Henrywood and Agarwal, Equation (3)

Average Error: 19.6 → 5.8

Time: 12.8s

Precision: binary64

Cost: 20036

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

FPCore

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

↓

(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) 0.0)
   (* (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)) c0)
   (if (<= (* V l) 2e+292)
     (/ c0 (/ (sqrt (* V l)) (sqrt A)))
     (* c0 (sqrt (/ (/ A V) 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) <= 0.0) {
		tmp = ((sqrt(-A) / sqrt(-V)) / sqrt(l)) * c0;
	} else if ((V * l) <= 2e+292) {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	} else {
		tmp = c0 * sqrt(((A / V) / l));
	}
	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) <= 0.0d0) then
        tmp = ((sqrt(-a) / sqrt(-v)) / sqrt(l)) * c0
    else if ((v * l) <= 2d+292) then
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    else
        tmp = c0 * sqrt(((a / v) / l))
    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) <= 0.0) {
		tmp = ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l)) * c0;
	} else if ((V * l) <= 2e+292) {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	} else {
		tmp = c0 * Math.sqrt(((A / V) / 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) <= 0.0:
		tmp = ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l)) * c0
	elif (V * l) <= 2e+292:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	else:
		tmp = c0 * math.sqrt(((A / V) / 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) <= 0.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)) * c0);
	elseif (Float64(V * l) <= 2e+292)
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / 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) <= 0.0)
		tmp = ((sqrt(-A) / sqrt(-V)) / sqrt(l)) * c0;
	elseif ((V * l) <= 2e+292)
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	else
		tmp = c0 * sqrt(((A / V) / 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], 0.0], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+292], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 0.0

   1. Initial program 23.2

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

   2. Applied egg-rr 15.9

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

   3. Simplified 14.5

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

      [Start] 15.9	\[ \frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}} \]
      *-commutative [=>] 15.9	\[ \frac{\color{blue}{\sqrt{\frac{A}{V}} \cdot c0}}{\sqrt{\ell}} \]
      associate-/l* [=>] 16.5	\[ \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}} \]
      associate-/r/ [=>] 14.5	\[ \color{blue}{\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0} \]

   4. Applied egg-rr 7.0

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

   if 0.0 < (*.f64 V l) < 2e292

   1. Initial program 10.4

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

   2. Applied egg-rr 0.7

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

   if 2e292 < (*.f64 V l)

   1. Initial program 39.5

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

   2. Applied egg-rr 23.6

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

   3. Simplified 23.6

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

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

   4. Taylor expanded in c0 around 0 39.5

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

   5. Simplified 22.7

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

      [Start] 39.5	\[ \sqrt{\frac{A}{V \cdot \ell}} \cdot c0 \]
      associate-/r* [=>] 22.7	\[ \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}} \cdot c0 \]

3. Recombined 3 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{\sqrt{-A}}{\sqrt{-V}}}{\sqrt{\ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.3
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+213}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{c0}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \frac{c0 \cdot \frac{c0}{V}}{\ell}}\\ \end{array} \]

Alternative 2
Error	14.3
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+213}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{c0}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot \frac{\frac{V}{c0}}{c0}}}\\ \end{array} \]

Alternative 3
Error	12.4
Cost	34640

\[\begin{array}{l} t_0 := c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ t_1 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot \frac{\frac{V}{c0}}{c0}}}\\ \end{array} \]

Alternative 4
Error	12.4
Cost	34640

\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ t_1 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot \frac{\frac{V}{c0}}{c0}}}\\ \end{array} \]

Alternative 5
Error	15.0
Cost	34514

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+264} \lor \neg \left(t_0 \leq -1 \cdot 10^{-323}\right) \land \left(t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+305}\right)\right):\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	15.4
Cost	34512

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ t_1 := \frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-236}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 7
Error	15.1
Cost	34512

\[\begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+213}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{c0}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 8
Error	8.9
Cost	14288

\[\begin{array}{l} t_0 := \frac{c0}{\sqrt{-V}} \cdot \sqrt{\frac{-A}{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+230}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-193}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 9
Error	8.7
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{-V}\\ t_1 := \sqrt{\frac{-A}{\ell}}\\ \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+230}:\\ \;\;\;\;\frac{c0}{t_0} \cdot t_1\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-176}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{t_0}{t_1}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 10
Error	8.1
Cost	14028

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-292}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \left(-\ell\right)}}{\sqrt{-A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 11
Error	9.9
Cost	13768

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array} \]

Alternative 12
Error	15.6
Cost	7625

\[\begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 10^{-287} \lor \neg \left(t_0 \leq 4 \cdot 10^{+295}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 13
Error	19.6
Cost	6848

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

Reproduce

herbie shell --seed 2023056 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))