Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.8%

Time: 9.0s

Alternatives: 6

Speedup: N/A×

• 52.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

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(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Initial Program: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

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(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Alternative 1: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(r \cdot w\right)}^{1}\\ \mathsf{fma}\left({r}^{-2}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, \frac{t\_0 \cdot t\_0}{1 - v}, 4.5\right) \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (pow (* r w) 1.0)))
   (-
    (fma (pow r -2.0) 2.0 3.0)
    (fma (* (fma -2.0 v 3.0) 0.125) (/ (* t_0 t_0) (- 1.0 v)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = pow((r * w), 1.0);
	return fma(pow(r, -2.0), 2.0, 3.0) - fma((fma(-2.0, v, 3.0) * 0.125), ((t_0 * t_0) / (1.0 - v)), 4.5);
}

Julia

function code(v, w, r)
	t_0 = Float64(r * w) ^ 1.0
	return Float64(fma((r ^ -2.0), 2.0, 3.0) - fma(Float64(fma(-2.0, v, 3.0) * 0.125), Float64(Float64(t_0 * t_0) / Float64(1.0 - v)), 4.5))
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[Power[N[(r * w), $MachinePrecision], 1.0], $MachinePrecision]}, N[(N[(N[Power[r, -2.0], $MachinePrecision] * 2.0 + 3.0), $MachinePrecision] - N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 84.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(r \cdot r\right)}^{-1}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, \frac{{\left(r \cdot w\right)}^{1} \cdot {\left(r \cdot w\right)}^{1}}{1 - v}, 4.5\right)} \]
4. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(r \cdot r\right)}^{-1}}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}, \frac{{\left(r \cdot w\right)}^{1} \cdot {\left(r \cdot w\right)}^{1}}{1 - v}, \frac{9}{2}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left(r \cdot r\right)}}^{-1}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}, \frac{{\left(r \cdot w\right)}^{1} \cdot {\left(r \cdot w\right)}^{1}}{1 - v}, \frac{9}{2}\right) \]

   3. pow2 N/A

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left({r}^{2}\right)}}^{-1}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}, \frac{{\left(r \cdot w\right)}^{1} \cdot {\left(r \cdot w\right)}^{1}}{1 - v}, \frac{9}{2}\right) \]

   4. pow-pow N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{r}^{\left(2 \cdot -1\right)}}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}, \frac{{\left(r \cdot w\right)}^{1} \cdot {\left(r \cdot w\right)}^{1}}{1 - v}, \frac{9}{2}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({r}^{\color{blue}{-2}}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot \frac{1}{8}, \frac{{\left(r \cdot w\right)}^{1} \cdot {\left(r \cdot w\right)}^{1}}{1 - v}, \frac{9}{2}\right) \]

   6. lower-pow.f64 99.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{{r}^{-2}}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, \frac{{\left(r \cdot w\right)}^{1} \cdot {\left(r \cdot w\right)}^{1}}{1 - v}, 4.5\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{r}^{-2}}, 2, 3\right) - \mathsf{fma}\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, \frac{{\left(r \cdot w\right)}^{1} \cdot {\left(r \cdot w\right)}^{1}}{1 - v}, 4.5\right) \]
6. Add Preprocessing

Alternative 2: 94.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= w 8.5e+25)
   (-
    (-
     (+ 3.0 (/ 2.0 (* r r)))
     (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* w (* w r)) r)) (- 1.0 v)))
    4.5)
   (-
    (*
     (*
      (fma
       (/ (* r (* r (fma v -2.0 3.0))) (- 1.0 v))
       -0.125
       (fma (pow (* w w) -1.0) 3.0 (* (pow (pow (* w r) 2.0) -1.0) 2.0)))
      w)
     w)
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (w <= 8.5e+25) {
		tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((fma(((r * (r * fma(v, -2.0, 3.0))) / (1.0 - v)), -0.125, fma(pow((w * w), -1.0), 3.0, (pow(pow((w * r), 2.0), -1.0) * 2.0))) * w) * w) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (w <= 8.5e+25)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(w * Float64(w * r)) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(r * Float64(r * fma(v, -2.0, 3.0))) / Float64(1.0 - v)), -0.125, fma((Float64(w * w) ^ -1.0), 3.0, Float64(((Float64(w * r) ^ 2.0) ^ -1.0) * 2.0))) * w) * w) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[w, 8.5e+25], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(r * N[(r * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Power[N[(w * w), $MachinePrecision], -1.0], $MachinePrecision] * 3.0 + N[(N[Power[N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 8.5000000000000007e25

