Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.1% → 95.9%

Time: 5.0s

Alternatives: 9

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|V\right|, \left|\ell\right|\right)\\ t_1 := \mathsf{max}\left(\left|V\right|, \left|\ell\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{c0}{\sqrt{t\_0}} \cdot \sqrt{\frac{\left|A\right|}{t\_1}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left|A\right|}}{\sqrt{t\_1 \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{t\_1}} \cdot \sqrt{\frac{\left|A\right|}{t\_0}}\\ \end{array} \]

FPCore

(FPCore (c0 A V l)
  :precision binary64
  (let* ((t_0 (fmin (fabs V) (fabs l)))
       (t_1 (fmax (fabs V) (fabs l)))
       (t_2 (* t_0 t_1)))
  (if (<= t_2 2e-309)
    (* (/ c0 (sqrt t_0)) (sqrt (/ (fabs A) t_1)))
    (if (<= t_2 5e+286)
      (* c0 (/ (sqrt (fabs A)) (sqrt (* t_1 t_0))))
      (* (/ c0 (sqrt t_1)) (sqrt (/ (fabs A) t_0)))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = fmin(fabs(V), fabs(l));
	double t_1 = fmax(fabs(V), fabs(l));
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 2e-309) {
		tmp = (c0 / sqrt(t_0)) * sqrt((fabs(A) / t_1));
	} else if (t_2 <= 5e+286) {
		tmp = c0 * (sqrt(fabs(A)) / sqrt((t_1 * t_0)));
	} else {
		tmp = (c0 / sqrt(t_1)) * sqrt((fabs(A) / t_0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(abs(v), abs(l))
    t_1 = fmax(abs(v), abs(l))
    t_2 = t_0 * t_1
    if (t_2 <= 2d-309) then
        tmp = (c0 / sqrt(t_0)) * sqrt((abs(a) / t_1))
    else if (t_2 <= 5d+286) then
        tmp = c0 * (sqrt(abs(a)) / sqrt((t_1 * t_0)))
    else
        tmp = (c0 / sqrt(t_1)) * sqrt((abs(a) / t_0))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = fmin(Math.abs(V), Math.abs(l));
	double t_1 = fmax(Math.abs(V), Math.abs(l));
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 2e-309) {
		tmp = (c0 / Math.sqrt(t_0)) * Math.sqrt((Math.abs(A) / t_1));
	} else if (t_2 <= 5e+286) {
		tmp = c0 * (Math.sqrt(Math.abs(A)) / Math.sqrt((t_1 * t_0)));
	} else {
		tmp = (c0 / Math.sqrt(t_1)) * Math.sqrt((Math.abs(A) / t_0));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = fmin(math.fabs(V), math.fabs(l))
	t_1 = fmax(math.fabs(V), math.fabs(l))
	t_2 = t_0 * t_1
	tmp = 0
	if t_2 <= 2e-309:
		tmp = (c0 / math.sqrt(t_0)) * math.sqrt((math.fabs(A) / t_1))
	elif t_2 <= 5e+286:
		tmp = c0 * (math.sqrt(math.fabs(A)) / math.sqrt((t_1 * t_0)))
	else:
		tmp = (c0 / math.sqrt(t_1)) * math.sqrt((math.fabs(A) / t_0))
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = fmin(abs(V), abs(l))
	t_1 = fmax(abs(V), abs(l))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= 2e-309)
		tmp = Float64(Float64(c0 / sqrt(t_0)) * sqrt(Float64(abs(A) / t_1)));
	elseif (t_2 <= 5e+286)
		tmp = Float64(c0 * Float64(sqrt(abs(A)) / sqrt(Float64(t_1 * t_0))));
	else
		tmp = Float64(Float64(c0 / sqrt(t_1)) * sqrt(Float64(abs(A) / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = min(abs(V), abs(l));
	t_1 = max(abs(V), abs(l));
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (t_2 <= 2e-309)
		tmp = (c0 / sqrt(t_0)) * sqrt((abs(A) / t_1));
	elseif (t_2 <= 5e+286)
		tmp = c0 * (sqrt(abs(A)) / sqrt((t_1 * t_0)));
	else
		tmp = (c0 / sqrt(t_1)) * sqrt((abs(A) / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Min[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-309], N[(N[(c0 / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+286], N[(c0 * N[(N[Sqrt[N[Abs[A], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < 1.9999999999999988e-309

