Henrywood and Agarwal, Equation (9a)

Average Error: 14.1 → 9.5

Time: 13.8s

Precision: binary64

Cost: 20804

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{\ell} \cdot \frac{M \cdot 0.25}{\frac{1}{h}}\right)}\\ \end{array} \]

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 (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 2e+302)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* D (* 0.5 (/ M d))) 2.0)) l))))
   (*
    w0
    (sqrt
     (- 1.0 (* (* (/ D d) (/ D d)) (* (/ M l) (/ (* M 0.25) (/ 1.0 h)))))))))

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 (pow(((M * D) / (2.0 * d)), 2.0) <= 2e+302) {
		tmp = w0 * sqrt((1.0 - ((h * pow((D * (0.5 * (M / d))), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 - (((D / d) * (D / d)) * ((M / l) * ((M * 0.25) / (1.0 / h))))));
	}
	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 ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 2d+302) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((d * (0.5d0 * (m / d_1))) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 - (((d / d_1) * (d / d_1)) * ((m / l) * ((m * 0.25d0) / (1.0d0 / h))))))
    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 (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 2e+302) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((D * (0.5 * (M / d))), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 - (((D / d) * (D / d)) * ((M / l) * ((M * 0.25) / (1.0 / h))))));
	}
	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 math.pow(((M * D) / (2.0 * d)), 2.0) <= 2e+302:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((D * (0.5 * (M / d))), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 - (((D / d) * (D / d)) * ((M / l) * ((M * 0.25) / (1.0 / h))))))
	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 ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 2e+302)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(D * Float64(0.5 * Float64(M / d))) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(Float64(M / l) * Float64(Float64(M * 0.25) / Float64(1.0 / h)))))));
	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 ((((M * D) / (2.0 * d)) ^ 2.0) <= 2e+302)
		tmp = w0 * sqrt((1.0 - ((h * ((D * (0.5 * (M / d))) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 - (((D / d) * (D / d)) * ((M / l) * ((M * 0.25) / (1.0 / h))))));
	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[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 2e+302], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(D * N[(0.5 * N[(M / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(N[(M / l), $MachinePrecision] * N[(N[(M * 0.25), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 2.0000000000000002e302

   1. Initial program 7.2

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

   2. Applied egg-rr 7.8

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

   3. Applied egg-rr 3.5

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

   if 2.0000000000000002e302 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 62.7

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

   2. Simplified 59.5

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

   3. Applied egg-rr 60.2

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

   4. Taylor expanded in M around 0 61.0

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

   5. Simplified 58.6

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

   6. Applied egg-rr 51.0

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{\ell} \cdot \frac{M \cdot 0.25}{\frac{1}{h}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.7
Cost	7876

\[\begin{array}{l} \mathbf{if}\;D \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;w0 \cdot \sqrt{1 - 0.25 \cdot \frac{\frac{\frac{h \cdot \left(D \cdot D\right)}{\frac{d}{M}}}{\frac{d}{M}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	13.7
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2022214 
(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))))))