Henrywood and Agarwal, Equation (3)

Average Error: 19.3 → 6.0

Time: 13.4s

Precision: binary64

Cost: 20036

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V} \cdot \sqrt{\ell}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-305} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -5e-321)
   (/ c0 (/ (* (sqrt (- V)) (sqrt l)) (sqrt (- A))))
   (if (or (<= (* V l) 5e-305) (not (<= (* V l) 2e+304)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (* (sqrt (/ 1.0 (* V l))) (sqrt A))))))

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) <= -5e-321) {
		tmp = c0 / ((sqrt(-V) * sqrt(l)) / sqrt(-A));
	} else if (((V * l) <= 5e-305) || !((V * l) <= 2e+304)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * (sqrt((1.0 / (V * l))) * sqrt(A));
	}
	return tmp;
}

Fortran

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

↓

real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d-321)) then
        tmp = c0 / ((sqrt(-v) * sqrt(l)) / sqrt(-a))
    else if (((v * l) <= 5d-305) .or. (.not. ((v * l) <= 2d+304))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * (sqrt((1.0d0 / (v * l))) * sqrt(a))
    end if
    code = tmp
end function

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) <= -5e-321) {
		tmp = c0 / ((Math.sqrt(-V) * Math.sqrt(l)) / Math.sqrt(-A));
	} else if (((V * l) <= 5e-305) || !((V * l) <= 2e+304)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * (Math.sqrt((1.0 / (V * l))) * Math.sqrt(A));
	}
	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) <= -5e-321:
		tmp = c0 / ((math.sqrt(-V) * math.sqrt(l)) / math.sqrt(-A))
	elif ((V * l) <= 5e-305) or not ((V * l) <= 2e+304):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * (math.sqrt((1.0 / (V * l))) * math.sqrt(A))
	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) <= -5e-321)
		tmp = Float64(c0 / Float64(Float64(sqrt(Float64(-V)) * sqrt(l)) / sqrt(Float64(-A))));
	elseif ((Float64(V * l) <= 5e-305) || !(Float64(V * l) <= 2e+304))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(1.0 / Float64(V * l))) * sqrt(A)));
	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) <= -5e-321)
		tmp = c0 / ((sqrt(-V) * sqrt(l)) / sqrt(-A));
	elseif (((V * l) <= 5e-305) || ~(((V * l) <= 2e+304)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * (sqrt((1.0 / (V * l))) * sqrt(A));
	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], -5e-321], N[(c0 / N[(N[(N[Sqrt[(-V)], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 5e-305], N[Not[LessEqual[N[(V * l), $MachinePrecision], 2e+304]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(1.0 / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99994e-321

   1. Initial program 15.1

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

   2. Applied egg-rr 17.4

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

   3. Simplified 18.2

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

      [Start] 17.4	\[ \frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}} \]
      associate-/r/ [=>] 18.2	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   4. Applied egg-rr 1.3

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

   if -4.99994e-321 < (*.f64 V l) < 4.99999999999999985e-305 or 1.9999999999999999e304 < (*.f64 V l)

   1. Initial program 53.0

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

   2. Applied egg-rr 31.9

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

   3. Simplified 31.9

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

      [Start] 31.9	\[ \frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}} \]
      associate-/r/ [=>] 31.9	\[ \frac{c0}{\sqrt{\color{blue}{\frac{V}{A} \cdot \ell}}} \]

   4. Taylor expanded in c0 around 0 53.0

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

   5. Simplified 32.2

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

      [Start] 53.0	\[ \sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \]
      associate-/r* [=>] 32.2	\[ \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \cdot c0 \]

   if 4.99999999999999985e-305 < (*.f64 V l) < 1.9999999999999999e304

   1. Initial program 9.8

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

   2. Applied egg-rr 0.4

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V} \cdot \sqrt{\ell}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-305} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.6
Cost	14417

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+204}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{-A} \cdot \frac{c0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-305} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \end{array} \]

Alternative 2
Error	6.4
Cost	14417

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+306}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-305} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \end{array} \]

Alternative 3
Error	6.4
Cost	14417

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-1}{V}}}{\sqrt{\frac{\ell}{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-305} \lor \neg \left(V \cdot \ell \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \end{array} \]

Alternative 4
Error	9.9
Cost	14416

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-156}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 5
Error	9.9
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-156}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{-305}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 6
Error	13.7
Cost	13508

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-156}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+190}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 7
Error	14.9
Cost	7890

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+306} \lor \neg \left(V \cdot \ell \leq -5 \cdot 10^{-309} \lor \neg \left(V \cdot \ell \leq 0\right) \land V \cdot \ell \leq 5 \cdot 10^{+190}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 8
Error	14.8
Cost	7688

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 9
Error	14.8
Cost	7688

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{1}{V}}{\frac{\ell}{A}}}\\ \mathbf{elif}\;t_0 \leq 10^{+291}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \end{array} \]

Alternative 10
Error	14.8
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+284}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 11
Error	19.3
Cost	6848

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

Reproduce

herbie shell --seed 2023045 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))