Henrywood and Agarwal, Equation (9a)

Average Error: 13.6 → 9.3

Time: 10.7s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4.3428642952663706 \cdot 10^{+269}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \left(D \cdot \frac{1}{\frac{d}{M}}\right)\right)}^{2}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1.8024388896205966 \cdot 10^{-164}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

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 (<= (/ h l) -4.3428642952663706e+269)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* 0.5 (* D (/ 1.0 (/ d M)))) 2.0)) l))))
   (if (<= (/ h l) -1.8024388896205966e-164)
     (* w0 (sqrt (- 1.0 (/ (pow (* M (* D (/ 0.5 d))) 2.0) (/ l h)))))
     w0)))

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 ((h / l) <= -4.3428642952663706e+269) {
		tmp = w0 * sqrt((1.0 - ((h * pow((0.5 * (D * (1.0 / (d / M)))), 2.0)) / l)));
	} else if ((h / l) <= -1.8024388896205966e-164) {
		tmp = w0 * sqrt((1.0 - (pow((M * (D * (0.5 / d))), 2.0) / (l / h))));
	} else {
		tmp = w0;
	}
	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 ((h / l) <= (-4.3428642952663706d+269)) then
        tmp = w0 * sqrt((1.0d0 - ((h * ((0.5d0 * (d * (1.0d0 / (d_1 / m)))) ** 2.0d0)) / l)))
    else if ((h / l) <= (-1.8024388896205966d-164)) then
        tmp = w0 * sqrt((1.0d0 - (((m * (d * (0.5d0 / d_1))) ** 2.0d0) / (l / h))))
    else
        tmp = w0
    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 ((h / l) <= -4.3428642952663706e+269) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow((0.5 * (D * (1.0 / (d / M)))), 2.0)) / l)));
	} else if ((h / l) <= -1.8024388896205966e-164) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow((M * (D * (0.5 / d))), 2.0) / (l / h))));
	} else {
		tmp = w0;
	}
	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 (h / l) <= -4.3428642952663706e+269:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow((0.5 * (D * (1.0 / (d / M)))), 2.0)) / l)))
	elif (h / l) <= -1.8024388896205966e-164:
		tmp = w0 * math.sqrt((1.0 - (math.pow((M * (D * (0.5 / d))), 2.0) / (l / h))))
	else:
		tmp = w0
	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(h / l) <= -4.3428642952663706e+269)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(0.5 * Float64(D * Float64(1.0 / Float64(d / M)))) ^ 2.0)) / l))));
	elseif (Float64(h / l) <= -1.8024388896205966e-164)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(M * Float64(D * Float64(0.5 / d))) ^ 2.0) / Float64(l / h)))));
	else
		tmp = w0;
	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 ((h / l) <= -4.3428642952663706e+269)
		tmp = w0 * sqrt((1.0 - ((h * ((0.5 * (D * (1.0 / (d / M)))) ^ 2.0)) / l)));
	elseif ((h / l) <= -1.8024388896205966e-164)
		tmp = w0 * sqrt((1.0 - (((M * (D * (0.5 / d))) ^ 2.0) / (l / h))));
	else
		tmp = w0;
	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[(h / l), $MachinePrecision], -4.3428642952663706e+269], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(0.5 * N[(D * N[(1.0 / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(h / l), $MachinePrecision], -1.8024388896205966e-164], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(M * N[(D * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]]

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 (/.f64 h l) < -4.3428642952663706e269

   1. Initial program 52.9

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

   2. Simplified 52.8

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

   3. Applied egg-rr 23.4

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

   4. Taylor expanded in M around 0 23.5

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

   5. Applied egg-rr 23.7

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

   if -4.3428642952663706e269 < (/.f64 h l) < -1.8024388896205966e-164

   1. Initial program 13.8

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

   2. Simplified 13.7

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

   3. Applied egg-rr 13.7

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

   if -1.8024388896205966e-164 < (/.f64 h l)

   1. Initial program 8.7

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

   2. Simplified 9.0

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

   3. Taylor expanded in M around 0 5.0

      \[\leadsto \color{blue}{w0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -4.3428642952663706 \cdot 10^{+269}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(0.5 \cdot \left(D \cdot \frac{1}{\frac{d}{M}}\right)\right)}^{2}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1.8024388896205966 \cdot 10^{-164}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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))))))