Henrywood and Agarwal, Equation (3)

Average Error: 19.4 → 5.4

Time: 5.1s

Precision: binary64

\[ \begin{array}{c}[V, l] = \mathsf{sort}([V, l])\\ \end{array} \]

Math

\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

↓

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -9.476766494841006 \cdot 10^{-290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 4.879326628049021 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

FPCore

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

↓

(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (/ (sqrt (/ A V)) (sqrt l)))))
   (if (<= (* V l) (- INFINITY))
     t_0
     (if (<= (* V l) -9.476766494841006e-290)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 4.879326628049021e+306)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (/ c0 (sqrt (/ l (/ A V))))))))))

C

double code(double c0, double A, double V, double l) {
	return c0 * sqrt((A / (V * l)));
}

↓

double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -9.476766494841006e-290) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 4.879326628049021e+306) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 / sqrt((l / (A / V)));
	}
	return tmp;
}

Java

public static double code(double c0, double A, double V, double l) {
	return c0 * Math.sqrt((A / (V * l)));
}

↓

public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -9.476766494841006e-290) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 4.879326628049021e+306) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 / Math.sqrt((l / (A / V)));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	return c0 * math.sqrt((A / (V * l)))

↓

def code(c0, A, V, l):
	t_0 = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -9.476766494841006e-290:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 4.879326628049021e+306:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 / math.sqrt((l / (A / V)))
	return tmp

Julia

function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

↓

function code(c0, A, V, l)
	t_0 = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -9.476766494841006e-290)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 4.879326628049021e+306)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 / sqrt(Float64(l / Float64(A / V))));
	end
	return tmp
end

MATLAB

function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

↓

function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * (sqrt((A / V)) / sqrt(l));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -9.476766494841006e-290)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 4.879326628049021e+306)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 / sqrt((l / (A / V)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -9.476766494841006e-290], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 4.879326628049021e+306], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -9.47676649484100611e-290 < (*.f64 V l) < 0.0

   1. Initial program 52.3

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Applied egg-rr 19.0

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}} \]

   if -inf.0 < (*.f64 V l) < -9.47676649484100611e-290

   1. Initial program 9.5

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Applied egg-rr 0.4

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}} \]

   if 0.0 < (*.f64 V l) < 4.879326628049021e306

   1. Initial program 10.3

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Applied egg-rr 15.5

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}} \]

   3. Applied egg-rr 15.8

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \frac{\ell}{A}}}} \]

   4. Applied egg-rr 15.5

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

   5. Applied egg-rr 0.7

      \[\leadsto \frac{c0}{\color{blue}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}} \]

   if 4.879326628049021e306 < (*.f64 V l)

   1. Initial program 42.8

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]

   2. Applied egg-rr 23.4

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}} \]

   3. Applied egg-rr 24.4

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V \cdot \frac{\ell}{A}}}} \]

   4. Applied egg-rr 24.4

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}} \]

   5. Taylor expanded in V around 0 42.8

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{V \cdot \ell}{A}}}} \]

   6. Simplified 24.4

      \[\leadsto \frac{c0}{\sqrt{\color{blue}{\frac{\ell}{\frac{A}{V}}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -9.476766494841006 \cdot 10^{-290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4.879326628049021 \cdot 10^{+306}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022151 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))