   1. Initial program 87.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\color{blue}{\left(w \cdot w\right)} \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      5. lower-*.f64 93.6

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Applied rewrites 93.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5 \]

   if 8.5000000000000007e25 < w

   1. Initial program 77.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   2. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)} - \frac{9}{2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left({\left(r \cdot r\right)}^{-1}, 2, 3\right)}{w \cdot w} - \frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(-2, v, 3\right)}{1 - v} \cdot 0.125\right) \cdot \left(w \cdot w\right)} - 4.5 \]

   6. Applied rewrites 99.1%

      \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot \color{blue}{w} - 4.5 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \left(v \cdot -2 + 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      4. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + -2 \cdot v\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + -2 \cdot v\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      11. lift-fma.f64 99.1

          \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5 \]

   8. Applied rewrites 99.1%

      \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 7.8 \cdot 10^{+25}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \mathsf{fma}\left(-2, v, 3\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= w 7.8e+25)
   (-
    (-
     (+ 3.0 (/ 2.0 (* r r)))
     (/ (* (* 0.125 (* r r)) (* (* w w) (fma -2.0 v 3.0))) (- 1.0 v)))
    4.5)
   (-
    (*
     (*
      (fma
       (/ (* r (* r (fma v -2.0 3.0))) (- 1.0 v))
       -0.125
       (fma (pow (* w w) -1.0) 3.0 (* (pow (pow (* w r) 2.0) -1.0) 2.0)))
      w)
     w)
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (w <= 7.8e+25) {
		tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (r * r)) * ((w * w) * fma(-2.0, v, 3.0))) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((fma(((r * (r * fma(v, -2.0, 3.0))) / (1.0 - v)), -0.125, fma(pow((w * w), -1.0), 3.0, (pow(pow((w * r), 2.0), -1.0) * 2.0))) * w) * w) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (w <= 7.8e+25)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(r * r)) * Float64(Float64(w * w) * fma(-2.0, v, 3.0))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(r * Float64(r * fma(v, -2.0, 3.0))) / Float64(1.0 - v)), -0.125, fma((Float64(w * w) ^ -1.0), 3.0, Float64(((Float64(w * r) ^ 2.0) ^ -1.0) * 2.0))) * w) * w) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[w, 7.8e+25], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(r * N[(r * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Power[N[(w * w), $MachinePrecision], -1.0], $MachinePrecision] * 3.0 + N[(N[Power[N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 7.8000000000000004e25

   1. Initial program 87.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   2. Taylor expanded in w around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{1}{8} \cdot \left({r}^{2} \cdot \left({w}^{2} \cdot \left(3 - 2 \cdot v\right)\right)\right)}}{1 - v}\right) - \frac{9}{2} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot {r}^{2}\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(3 - 2 \cdot v\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot {r}^{2}\right) \cdot \color{blue}{\left({w}^{2} \cdot \left(3 - 2 \cdot v\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot {r}^{2}\right) \cdot \left(\color{blue}{{w}^{2}} \cdot \left(3 - 2 \cdot v\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      4. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(r \cdot r\right)\right) \cdot \left({w}^{\color{blue}{2}} \cdot \left(3 - 2 \cdot v\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(r \cdot r\right)\right) \cdot \left({w}^{\color{blue}{2}} \cdot \left(3 - 2 \cdot v\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(r \cdot r\right)\right) \cdot \left({w}^{2} \cdot \color{blue}{\left(3 - 2 \cdot v\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. pow2 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\color{blue}{3} - 2 \cdot v\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\color{blue}{3} - 2 \cdot v\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \left(3 - \left(\mathsf{neg}\left(-2\right)\right) \cdot v\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \left(3 + \color{blue}{-2 \cdot v}\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \left(-2 \cdot v + \color{blue}{3}\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      12. lower-fma.f64 78.4

