Henrywood and Agarwal, Equation (9a)

Average Error: 14.1 → 9.5

Time: 9.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+286}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{\left(\left(M \cdot \frac{M}{\ell}\right) \cdot \frac{h}{d \cdot d}\right) \cdot -0.25}\right)\\ \end{array} \]

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
 (let* ((t_0 (pow (/ (* M D) (* 2.0 d)) 2.0)))
   (if (<= t_0 2e+57)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (* (* M 0.5) (/ D d)) 2.0)) l))))
     (if (<= t_0 1e+286)
       (* w0 (sqrt (- 1.0 (* t_0 (/ h l)))))
       (* w0 (* D (sqrt (* (* (* M (/ M l)) (/ h (* 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 <= 2e+57) {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	} else if (t_0 <= 1e+286) {
		tmp = w0 * sqrt((1.0 - (t_0 * (h / l))));
	} else {
		tmp = w0 * (D * sqrt((((M * (M / l)) * (h / (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 <= 2d+57) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((m * 0.5d0) * (d / d_1)) ** 2.0d0)) / l)))
    else if (t_0 <= 1d+286) then
        tmp = w0 * sqrt((1.0d0 - (t_0 * (h / l))))
    else
        tmp = w0 * (d * sqrt((((m * (m / l)) * (h / (d_1 * d_1))) * (-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 <= 2e+57) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	} else if (t_0 <= 1e+286) {
		tmp = w0 * Math.sqrt((1.0 - (t_0 * (h / l))));
	} else {
		tmp = w0 * (D * Math.sqrt((((M * (M / l)) * (h / (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 <= 2e+57:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M * 0.5) * (D / d)), 2.0)) / l)))
	elif t_0 <= 1e+286:
		tmp = w0 * math.sqrt((1.0 - (t_0 * (h / l))))
	else:
		tmp = w0 * (D * math.sqrt((((M * (M / l)) * (h / (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 <= 2e+57)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / l))));
	elseif (t_0 <= 1e+286)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(h / l)))));
	else
		tmp = Float64(w0 * Float64(D * sqrt(Float64(Float64(Float64(M * Float64(M / l)) * Float64(h / 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 <= 2e+57)
		tmp = w0 * sqrt((1.0 - ((h * (((M * 0.5) * (D / d)) ^ 2.0)) / l)));
	elseif (t_0 <= 1e+286)
		tmp = w0 * sqrt((1.0 - (t_0 * (h / l))));
	else
		tmp = w0 * (D * sqrt((((M * (M / l)) * (h / (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, 2e+57], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+286], N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(D * N[Sqrt[N[(N[(N[(M * N[(M / l), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 2.0000000000000001e57

   1. Initial program 6.4

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

   2. Applied egg-rr 1.4

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

   if 2.0000000000000001e57 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 1.00000000000000003e286

   1. Initial program 9.7

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

   if 1.00000000000000003e286 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 61.8

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

   2. Simplified 58.6

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

   3. Applied egg-rr 59.6

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{\ell}}}} \]

   4. Applied egg-rr 59.6

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

   5. Taylor expanded in D around inf 57.3

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

   6. Simplified 55.8

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 10^{+286}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(D \cdot \sqrt{\left(\left(M \cdot \frac{M}{\ell}\right) \cdot \frac{h}{d \cdot d}\right) \cdot -0.25}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022171 
(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))))))