Henrywood and Agarwal, Equation (9a)

Average Error: 14.3 → 8.9

Time: 8.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 10^{+304}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;w0 \cdot \left(M \cdot \left(-\sqrt{\frac{\left(D \cdot D\right) \cdot -0.25}{\frac{d \cdot \left(d \cdot \ell\right)}{h}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

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
 (let* ((t_0 (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
   (if (<= t_0 1e+304)
     (* w0 (sqrt t_0))
     (if (<= t_0 INFINITY)
       (* w0 (* M (- (sqrt (/ (* (* D D) -0.25) (/ (* d (* d 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 t_0 = 1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+304) {
		tmp = w0 * sqrt(t_0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = w0 * (M * -sqrt((((D * D) * -0.25) / ((d * (d * l)) / h))));
	} else {
		tmp = w0;
	}
	return tmp;
}

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 = 1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l));
	double tmp;
	if (t_0 <= 1e+304) {
		tmp = w0 * Math.sqrt(t_0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = w0 * (M * -Math.sqrt((((D * D) * -0.25) / ((d * (d * 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):
	t_0 = 1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))
	tmp = 0
	if t_0 <= 1e+304:
		tmp = w0 * math.sqrt(t_0)
	elif t_0 <= math.inf:
		tmp = w0 * (M * -math.sqrt((((D * D) * -0.25) / ((d * (d * 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)
	t_0 = Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 1e+304)
		tmp = Float64(w0 * sqrt(t_0));
	elseif (t_0 <= Inf)
		tmp = Float64(w0 * Float64(M * Float64(-sqrt(Float64(Float64(Float64(D * D) * -0.25) / Float64(Float64(d * Float64(d * 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)
	t_0 = 1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (t_0 <= 1e+304)
		tmp = w0 * sqrt(t_0);
	elseif (t_0 <= Inf)
		tmp = w0 * (M * -sqrt((((D * D) * -0.25) / ((d * (d * 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_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 1e+304], N[(w0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(w0 * N[(M * (-N[Sqrt[N[(N[(N[(D * D), $MachinePrecision] * -0.25), $MachinePrecision] / N[(N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision] / 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 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 9.9999999999999994e303

   1. Initial program 0.3

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

   if 9.9999999999999994e303 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < +inf.0

   1. Initial program 63.9

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

   2. Applied egg-rr 58.6

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

   3. Taylor expanded in M around 0 59.8

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

   4. Taylor expanded in M around -inf 57.1

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

   5. Simplified 55.4

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

   if +inf.0 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 64.0

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

   2. Taylor expanded in M around 0 15.5

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+304}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;w0 \cdot \left(M \cdot \left(-\sqrt{\frac{\left(D \cdot D\right) \cdot -0.25}{\frac{d \cdot \left(d \cdot \ell\right)}{h}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Reproduce

herbie shell --seed 2022169 
(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))))))