Henrywood and Agarwal, Equation (3)

Average Error: 18.9 → 5.5

Time: 13.6s

Precision: binary64

Cost: 14288

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+297}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+293}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

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 l) (sqrt (/ A V))))))
   (if (<= (* V l) -5e+297)
     t_0
     (if (<= (* V l) -4e-296)
       (/ c0 (/ (sqrt (* V (- l))) (sqrt (- A))))
       (if (<= (* V l) 0.0)
         t_0
         (if (<= (* V l) 1e+293)
           (/ c0 (/ (sqrt (* V l)) (sqrt A)))
           (* c0 (sqrt (/ (/ A l) 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(l) / sqrt((A / V)));
	double tmp;
	if ((V * l) <= -5e+297) {
		tmp = t_0;
	} else if ((V * l) <= -4e-296) {
		tmp = c0 / (sqrt((V * -l)) / sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+293) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / l) / V));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = c0 / (sqrt(l) / sqrt((a / v)))
    if ((v * l) <= (-5d+297)) then
        tmp = t_0
    else if ((v * l) <= (-4d-296)) then
        tmp = c0 / (sqrt((v * -l)) / sqrt(-a))
    else if ((v * l) <= 0.0d0) then
        tmp = t_0
    else if ((v * l) <= 1d+293) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((a / l) / v))
    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 t_0 = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	double tmp;
	if ((V * l) <= -5e+297) {
		tmp = t_0;
	} else if ((V * l) <= -4e-296) {
		tmp = c0 / (Math.sqrt((V * -l)) / Math.sqrt(-A));
	} else if ((V * l) <= 0.0) {
		tmp = t_0;
	} else if ((V * l) <= 1e+293) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / l) / 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(l) / math.sqrt((A / V)))
	tmp = 0
	if (V * l) <= -5e+297:
		tmp = t_0
	elif (V * l) <= -4e-296:
		tmp = c0 / (math.sqrt((V * -l)) / math.sqrt(-A))
	elif (V * l) <= 0.0:
		tmp = t_0
	elif (V * l) <= 1e+293:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / l) / 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(l) / sqrt(Float64(A / V))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+297)
		tmp = t_0;
	elseif (Float64(V * l) <= -4e-296)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * Float64(-l))) / sqrt(Float64(-A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = t_0;
	elseif (Float64(V * l) <= 1e+293)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / l) / 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(l) / sqrt((A / V)));
	tmp = 0.0;
	if ((V * l) <= -5e+297)
		tmp = t_0;
	elseif ((V * l) <= -4e-296)
		tmp = c0 / (sqrt((V * -l)) / sqrt(-A));
	elseif ((V * l) <= 0.0)
		tmp = t_0;
	elseif ((V * l) <= 1e+293)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((A / l) / 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[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+297], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -4e-296], N[(c0 / N[(N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 1e+293], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -4.9999999999999998e297 or -4e-296 < (*.f64 V l) < -0.0

   1. Initial program 51.5

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

   2. Applied egg-rr 19.7

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

   3. Simplified 19.3

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

      [Start] 19.7	\[ \frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}} \]
      associate-/l* [=>] 19.3	\[ \color{blue}{\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}} \]

   if -4.9999999999999998e297 < (*.f64 V l) < -4e-296

   1. Initial program 8.9

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

   2. Applied egg-rr 2.9

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

   3. Simplified 0.4

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

      [Start] 2.9	\[ \frac{c0 \cdot \sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}} \]
      associate-/l* [=>] 0.4	\[ \color{blue}{\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}} \]

   if -0.0 < (*.f64 V l) < 9.9999999999999992e292

   1. Initial program 9.8

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

   2. Applied egg-rr 0.7

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

   if 9.9999999999999992e292 < (*.f64 V l)

   1. Initial program 40.8

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

   2. Applied egg-rr 38.8

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

   3. Simplified 38.8

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

      [Start] 38.8	\[ c0 \cdot \left(\sqrt{A} \cdot \frac{1}{\sqrt{V \cdot \ell}}\right) \]
      *-commutative [<=] 38.8	\[ c0 \cdot \color{blue}{\left(\frac{1}{\sqrt{V \cdot \ell}} \cdot \sqrt{A}\right)} \]
      associate-*l/ [=>] 38.8	\[ c0 \cdot \color{blue}{\frac{1 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}} \]
      *-lft-identity [=>] 38.8	\[ c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \ell}} \]

   4. Applied egg-rr 24.2

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

   5. Taylor expanded in c0 around 0 40.8

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

   6. Simplified 23.6

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

      [Start] 40.8	\[ \sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \]
      *-commutative [=>] 40.8	\[ \color{blue}{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \]
      *-commutative [=>] 40.8	\[ c0 \cdot \sqrt{\frac{A}{\color{blue}{\ell \cdot V}}} \]
      associate-/r* [=>] 23.6	\[ c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+297}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq -4 \cdot 10^{-296}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+293}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.7
Cost	20681

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-228} \lor \neg \left(t_0 \leq 5 \cdot 10^{-192}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 2
Error	5.7
Cost	20036

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{-A}} \cdot \sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+293}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 3
Error	10.6
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+271}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-227}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+293}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 4
Error	8.1
Cost	14288

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-197}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+293}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 5
Error	8.2
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-197}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+293}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 6
Error	8.2
Cost	14288

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+77}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-197}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{+293}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Alternative 7
Error	14.8
Cost	7889

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+119}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-233} \lor \neg \left(V \cdot \ell \leq 4 \cdot 10^{-158}\right) \land V \cdot \ell \leq 5 \cdot 10^{+113}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 8
Error	14.7
Cost	7888

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+119}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-251}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-217}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+113}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 9
Error	14.0
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 10
Error	14.0
Cost	7624

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]

Alternative 11
Error	17.0
Cost	7369

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -4 \cdot 10^{+119} \lor \neg \left(V \cdot \ell \leq 5 \cdot 10^{+113}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 12
Error	18.9
Cost	6848

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

Alternative 13
Error	54.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))