Henrywood and Agarwal, Equation (9a)

Average Error: 14.5 → 8.8

Time: 11.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h}{d \cdot d}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{\ell}{h}}}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;\frac{\sqrt{-0.25 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{\ell}{h}}}}{\frac{d}{w0}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

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
         (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))))
   (if (<= t_0 (- INFINITY))
     (* w0 (* (sqrt (* (* (/ D (/ l D)) (/ h (* d d))) -0.25)) (- M)))
     (if (<= t_0 5e+306)
       (* w0 (sqrt (- 1.0 (/ (pow (* (* M D) (/ 0.5 d)) 2.0) (/ l h)))))
       (if (<= t_0 INFINITY)
         (/ (sqrt (* -0.25 (/ (* (* M D) (* M D)) (/ l h)))) (/ d w0))
         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 = w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = w0 * (sqrt((((D / (l / D)) * (h / (d * d))) * -0.25)) * -M);
	} else if (t_0 <= 5e+306) {
		tmp = w0 * sqrt((1.0 - (pow(((M * D) * (0.5 / d)), 2.0) / (l / h))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = sqrt((-0.25 * (((M * D) * (M * D)) / (l / h)))) / (d / w0);
	} 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 = w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = w0 * (Math.sqrt((((D / (l / D)) * (h / (d * d))) * -0.25)) * -M);
	} else if (t_0 <= 5e+306) {
		tmp = w0 * Math.sqrt((1.0 - (Math.pow(((M * D) * (0.5 / d)), 2.0) / (l / h))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((-0.25 * (((M * D) * (M * D)) / (l / h)))) / (d / w0);
	} 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 = w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = w0 * (math.sqrt((((D / (l / D)) * (h / (d * d))) * -0.25)) * -M)
	elif t_0 <= 5e+306:
		tmp = w0 * math.sqrt((1.0 - (math.pow(((M * D) * (0.5 / d)), 2.0) / (l / h))))
	elif t_0 <= math.inf:
		tmp = math.sqrt((-0.25 * (((M * D) * (M * D)) / (l / h)))) / (d / w0)
	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(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(w0 * Float64(sqrt(Float64(Float64(Float64(D / Float64(l / D)) * Float64(h / Float64(d * d))) * -0.25)) * Float64(-M)));
	elseif (t_0 <= 5e+306)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) * Float64(0.5 / d)) ^ 2.0) / Float64(l / h)))));
	elseif (t_0 <= Inf)
		tmp = Float64(sqrt(Float64(-0.25 * Float64(Float64(Float64(M * D) * Float64(M * D)) / Float64(l / h)))) / Float64(d / w0));
	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 = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = w0 * (sqrt((((D / (l / D)) * (h / (d * d))) * -0.25)) * -M);
	elseif (t_0 <= 5e+306)
		tmp = w0 * sqrt((1.0 - ((((M * D) * (0.5 / d)) ^ 2.0) / (l / h))));
	elseif (t_0 <= Inf)
		tmp = sqrt((-0.25 * (((M * D) * (M * D)) / (l / h)))) / (d / w0);
	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[(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]}, If[LessEqual[t$95$0, (-Infinity)], N[(w0 * N[(N[Sqrt[N[(N[(N[(D / N[(l / D), $MachinePrecision]), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision] * (-M)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[Sqrt[N[(-0.25 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(d / w0), $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 4 regimes

2. if (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < -inf.0

   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 -inf 56.4

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

   3. Simplified 54.1

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

   if -inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 4.99999999999999993e306

   1. Initial program 0.1

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.1

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

   if 4.99999999999999993e306 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

   1. Initial program 63.1

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

   2. Applied egg-rr 60.2

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

   3. Taylor expanded in d around 0 60.0

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

   4. Simplified 55.0

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

   if +inf.0 < (*.f64 w0 (sqrt.f64 (-.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 15.7

      \[\leadsto \color{blue}{w0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq -\infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{D}{\frac{\ell}{D}} \cdot \frac{h}{d \cdot d}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\frac{\ell}{h}}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{-0.25 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{\frac{\ell}{h}}}}{\frac{d}{w0}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))))