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 42.7%

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

   3. Applied rewrites 42.7%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{\ell}}}{\sqrt{V}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 41.4%

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

   5. Applied rewrites 41.4%

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

   if 1.9999999999999988e-309 < (*.f64 V l) < 5.0000000000000004e286

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-sqrt.f64 N/A

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

      6. lower-unsound-sqrt.f64 42.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.0%

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

   3. Applied rewrites 42.0%

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

   if 5.0000000000000004e286 < (*.f64 V l)

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-unsound-sqrt.f64 42.6%

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

   3. Applied rewrites 42.6%

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

   5. Applied rewrites 41.5%

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

Alternative 2: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|V\right|, \left|\ell\right|\right)\\ t_1 := \mathsf{max}\left(\left|V\right|, \left|\ell\right|\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \frac{c0}{\sqrt{t\_0}} \cdot \sqrt{\frac{\left|A\right|}{t\_1}}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left|A\right|}}{\sqrt{t\_1 \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (c0 A V l)
  :precision binary64
  (let* ((t_0 (fmin (fabs V) (fabs l)))
       (t_1 (fmax (fabs V) (fabs l)))
       (t_2 (* t_0 t_1))
       (t_3 (* (/ c0 (sqrt t_0)) (sqrt (/ (fabs A) t_1)))))
  (if (<= t_2 2e-309)
    t_3
    (if (<= t_2 2e+302)
      (* c0 (/ (sqrt (fabs A)) (sqrt (* t_1 t_0))))
      t_3))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = fmin(fabs(V), fabs(l));
	double t_1 = fmax(fabs(V), fabs(l));
	double t_2 = t_0 * t_1;
	double t_3 = (c0 / sqrt(t_0)) * sqrt((fabs(A) / t_1));
	double tmp;
	if (t_2 <= 2e-309) {
		tmp = t_3;
	} else if (t_2 <= 2e+302) {
		tmp = c0 * (sqrt(fabs(A)) / sqrt((t_1 * t_0)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(v), abs(l))
    t_1 = fmax(abs(v), abs(l))
    t_2 = t_0 * t_1
    t_3 = (c0 / sqrt(t_0)) * sqrt((abs(a) / t_1))
    if (t_2 <= 2d-309) then
        tmp = t_3
    else if (t_2 <= 2d+302) then
        tmp = c0 * (sqrt(abs(a)) / sqrt((t_1 * t_0)))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = fmin(Math.abs(V), Math.abs(l));
	double t_1 = fmax(Math.abs(V), Math.abs(l));
	double t_2 = t_0 * t_1;
	double t_3 = (c0 / Math.sqrt(t_0)) * Math.sqrt((Math.abs(A) / t_1));
	double tmp;
	if (t_2 <= 2e-309) {
		tmp = t_3;
	} else if (t_2 <= 2e+302) {
		tmp = c0 * (Math.sqrt(Math.abs(A)) / Math.sqrt((t_1 * t_0)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = fmin(math.fabs(V), math.fabs(l))
	t_1 = fmax(math.fabs(V), math.fabs(l))
	t_2 = t_0 * t_1
	t_3 = (c0 / math.sqrt(t_0)) * math.sqrt((math.fabs(A) / t_1))
	tmp = 0
	if t_2 <= 2e-309:
		tmp = t_3
	elif t_2 <= 2e+302:
		tmp = c0 * (math.sqrt(math.fabs(A)) / math.sqrt((t_1 * t_0)))
	else:
		tmp = t_3
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = fmin(abs(V), abs(l))
	t_1 = fmax(abs(V), abs(l))
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(Float64(c0 / sqrt(t_0)) * sqrt(Float64(abs(A) / t_1)))
	tmp = 0.0
	if (t_2 <= 2e-309)
		tmp = t_3;
	elseif (t_2 <= 2e+302)
		tmp = Float64(c0 * Float64(sqrt(abs(A)) / sqrt(Float64(t_1 * t_0))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = min(abs(V), abs(l));
	t_1 = max(abs(V), abs(l));
	t_2 = t_0 * t_1;
	t_3 = (c0 / sqrt(t_0)) * sqrt((abs(A) / t_1));
	tmp = 0.0;
	if (t_2 <= 2e-309)
		tmp = t_3;
	elseif (t_2 <= 2e+302)
		tmp = c0 * (sqrt(abs(A)) / sqrt((t_1 * t_0)));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Min[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-309], t$95$3, If[LessEqual[t$95$2, 2e+302], N[(c0 * N[(N[Sqrt[N[Abs[A], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 V l) < 1.9999999999999988e-309 or 2.0000000000000002e302 < (*.f64 V l)

