}}}}} \]
      associate-/l* [<=] 23.4	\[ w0 \cdot \sqrt{1 + \color{blue}{\frac{\left(\frac{M}{\ell} \cdot h\right) \cdot \left(D \cdot \frac{M}{d}\right)}{\frac{d \cdot -4}{D}}}} \]

   6. Taylor expanded in M around 0 21.9

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

   7. Simplified 20.8

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

      [Start] 21.9	\[ w0 \cdot \sqrt{1 + D \cdot \left(\left(\frac{M}{d} \cdot \frac{D}{-4}\right) \cdot \frac{M \cdot h}{d \cdot \ell}\right)} \]
      times-frac [=>] 20.8	\[ w0 \cdot \sqrt{1 + D \cdot \left(\left(\frac{M}{d} \cdot \frac{D}{-4}\right) \cdot \color{blue}{\left(\frac{M}{d} \cdot \frac{h}{\ell}\right)}\right)} \]
      *-commutative [=>] 20.8	\[ w0 \cdot \sqrt{1 + D \cdot \left(\left(\frac{M}{d} \cdot \frac{D}{-4}\right) \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \frac{M}{d}\right)}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 2.1 \cdot 10^{+155}:\\ \;\;\;\;w0 \cdot \sqrt{1 - M \cdot \left(\frac{h}{\frac{d}{D \cdot M} \cdot \frac{\ell}{0.25}} \cdot \frac{D}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \left(\left(\frac{M}{d} \cdot \frac{D}{-4}\right) \cdot \left(\frac{M}{d} \cdot \frac{h}{\ell}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1.2 \cdot 10^{+253}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq 2 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \left(\left(\frac{M}{d} \cdot \frac{D}{-4}\right) \cdot \left(\frac{M}{d} \cdot \frac{h}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	11.1
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -5 \cdot 10^{+266}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq 5 \cdot 10^{+241}:\\ \;\;\;\;w0 \cdot \sqrt{1 + M \cdot \left(\left(\frac{D}{d} \cdot \frac{h}{\ell}\right) \cdot \left(\frac{M}{\frac{d}{D}} \cdot -0.25\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(1 + \left(h \cdot \frac{\frac{\frac{M}{d}}{\frac{\ell}{D}} \cdot \left(D \cdot M\right)}{d}\right) \cdot -0.125\right)\\ \end{array} \]

Alternative 3
Error	14.4
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023011 
(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))))))