Henrywood and Agarwal, Equation (3)

Average Error: 18.8 → 6.4

Time: 14.8s

Precision: binary64

Cost: 20688

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+269}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \frac{\sqrt{A}}{\sqrt{\ell}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) (- INFINITY))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (if (<= (* V l) -1e-296)
     (* c0 (/ (sqrt (- A)) (sqrt (* l (- V)))))
     (if (<= (* V l) 0.0)
       (* c0 (/ 1.0 (* (sqrt l) (sqrt (/ V A)))))
       (if (<= (* V l) 1e+269)
         (* c0 (/ (sqrt A) (sqrt (* V l))))
         (* (/ c0 (sqrt V)) (/ (sqrt A) (sqrt l))))))))

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 tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-296) {
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (1.0 / (sqrt(l) * sqrt((V / A))));
	} else if ((V * l) <= 1e+269) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0 / sqrt(V)) * (sqrt(A) / sqrt(l));
	}
	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 tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-296) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (1.0 / (Math.sqrt(l) * Math.sqrt((V / A))));
	} else if ((V * l) <= 1e+269) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0 / Math.sqrt(V)) * (Math.sqrt(A) / Math.sqrt(l));
	}
	return tmp;
}

Python

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

↓

def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-296:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (V * l) <= 0.0:
		tmp = c0 * (1.0 / (math.sqrt(l) * math.sqrt((V / A))))
	elif (V * l) <= 1e+269:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0 / math.sqrt(V)) * (math.sqrt(A) / math.sqrt(l))
	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)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-296)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 * Float64(1.0 / Float64(sqrt(l) * sqrt(Float64(V / A)))));
	elseif (Float64(V * l) <= 1e+269)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0 / sqrt(V)) * Float64(sqrt(A) / sqrt(l)));
	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)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-296)
		tmp = c0 * (sqrt(-A) / sqrt((l * -V)));
	elseif ((V * l) <= 0.0)
		tmp = c0 * (1.0 / (sqrt(l) * sqrt((V / A))));
	elseif ((V * l) <= 1e+269)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0 / sqrt(V)) * (sqrt(A) / sqrt(l));
	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_] := If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-296], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 * N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e+269], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -inf.0

   1. Initial program 41.6

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

   2. Applied egg-rr 33.2

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

   3. Applied egg-rr 9.7

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

   4. Applied egg-rr 9.0

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

   if -inf.0 < (*.f64 V l) < -1e-296

   1. Initial program 9.9

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

   2. Applied egg-rr 0.4

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

   if -1e-296 < (*.f64 V l) < 0.0

   1. Initial program 61.3

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

   2. Applied egg-rr 37.4

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

   3. Applied egg-rr 28.5

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

   if 0.0 < (*.f64 V l) < 1e269

   1. Initial program 9.2

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

   2. Applied egg-rr 24.1

      \[\leadsto c0 \cdot \color{blue}{\sqrt[3]{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{1.5}}} \]

   3. Taylor expanded in c0 around 0 9.2

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

   4. Simplified 15.5

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 (/.f64 A V) l)) c0): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 A (*.f64 V l)))) c0): 28 points increase in error, 25 points decrease in error

   5. Applied egg-rr 0.7

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

   if 1e269 < (*.f64 V l)

   1. Initial program 34.2

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

   2. Applied egg-rr 31.8

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

   3. Applied egg-rr 31.8

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V}} \cdot \frac{\sqrt{A}}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+269}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \frac{\sqrt{A}}{\sqrt{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.3
Cost	20168

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}} \cdot c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+269}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \frac{\sqrt{A}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 2
Error	9.9
Cost	14416

\[\begin{array}{l} t_0 := c0 \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+248}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	7.0
Cost	14416

\[\begin{array}{l} t_0 := c0 \cdot \frac{1}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-296}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+248}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	10.9
Cost	14288

\[\begin{array}{l} t_0 := \ell \cdot \frac{V}{A}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+278}:\\ \;\;\;\;\frac{c0}{\sqrt{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-307}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot {t_0}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+269}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 5
Error	10.6
Cost	14288

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+248}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	10.2
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ t_1 := c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+189}:\\ \;\;\;\;c0 \cdot \left(t_0 \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-111}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+248}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	14.5
Cost	14024

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 10^{-317}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+306}:\\ \;\;\;\;\frac{c0}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V}}{\sqrt{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 8
Error	14.0
Cost	7952

\[\begin{array}{l} t_0 := \ell \cdot \frac{V}{A}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+278}:\\ \;\;\;\;\frac{c0}{\sqrt{t_0}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-307}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-291}:\\ \;\;\;\;c0 \cdot {t_0}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+178}:\\ \;\;\;\;\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 9
Error	14.1
Cost	7888

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ t_1 := \frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+278}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	14.1
Cost	7888

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ t_1 := c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-282}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	14.1
Cost	7888

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+278}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-291}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 12
Error	14.3
Cost	7888

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+178}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 13
Error	18.7
Cost	6848

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

Reproduce

herbie shell --seed 2022296 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))