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 42.7%

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

   3. Applied rewrites 42.7%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{\ell}}}{\sqrt{V}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 41.4%

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

   5. Applied rewrites 41.4%

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

   if 1.9999999999999988e-309 < (*.f64 V l) < 2.0000000000000002e302

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-sqrt.f64 N/A

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

      6. lower-unsound-sqrt.f64 42.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.0%

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

   3. Applied rewrites 42.0%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{\ell \cdot V}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|V\right|, \left|\ell\right|\right)\\ t_1 := \mathsf{min}\left(\left|V\right|, \left|\ell\right|\right)\\ \mathbf{if}\;\left|A\right| \leq 10^{-25}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{\left|A\right|}{t\_1}}}{\sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\left|A\right|}{t\_0}} \cdot \sqrt{\frac{1}{t\_1}}\right)\\ \end{array} \]

FPCore

(FPCore (c0 A V l)
  :precision binary64
  (let* ((t_0 (fmax (fabs V) (fabs l))) (t_1 (fmin (fabs V) (fabs l))))
  (if (<= (fabs A) 1e-25)
    (* c0 (/ (sqrt (/ (fabs A) t_1)) (sqrt t_0)))
    (* c0 (* (sqrt (/ (fabs A) t_0)) (sqrt (/ 1.0 t_1)))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = fmax(fabs(V), fabs(l));
	double t_1 = fmin(fabs(V), fabs(l));
	double tmp;
	if (fabs(A) <= 1e-25) {
		tmp = c0 * (sqrt((fabs(A) / t_1)) / sqrt(t_0));
	} else {
		tmp = c0 * (sqrt((fabs(A) / t_0)) * sqrt((1.0 / t_1)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: tmp
    t_0 = fmax(abs(v), abs(l))
    t_1 = fmin(abs(v), abs(l))
    if (abs(a) <= 1d-25) then
        tmp = c0 * (sqrt((abs(a) / t_1)) / sqrt(t_0))
    else
        tmp = c0 * (sqrt((abs(a) / t_0)) * sqrt((1.0d0 / t_1)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = fmax(Math.abs(V), Math.abs(l));
	double t_1 = fmin(Math.abs(V), Math.abs(l));
	double tmp;
	if (Math.abs(A) <= 1e-25) {
		tmp = c0 * (Math.sqrt((Math.abs(A) / t_1)) / Math.sqrt(t_0));
	} else {
		tmp = c0 * (Math.sqrt((Math.abs(A) / t_0)) * Math.sqrt((1.0 / t_1)));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = fmax(math.fabs(V), math.fabs(l))
	t_1 = fmin(math.fabs(V), math.fabs(l))
	tmp = 0
	if math.fabs(A) <= 1e-25:
		tmp = c0 * (math.sqrt((math.fabs(A) / t_1)) / math.sqrt(t_0))
	else:
		tmp = c0 * (math.sqrt((math.fabs(A) / t_0)) * math.sqrt((1.0 / t_1)))
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = fmax(abs(V), abs(l))
	t_1 = fmin(abs(V), abs(l))
	tmp = 0.0
	if (abs(A) <= 1e-25)
		tmp = Float64(c0 * Float64(sqrt(Float64(abs(A) / t_1)) / sqrt(t_0)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(abs(A) / t_0)) * sqrt(Float64(1.0 / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = max(abs(V), abs(l));
	t_1 = min(abs(V), abs(l));
	tmp = 0.0;
	if (abs(A) <= 1e-25)
		tmp = c0 * (sqrt((abs(A) / t_1)) / sqrt(t_0));
	else
		tmp = c0 * (sqrt((abs(A) / t_0)) * sqrt((1.0 / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Max[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[A], $MachinePrecision], 1e-25], N[(c0 * N[(N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if A < 1e-25

