Henrywood and Agarwal, Equation (9a)

Average Error: 13.6 → 9.0

Time: 19.2s

Precision: binary64

Cost: 27784

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-260}:\\ \;\;\;\;w0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+234}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)}{\frac{d}{D}} \cdot -0.25}\\ \end{array} \]

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 (pow (/ (* M D) (* 2.0 d)) 2.0)))
   (if (<= t_0 5e-260)
     w0
     (if (<= t_0 5e+234)
       (* w0 (sqrt (- 1.0 (* t_0 (/ h l)))))
       (*
        w0
        (sqrt
         (+ 1.0 (* (/ (* (/ D d) (* M (* h (/ M l)))) (/ d D)) -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 = pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (t_0 <= 5e-260) {
		tmp = w0;
	} else if (t_0 <= 5e+234) {
		tmp = w0 * sqrt((1.0 - (t_0 * (h / l))));
	} else {
		tmp = w0 * sqrt((1.0 + ((((D / d) * (M * (h * (M / l)))) / (d / D)) * -0.25)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = ((m * d) / (2.0d0 * d_1)) ** 2.0d0
    if (t_0 <= 5d-260) then
        tmp = w0
    else if (t_0 <= 5d+234) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (h / l))))
    else
        tmp = w0 * sqrt((1.0d0 + ((((d / d_1) * (m * (h * (m / l)))) / (d_1 / d)) * (-0.25d0))))
    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 t_0 = Math.pow(((M * D) / (2.0 * d)), 2.0);
	double tmp;
	if (t_0 <= 5e-260) {
		tmp = w0;
	} else if (t_0 <= 5e+234) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (h / l))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((((D / d) * (M * (h * (M / l)))) / (d / D)) * -0.25)));
	}
	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):
	t_0 = math.pow(((M * D) / (2.0 * d)), 2.0)
	tmp = 0
	if t_0 <= 5e-260:
		tmp = w0
	elif t_0 <= 5e+234:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (h / l))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((((D / d) * (M * (h * (M / l)))) / (d / D)) * -0.25)))
	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)
	t_0 = Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0
	tmp = 0.0
	if (t_0 <= 5e-260)
		tmp = w0;
	elseif (t_0 <= 5e+234)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(h / l)))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(D / d) * Float64(M * Float64(h * Float64(M / l)))) / Float64(d / D)) * -0.25))));
	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)
	t_0 = ((M * D) / (2.0 * d)) ^ 2.0;
	tmp = 0.0;
	if (t_0 <= 5e-260)
		tmp = w0;
	elseif (t_0 <= 5e+234)
		tmp = w0 * sqrt((1.0 - (t_0 * (h / l))));
	else
		tmp = w0 * sqrt((1.0 + ((((D / d) * (M * (h * (M / l)))) / (d / D)) * -0.25)));
	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_] := Block[{t$95$0 = N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, 5e-260], w0, If[LessEqual[t$95$0, 5e+234], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(D / d), $MachinePrecision] * N[(M * N[(h * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5.0000000000000003e-260

   1. Initial program 7.0

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

   2. Taylor expanded in M around 0 1.1

      \[\leadsto \color{blue}{w0} \]

   if 5.0000000000000003e-260 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5.0000000000000003e234

   1. Initial program 5.1

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

   if 5.0000000000000003e234 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 55.7

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

   2. Taylor expanded in M around 0 61.0

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

   3. Simplified 57.3

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

   4. Applied egg-rr 48.2

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

   5. Applied egg-rr 45.6

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{-260}:\\ \;\;\;\;w0\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)}{\frac{d}{D}} \cdot -0.25}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.1
Cost	20932

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{-1}{\frac{\ell}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)}{\frac{d}{D}} \cdot -0.25}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	14344

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 + \frac{\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)}{\frac{d}{D}} \cdot -0.25}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-203}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}{\frac{\ell}{h}}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	10.5
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-231}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M \cdot \left(M \cdot \frac{D}{d}\right)}{\frac{\ell}{h}}}{\frac{d}{D}} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Error	10.8
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{D}{d} \cdot \left(M \cdot \left(h \cdot \frac{M}{\ell}\right)\right)}{\frac{d}{D}} \cdot -0.25}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-231}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\frac{M \cdot \left(M \cdot \frac{D}{d}\right)}{\frac{\ell}{h}}}{\frac{d}{D}} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Error	13.6
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2022284 
(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))))))