Henrywood and Agarwal, Equation (9a)

Average Error: 14.3 → 9.3

Time: 12.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;w0 \cdot {\left({\left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(M \cdot \left(-\sqrt{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h}{\ell}\right) \cdot -0.25}\right)\right)\\ \end{array} \]

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) 5e+259)
   (*
    w0
    (pow (pow (- 1.0 (/ (* (pow (* 0.5 (/ (* M D) d)) 2.0) h) l)) 0.25) 2.0))
   (* w0 (* M (- (sqrt (* (* (* (/ D d) (/ D d)) (/ h l)) -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 tmp;
	if (pow(((M * D) / (2.0 * d)), 2.0) <= 5e+259) {
		tmp = w0 * pow(pow((1.0 - ((pow((0.5 * ((M * D) / d)), 2.0) * h) / l)), 0.25), 2.0);
	} else {
		tmp = w0 * (M * -sqrt(((((D / d) * (D / d)) * (h / l)) * -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) :: tmp
    if ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) <= 5d+259) then
        tmp = w0 * (((1.0d0 - ((((0.5d0 * ((m * d) / d_1)) ** 2.0d0) * h) / l)) ** 0.25d0) ** 2.0d0)
    else
        tmp = w0 * (m * -sqrt(((((d / d_1) * (d / d_1)) * (h / l)) * (-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 tmp;
	if (Math.pow(((M * D) / (2.0 * d)), 2.0) <= 5e+259) {
		tmp = w0 * Math.pow(Math.pow((1.0 - ((Math.pow((0.5 * ((M * D) / d)), 2.0) * h) / l)), 0.25), 2.0);
	} else {
		tmp = w0 * (M * -Math.sqrt(((((D / d) * (D / d)) * (h / l)) * -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):
	tmp = 0
	if math.pow(((M * D) / (2.0 * d)), 2.0) <= 5e+259:
		tmp = w0 * math.pow(math.pow((1.0 - ((math.pow((0.5 * ((M * D) / d)), 2.0) * h) / l)), 0.25), 2.0)
	else:
		tmp = w0 * (M * -math.sqrt(((((D / d) * (D / d)) * (h / l)) * -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)
	tmp = 0.0
	if ((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) <= 5e+259)
		tmp = Float64(w0 * ((Float64(1.0 - Float64(Float64((Float64(0.5 * Float64(Float64(M * D) / d)) ^ 2.0) * h) / l)) ^ 0.25) ^ 2.0));
	else
		tmp = Float64(w0 * Float64(M * Float64(-sqrt(Float64(Float64(Float64(Float64(D / d) * Float64(D / d)) * Float64(h / l)) * -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)
	tmp = 0.0;
	if ((((M * D) / (2.0 * d)) ^ 2.0) <= 5e+259)
		tmp = w0 * (((1.0 - ((((0.5 * ((M * D) / d)) ^ 2.0) * h) / l)) ^ 0.25) ^ 2.0);
	else
		tmp = w0 * (M * -sqrt(((((D / d) * (D / d)) * (h / l)) * -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_] := If[LessEqual[N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 5e+259], N[(w0 * N[Power[N[Power[N[(1.0 - N[(N[(N[Power[N[(0.5 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(w0 * N[(M * (-N[Sqrt[N[(N[(N[(N[(D / d), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(h / l), $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 2 regimes

2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 5.00000000000000033e259

   1. Initial program 7.1

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

   2. Simplified 7.6

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

   3. Applied egg-rr 3.3

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

   4. Applied egg-rr 3.2

      \[\leadsto w0 \cdot \color{blue}{{\left({\left(1 - \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in M around 0 2.8

      \[\leadsto w0 \cdot {\left({\left(1 - \frac{{\color{blue}{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}}^{2} \cdot h}{\ell}\right)}^{0.25}\right)}^{2} \]

   if 5.00000000000000033e259 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

   1. Initial program 58.8

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

   2. Simplified 54.6

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

   3. Taylor expanded in M around -inf 57.7

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

   4. Simplified 49.4

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;w0 \cdot {\left({\left(1 - \frac{{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}^{2} \cdot h}{\ell}\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(M \cdot \left(-\sqrt{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{h}{\ell}\right) \cdot -0.25}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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))))))