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-unsound-sqrt.f64 42.6%

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

   3. Applied rewrites 42.6%

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

   if 1e-25 < A

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. mult-flip N/A

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

      12. associate-*l/ N/A

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

      13. sqrt-prod N/A

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

      14. lower-unsound-*.f64 N/A

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

      15. lower-unsound-sqrt.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-unsound-sqrt.f64 N/A

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

      18. lower-/.f64 42.7%

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

   3. Applied rewrites 42.7%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{\ell}} \cdot \sqrt{\frac{1}{V}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|V\right|, \left|\ell\right|\right)\\ t_1 := \mathsf{min}\left(\left|V\right|, \left|\ell\right|\right)\\ \mathbf{if}\;\sqrt{\frac{\left|A\right|}{t\_1 \cdot t\_0}} \leq 5:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{\left|A\right|}{t\_1}}}{\sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{\left|A\right|}{t\_0}}}{\sqrt{t\_1}}\\ \end{array} \]

FPCore

(FPCore (c0 A V l)
  :precision binary64
  (let* ((t_0 (fmax (fabs V) (fabs l))) (t_1 (fmin (fabs V) (fabs l))))
  (if (<= (sqrt (/ (fabs A) (* t_1 t_0))) 5.0)
    (* c0 (/ (sqrt (/ (fabs A) t_1)) (sqrt t_0)))
    (* c0 (/ (sqrt (/ (fabs A) t_0)) (sqrt t_1))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = fmax(fabs(V), fabs(l));
	double t_1 = fmin(fabs(V), fabs(l));
	double tmp;
	if (sqrt((fabs(A) / (t_1 * t_0))) <= 5.0) {
		tmp = c0 * (sqrt((fabs(A) / t_1)) / sqrt(t_0));
	} else {
		tmp = c0 * (sqrt((fabs(A) / t_0)) / sqrt(t_1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: tmp
    t_0 = fmax(abs(v), abs(l))
    t_1 = fmin(abs(v), abs(l))
    if (sqrt((abs(a) / (t_1 * t_0))) <= 5.0d0) then
        tmp = c0 * (sqrt((abs(a) / t_1)) / sqrt(t_0))
    else
        tmp = c0 * (sqrt((abs(a) / t_0)) / sqrt(t_1))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = fmax(Math.abs(V), Math.abs(l));
	double t_1 = fmin(Math.abs(V), Math.abs(l));
	double tmp;
	if (Math.sqrt((Math.abs(A) / (t_1 * t_0))) <= 5.0) {
		tmp = c0 * (Math.sqrt((Math.abs(A) / t_1)) / Math.sqrt(t_0));
	} else {
		tmp = c0 * (Math.sqrt((Math.abs(A) / t_0)) / Math.sqrt(t_1));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = fmax(math.fabs(V), math.fabs(l))
	t_1 = fmin(math.fabs(V), math.fabs(l))
	tmp = 0
	if math.sqrt((math.fabs(A) / (t_1 * t_0))) <= 5.0:
		tmp = c0 * (math.sqrt((math.fabs(A) / t_1)) / math.sqrt(t_0))
	else:
		tmp = c0 * (math.sqrt((math.fabs(A) / t_0)) / math.sqrt(t_1))
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = fmax(abs(V), abs(l))
	t_1 = fmin(abs(V), abs(l))
	tmp = 0.0
	if (sqrt(Float64(abs(A) / Float64(t_1 * t_0))) <= 5.0)
		tmp = Float64(c0 * Float64(sqrt(Float64(abs(A) / t_1)) / sqrt(t_0)));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(abs(A) / t_0)) / sqrt(t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = max(abs(V), abs(l));
	t_1 = min(abs(V), abs(l));
	tmp = 0.0;
	if (sqrt((abs(A) / (t_1 * t_0))) <= 5.0)
		tmp = c0 * (sqrt((abs(A) / t_1)) / sqrt(t_0));
	else
		tmp = c0 * (sqrt((abs(A) / t_0)) / sqrt(t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Max[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[Abs[A], $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5.0], N[(c0 * N[(N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 5

