Average Error: 13.8 → 9.1
Time: 7.1s
Precision: binary64
\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{D}{2 \cdot \frac{d}{M}}\\ w0 \cdot \sqrt{1 + h \cdot \left(t_0 \cdot \left(t_0 \cdot \frac{-1}{\ell}\right)\right)} \end{array} \]
(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
 (let* ((t_0 (/ D (* 2.0 (/ d M)))))
   (* w0 (sqrt (+ 1.0 (* h (* t_0 (* t_0 (/ -1.0 l)))))))))
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 t_0 = D / (2.0 * (d / M));
	return w0 * sqrt((1.0 + (h * (t_0 * (t_0 * (-1.0 / l))))));
}

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) :: t_0
    t_0 = d / (2.0d0 * (d_1 / m))
    code = w0 * sqrt((1.0d0 + (h * (t_0 * (t_0 * ((-1.0d0) / l))))))
end function

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 t_0 = D / (2.0 * (d / M));
	return w0 * Math.sqrt((1.0 + (h * (t_0 * (t_0 * (-1.0 / l))))));
}

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):
	t_0 = D / (2.0 * (d / M))
	return w0 * math.sqrt((1.0 + (h * (t_0 * (t_0 * (-1.0 / l))))))

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)
	t_0 = Float64(D / Float64(2.0 * Float64(d / M)))
	return Float64(w0 * sqrt(Float64(1.0 + Float64(h * Float64(t_0 * Float64(t_0 * Float64(-1.0 / l)))))))
end

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 = code(w0, M, D, h, l, d)
	t_0 = D / (2.0 * (d / M));
	tmp = w0 * sqrt((1.0 + (h * (t_0 * (t_0 * (-1.0 / l))))));
end

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_] := Block[{t$95$0 = N[(D / N[(2.0 * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 + N[(h * N[(t$95$0 * N[(t$95$0 * N[(-1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := \frac{D}{2 \cdot \frac{d}{M}}\\
w0 \cdot \sqrt{1 + h \cdot \left(t_0 \cdot \left(t_0 \cdot \frac{-1}{\ell}\right)\right)}
\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.8

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

  2. Applied egg-rr10.9

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

  3. Applied egg-rr10.7

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

  4. Applied egg-rr10.7

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

  5. Applied egg-rr9.1

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

  6. Final simplification9.1

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

Reproduce

herbie shell --seed 2022210 
(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))))))