Henrywood and Agarwal, Equation (9a)

Average Error: 21.51% → 13.31%

Time: 15.1s

Precision: binary64

Cost: 7744

Math

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

↓

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

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
 (let* ((t_0 (* (/ D d) M)))
   (* w0 (sqrt (+ 1.0 (* (* t_0 (/ h (/ l t_0))) -0.25))))))

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 t_0 = (D / d) * M;
	return w0 * sqrt((1.0 + ((t_0 * (h / (l / t_0))) * -0.25)));
}

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) :: t_0
    t_0 = (d / d_1) * m
    code = w0 * sqrt((1.0d0 + ((t_0 * (h / (l / t_0))) * (-0.25d0))))
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 t_0 = (D / d) * M;
	return w0 * Math.sqrt((1.0 + ((t_0 * (h / (l / t_0))) * -0.25)));
}

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):
	t_0 = (D / d) * M
	return w0 * math.sqrt((1.0 + ((t_0 * (h / (l / t_0))) * -0.25)))

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)
	t_0 = Float64(Float64(D / d) * M)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(t_0 * Float64(h / Float64(l / t_0))) * -0.25))))
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 = code(w0, M, D, h, l, d)
	t_0 = (D / d) * M;
	tmp = w0 * sqrt((1.0 + ((t_0 * (h / (l / t_0))) * -0.25)));
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_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 + N[(N[(t$95$0 * N[(h / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 21.51

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

2. Simplified 21.62

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

   [Start] 21.51	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   associate-*l/ [<=] 21.62	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
   *-commutative [=>] 21.62	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

3. Taylor expanded in D around 0 50.39

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

4. Simplified 27.51

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

   [Start] 50.39	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}} \]
   *-commutative [=>] 50.39	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{\ell \cdot {d}^{2}}} \]
   *-commutative [=>] 50.39	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{\color{blue}{{d}^{2} \cdot \ell}}} \]
   *-commutative [<=] 50.39	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{{D}^{2} \cdot \color{blue}{\left({M}^{2} \cdot h\right)}}{{d}^{2} \cdot \ell}} \]
   associate-*r* [=>] 49.07	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left({D}^{2} \cdot {M}^{2}\right) \cdot h}}{{d}^{2} \cdot \ell}} \]
   unpow2 [=>] 49.07	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot {M}^{2}\right) \cdot h}{{d}^{2} \cdot \ell}} \]
   unpow2 [=>] 49.07	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(\left(D \cdot D\right) \cdot \color{blue}{\left(M \cdot M\right)}\right) \cdot h}{{d}^{2} \cdot \ell}} \]
   swap-sqr [<=] 35	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right)} \cdot h}{{d}^{2} \cdot \ell}} \]
   associate-*l* [=>] 32.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}}{{d}^{2} \cdot \ell}} \]
   unpow2 [=>] 32.4	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}{\color{blue}{\left(d \cdot d\right)} \cdot \ell}} \]
   associate-*l* [=>] 27.51	\[ w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\left(D \cdot M\right) \cdot \left(\left(D \cdot M\right) \cdot h\right)}{\color{blue}{d \cdot \left(d \cdot \ell\right)}}} \]

5. Applied egg-rr 19.61

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

6. Applied egg-rr 13.31

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

7. Final simplification 13.31

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

Alternatives

Alternative 1
Error	15.53%
Cost	8264

\[\begin{array}{l} t_0 := \frac{D}{d} \cdot M\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -4 \cdot 10^{-136}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(t_0 \cdot \left(t_0 \cdot \frac{h}{\ell}\right)\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	21.63%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023115 
(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))))))