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-unsound-sqrt.f64 42.6%

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

   3. Applied rewrites 42.6%

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

   if 5 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 42.7%

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

   3. Applied rewrites 42.7%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{\ell}}}{\sqrt{V}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|V\right|, \left|\ell\right|\right)\\ t_1 := \mathsf{min}\left(\left|V\right|, \left|\ell\right|\right)\\ \mathbf{if}\;\left|A\right| \leq 1.45 \cdot 10^{+70}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{\left|A\right|}{t\_1}}}{\sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{t\_1}} \cdot \sqrt{\frac{\left|A\right|}{t\_0}}\\ \end{array} \]

FPCore

(FPCore (c0 A V l)
  :precision binary64
  (let* ((t_0 (fmax (fabs V) (fabs l))) (t_1 (fmin (fabs V) (fabs l))))
  (if (<= (fabs A) 1.45e+70)
    (* c0 (/ (sqrt (/ (fabs A) t_1)) (sqrt t_0)))
    (* (/ c0 (sqrt t_1)) (sqrt (/ (fabs A) t_0))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = fmax(fabs(V), fabs(l));
	double t_1 = fmin(fabs(V), fabs(l));
	double tmp;
	if (fabs(A) <= 1.45e+70) {
		tmp = c0 * (sqrt((fabs(A) / t_1)) / sqrt(t_0));
	} else {
		tmp = (c0 / sqrt(t_1)) * sqrt((fabs(A) / t_0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: tmp
    t_0 = fmax(abs(v), abs(l))
    t_1 = fmin(abs(v), abs(l))
    if (abs(a) <= 1.45d+70) then
        tmp = c0 * (sqrt((abs(a) / t_1)) / sqrt(t_0))
    else
        tmp = (c0 / sqrt(t_1)) * sqrt((abs(a) / t_0))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = fmax(Math.abs(V), Math.abs(l));
	double t_1 = fmin(Math.abs(V), Math.abs(l));
	double tmp;
	if (Math.abs(A) <= 1.45e+70) {
		tmp = c0 * (Math.sqrt((Math.abs(A) / t_1)) / Math.sqrt(t_0));
	} else {
		tmp = (c0 / Math.sqrt(t_1)) * Math.sqrt((Math.abs(A) / t_0));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = fmax(math.fabs(V), math.fabs(l))
	t_1 = fmin(math.fabs(V), math.fabs(l))
	tmp = 0
	if math.fabs(A) <= 1.45e+70:
		tmp = c0 * (math.sqrt((math.fabs(A) / t_1)) / math.sqrt(t_0))
	else:
		tmp = (c0 / math.sqrt(t_1)) * math.sqrt((math.fabs(A) / t_0))
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = fmax(abs(V), abs(l))
	t_1 = fmin(abs(V), abs(l))
	tmp = 0.0
	if (abs(A) <= 1.45e+70)
		tmp = Float64(c0 * Float64(sqrt(Float64(abs(A) / t_1)) / sqrt(t_0)));
	else
		tmp = Float64(Float64(c0 / sqrt(t_1)) * sqrt(Float64(abs(A) / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = max(abs(V), abs(l));
	t_1 = min(abs(V), abs(l));
	tmp = 0.0;
	if (abs(A) <= 1.45e+70)
		tmp = c0 * (sqrt((abs(A) / t_1)) / sqrt(t_0));
	else
		tmp = (c0 / sqrt(t_1)) * sqrt((abs(A) / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Max[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[A], $MachinePrecision], 1.45e+70], N[(c0 * N[(N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Abs[A], $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if A < 1.4499999999999999e70

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. sqrt-div N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-unsound-sqrt.f64 42.6%

