Henrywood and Agarwal, Equation (9a)

Average Error: 13.9 → 8.7

Time: 15.8s

Precision: binary64

Cost: 7872

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(D \cdot M\right) \cdot 0.5}{d}\\ w0 \cdot \sqrt{1 - \frac{h \cdot t_0}{\frac{\ell}{t_0}}} \end{array} \]

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 M) 0.5) d)))
   (* w0 (sqrt (- 1.0 (/ (* h t_0) (/ l t_0)))))))

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 * M) * 0.5) / d;
	return w0 * sqrt((1.0 - ((h * t_0) / (l / t_0))));
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    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 * m) * 0.5d0) / d_1
    code = w0 * sqrt((1.0d0 - ((h * t_0) / (l / t_0))))
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 * M) * 0.5) / d;
	return w0 * Math.sqrt((1.0 - ((h * t_0) / (l / t_0))));
}

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 * M) * 0.5) / d
	return w0 * math.sqrt((1.0 - ((h * t_0) / (l / t_0))))

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(Float64(D * M) * 0.5) / d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * t_0) / Float64(l / t_0)))))
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 * M) * 0.5) / d;
	tmp = w0 * sqrt((1.0 - ((h * t_0) / (l / t_0))));
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[(N[(D * M), $MachinePrecision] * 0.5), $MachinePrecision] / d), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * t$95$0), $MachinePrecision] / N[(l / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 13.9

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

2. Simplified 14.1

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

   [Start] 13.9	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   associate-*l/ [<=] 14.1	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2 \cdot d} \cdot D\right)}}^{2} \cdot \frac{h}{\ell}} \]
   *-commutative [=>] 14.1	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(D \cdot \frac{M}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

3. Applied egg-rr 10.6

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

4. Simplified 10.6

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

   [Start] 10.6	\[ w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot h}}} \]
   associate-/r/ [=>] 10.6	\[ w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot h\right)}} \]
   *-commutative [=>] 10.6	\[ w0 \cdot \sqrt{1 - \frac{1}{\ell} \cdot \color{blue}{\left(h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}} \]

5. Applied egg-rr 8.7

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

6. Final simplification 8.7

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

Alternatives

Alternative 1
Error	9.0
Cost	7876

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;D \leq 1.6 \cdot 10^{+85}:\\ \;\;\;\;w0 \cdot \sqrt{1 - h \cdot \left(\frac{\frac{M}{\frac{d}{D}}}{\ell} \cdot \frac{M}{\frac{d}{D} \cdot 4}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(t_0 \cdot \left(0.25 \cdot \frac{h}{\ell}\right)\right)}\\ \end{array} \]

Alternative 2
Error	9.2
Cost	7744

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

Alternative 3
Error	13.7
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023056 
(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))))))