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

   3. Applied rewrites 42.6%

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

   if 1.4499999999999999e70 < A

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 42.7%

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

   3. Applied rewrites 42.7%

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{A}{\ell}}}{\sqrt{V}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 41.4%

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

   5. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{c0}{\sqrt{V}} \cdot \sqrt{\frac{A}{\ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left|V\right| \cdot \left|\ell\right|\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{\left|A\right|}{\left|\ell\right|}}{\left|V\right|}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+288}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left|A\right|}}{\sqrt{\left|\ell\right| \cdot \left|V\right|}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{\left|A\right|}{\left|V\right|}}{\left|\ell\right|}}\\ \end{array} \]

FPCore

(FPCore (c0 A V l)
  :precision binary64
  (let* ((t_0 (* (fabs V) (fabs l))))
  (if (<= t_0 0.0)
    (* c0 (sqrt (/ (/ (fabs A) (fabs l)) (fabs V))))
    (if (<= t_0 2e+288)
      (* c0 (/ (sqrt (fabs A)) (sqrt (* (fabs l) (fabs V)))))
      (* c0 (sqrt (/ (/ (fabs A) (fabs V)) (fabs l))))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = fabs(V) * fabs(l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((fabs(A) / fabs(l)) / fabs(V)));
	} else if (t_0 <= 2e+288) {
		tmp = c0 * (sqrt(fabs(A)) / sqrt((fabs(l) * fabs(V))));
	} else {
		tmp = c0 * sqrt(((fabs(A) / fabs(V)) / fabs(l)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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 = abs(v) * abs(l)
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((abs(a) / abs(l)) / abs(v)))
    else if (t_0 <= 2d+288) then
        tmp = c0 * (sqrt(abs(a)) / sqrt((abs(l) * abs(v))))
    else
        tmp = c0 * sqrt(((abs(a) / abs(v)) / abs(l)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.abs(V) * Math.abs(l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((Math.abs(A) / Math.abs(l)) / Math.abs(V)));
	} else if (t_0 <= 2e+288) {
		tmp = c0 * (Math.sqrt(Math.abs(A)) / Math.sqrt((Math.abs(l) * Math.abs(V))));
	} else {
		tmp = c0 * Math.sqrt(((Math.abs(A) / Math.abs(V)) / Math.abs(l)));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = math.fabs(V) * math.fabs(l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((math.fabs(A) / math.fabs(l)) / math.fabs(V)))
	elif t_0 <= 2e+288:
		tmp = c0 * (math.sqrt(math.fabs(A)) / math.sqrt((math.fabs(l) * math.fabs(V))))
	else:
		tmp = c0 * math.sqrt(((math.fabs(A) / math.fabs(V)) / math.fabs(l)))
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = Float64(abs(V) * abs(l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(abs(A) / abs(l)) / abs(V))));
	elseif (t_0 <= 2e+288)
		tmp = Float64(c0 * Float64(sqrt(abs(A)) / sqrt(Float64(abs(l) * abs(V)))));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(abs(A) / abs(V)) / abs(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = abs(V) * abs(l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((abs(A) / abs(l)) / abs(V)));
	elseif (t_0 <= 2e+288)
		tmp = c0 * (sqrt(abs(A)) / sqrt((abs(l) * abs(V))));
	else
		tmp = c0 * sqrt(((abs(A) / abs(V)) / abs(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(N[Abs[V], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(N[Abs[A], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] / N[Abs[V], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+288], N[(c0 * N[(N[Sqrt[N[Abs[A], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Abs[l], $MachinePrecision] * N[Abs[V], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(N[Abs[A], $MachinePrecision] / N[Abs[V], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 73.8%

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

   3. Applied rewrites 73.8%

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

   if 0.0 < (*.f64 V l) < 2e288

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-sqrt.f64 N/A

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

      6. lower-unsound-sqrt.f64 42.0%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.0%

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

   3. Applied rewrites 42.0%

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

   if 2e288 < (*.f64 V l)

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.4%

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

   3. Applied rewrites 73.4%

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

Alternative 7: 80.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|V\right|, \left|\ell\right|\right)\\ t_1 := \mathsf{min}\left(\left|V\right|, \left|\ell\right|\right)\\ \mathbf{if}\;\sqrt{\frac{\left|A\right|}{t\_1 \cdot t\_0}} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{\left|A\right|}{t\_1}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{\left|A\right|}{t\_0}}{t\_1}}\\ \end{array} \]

FPCore

(FPCore (c0 A V l)
  :precision binary64
  (let* ((t_0 (fmax (fabs V) (fabs l))) (t_1 (fmin (fabs V) (fabs l))))
  (if (<= (sqrt (/ (fabs A) (* t_1 t_0))) 5e-5)
    (* c0 (sqrt (/ (/ (fabs A) t_1) t_0)))
    (* c0 (sqrt (/ (/ (fabs A) t_0) t_1))))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = fmax(fabs(V), fabs(l));
	double t_1 = fmin(fabs(V), fabs(l));
	double tmp;
	if (sqrt((fabs(A) / (t_1 * t_0))) <= 5e-5) {
		tmp = c0 * sqrt(((fabs(A) / t_1) / t_0));
	} else {
		tmp = c0 * sqrt(((fabs(A) / t_0) / t_1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: tmp
    t_0 = fmax(abs(v), abs(l))
    t_1 = fmin(abs(v), abs(l))
    if (sqrt((abs(a) / (t_1 * t_0))) <= 5d-5) then
        tmp = c0 * sqrt(((abs(a) / t_1) / t_0))
    else
        tmp = c0 * sqrt(((abs(a) / t_0) / t_1))
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = fmax(Math.abs(V), Math.abs(l));
	double t_1 = fmin(Math.abs(V), Math.abs(l));
	double tmp;
	if (Math.sqrt((Math.abs(A) / (t_1 * t_0))) <= 5e-5) {
		tmp = c0 * Math.sqrt(((Math.abs(A) / t_1) / t_0));
	} else {
		tmp = c0 * Math.sqrt(((Math.abs(A) / t_0) / t_1));
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = fmax(math.fabs(V), math.fabs(l))
	t_1 = fmin(math.fabs(V), math.fabs(l))
	tmp = 0
	if math.sqrt((math.fabs(A) / (t_1 * t_0))) <= 5e-5:
		tmp = c0 * math.sqrt(((math.fabs(A) / t_1) / t_0))
	else:
		tmp = c0 * math.sqrt(((math.fabs(A) / t_0) / t_1))
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = fmax(abs(V), abs(l))
	t_1 = fmin(abs(V), abs(l))
	tmp = 0.0
	if (sqrt(Float64(abs(A) / Float64(t_1 * t_0))) <= 5e-5)
		tmp = Float64(c0 * sqrt(Float64(Float64(abs(A) / t_1) / t_0)));
	else
		tmp = Float64(c0 * sqrt(Float64(Float64(abs(A) / t_0) / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = max(abs(V), abs(l));
	t_1 = min(abs(V), abs(l));
	tmp = 0.0;
	if (sqrt((abs(A) / (t_1 * t_0))) <= 5e-5)
		tmp = c0 * sqrt(((abs(A) / t_1) / t_0));
	else
		tmp = c0 * sqrt(((abs(A) / t_0) / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Max[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[Abs[A], $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e-5], N[(c0 * N[Sqrt[N[(N[(N[Abs[A], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[N[(N[(N[Abs[A], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 5.0000000000000002e-5

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.4%

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

   3. Applied rewrites 73.4%

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

   if 5.0000000000000002e-5 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 73.8%

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

   3. Applied rewrites 73.8%

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{\ell}}{V}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|V\right|, \left|\ell\right|\right)\\ t_1 := \mathsf{min}\left(\left|V\right|, \left|\ell\right|\right)\\ t_2 := c0 \cdot \sqrt{\frac{\frac{\left|A\right|}{t\_1}}{t\_0}}\\ t_3 := \sqrt{\frac{\left|A\right|}{t\_1 \cdot t\_0}}\\ \mathbf{if}\;t\_3 \leq 4 \cdot 10^{-17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;c0 \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (c0 A V l)
  :precision binary64
  (let* ((t_0 (fmax (fabs V) (fabs l)))
       (t_1 (fmin (fabs V) (fabs l)))
       (t_2 (* c0 (sqrt (/ (/ (fabs A) t_1) t_0))))
       (t_3 (sqrt (/ (fabs A) (* t_1 t_0)))))
  (if (<= t_3 4e-17) t_2 (if (<= t_3 2e+149) (* c0 t_3) t_2))))

C

double code(double c0, double A, double V, double l) {
	double t_0 = fmax(fabs(V), fabs(l));
	double t_1 = fmin(fabs(V), fabs(l));
	double t_2 = c0 * sqrt(((fabs(A) / t_1) / t_0));
	double t_3 = sqrt((fabs(A) / (t_1 * t_0)));
	double tmp;
	if (t_3 <= 4e-17) {
		tmp = t_2;
	} else if (t_3 <= 2e+149) {
		tmp = c0 * t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, a, v, l)
use fmin_fmax_functions
    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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(v), abs(l))
    t_1 = fmin(abs(v), abs(l))
    t_2 = c0 * sqrt(((abs(a) / t_1) / t_0))
    t_3 = sqrt((abs(a) / (t_1 * t_0)))
    if (t_3 <= 4d-17) then
        tmp = t_2
    else if (t_3 <= 2d+149) then
        tmp = c0 * t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double c0, double A, double V, double l) {
	double t_0 = fmax(Math.abs(V), Math.abs(l));
	double t_1 = fmin(Math.abs(V), Math.abs(l));
	double t_2 = c0 * Math.sqrt(((Math.abs(A) / t_1) / t_0));
	double t_3 = Math.sqrt((Math.abs(A) / (t_1 * t_0)));
	double tmp;
	if (t_3 <= 4e-17) {
		tmp = t_2;
	} else if (t_3 <= 2e+149) {
		tmp = c0 * t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(c0, A, V, l):
	t_0 = fmax(math.fabs(V), math.fabs(l))
	t_1 = fmin(math.fabs(V), math.fabs(l))
	t_2 = c0 * math.sqrt(((math.fabs(A) / t_1) / t_0))
	t_3 = math.sqrt((math.fabs(A) / (t_1 * t_0)))
	tmp = 0
	if t_3 <= 4e-17:
		tmp = t_2
	elif t_3 <= 2e+149:
		tmp = c0 * t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(c0, A, V, l)
	t_0 = fmax(abs(V), abs(l))
	t_1 = fmin(abs(V), abs(l))
	t_2 = Float64(c0 * sqrt(Float64(Float64(abs(A) / t_1) / t_0)))
	t_3 = sqrt(Float64(abs(A) / Float64(t_1 * t_0)))
	tmp = 0.0
	if (t_3 <= 4e-17)
		tmp = t_2;
	elseif (t_3 <= 2e+149)
		tmp = Float64(c0 * t_3);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, A, V, l)
	t_0 = max(abs(V), abs(l));
	t_1 = min(abs(V), abs(l));
	t_2 = c0 * sqrt(((abs(A) / t_1) / t_0));
	t_3 = sqrt((abs(A) / (t_1 * t_0)));
	tmp = 0.0;
	if (t_3 <= 4e-17)
		tmp = t_2;
	elseif (t_3 <= 2e+149)
		tmp = c0 * t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, A_, V_, l_] := Block[{t$95$0 = N[Max[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[V], $MachinePrecision], N[Abs[l], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(c0 * N[Sqrt[N[(N[(N[Abs[A], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Abs[A], $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 4e-17], t$95$2, If[LessEqual[t$95$3, 2e+149], N[(c0 * t$95$3), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (/.f64 A (*.f64 V l))) < 4.0000000000000003e-17 or 2.0000000000000001e149 < (sqrt.f64 (/.f64 A (*.f64 V l)))

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 73.4%

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

   3. Applied rewrites 73.4%

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

   if 4.0000000000000003e-17 < (sqrt.f64 (/.f64 A (*.f64 V l))) < 2.0000000000000001e149

   1. Initial program 74.1%

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
2. Add Preprocessing

Reproduce

herbie shell --seed 2025213 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))