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \mathsf{fma}\left(-2, \color{blue}{v}, 3\right)\right)}{1 - v}\right) - 4.5 \]

   4. Applied rewrites 78.4%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(0.125 \cdot \left(r \cdot r\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \mathsf{fma}\left(-2, v, 3\right)\right)}}{1 - v}\right) - 4.5 \]

   if 7.8000000000000004e25 < w

   1. Initial program 77.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   2. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)} - \frac{9}{2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left({\left(r \cdot r\right)}^{-1}, 2, 3\right)}{w \cdot w} - \frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(-2, v, 3\right)}{1 - v} \cdot 0.125\right) \cdot \left(w \cdot w\right)} - 4.5 \]

   6. Applied rewrites 99.1%

      \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot \color{blue}{w} - 4.5 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \left(v \cdot -2 + 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      4. associate-*l* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + -2 \cdot v\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + -2 \cdot v\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

      11. lift-fma.f64 99.1

          \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5 \]

   8. Applied rewrites 99.1%

      \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -r \cdot r\\ \mathbf{if}\;w \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;\left(\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(r \cdot r\right), w \cdot w, \left(\left(r \cdot r\right) \cdot 3\right) \cdot \left(w \cdot w\right)\right) \cdot 0.125, t\_0, -2 \cdot v\right)}{v \cdot t\_0} + 3\right) - \left(0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- (* r r))))
   (if (<= w 2.1e-121)
     (-
      (-
       (+
        (/
         (fma
          (* (fma (* -2.0 (* r r)) (* w w) (* (* (* r r) 3.0) (* w w))) 0.125)
          t_0
          (* -2.0 v))
         (* v t_0))
        3.0)
       (* (* 0.25 (* r r)) (* w w)))
      4.5)
     (-
      (*
       (*
        (fma
         (/ (* (* r r) (fma v -2.0 3.0)) (- 1.0 v))
         -0.125
         (fma (pow (* w w) -1.0) 3.0 (* (pow (pow (* w r) 2.0) -1.0) 2.0)))
        w)
       w)
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = -(r * r);
	double tmp;
	if (w <= 2.1e-121) {
		tmp = (((fma((fma((-2.0 * (r * r)), (w * w), (((r * r) * 3.0) * (w * w))) * 0.125), t_0, (-2.0 * v)) / (v * t_0)) + 3.0) - ((0.25 * (r * r)) * (w * w))) - 4.5;
	} else {
		tmp = ((fma((((r * r) * fma(v, -2.0, 3.0)) / (1.0 - v)), -0.125, fma(pow((w * w), -1.0), 3.0, (pow(pow((w * r), 2.0), -1.0) * 2.0))) * w) * w) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(-Float64(r * r))
	tmp = 0.0
	if (w <= 2.1e-121)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(fma(Float64(-2.0 * Float64(r * r)), Float64(w * w), Float64(Float64(Float64(r * r) * 3.0) * Float64(w * w))) * 0.125), t_0, Float64(-2.0 * v)) / Float64(v * t_0)) + 3.0) - Float64(Float64(0.25 * Float64(r * r)) * Float64(w * w))) - 4.5);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(r * r) * fma(v, -2.0, 3.0)) / Float64(1.0 - v)), -0.125, fma((Float64(w * w) ^ -1.0), 3.0, Float64(((Float64(w * r) ^ 2.0) ^ -1.0) * 2.0))) * w) * w) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = (-N[(r * r), $MachinePrecision])}, If[LessEqual[w, 2.1e-121], N[(N[(N[(N[(N[(N[(N[(N[(-2.0 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision] + N[(N[(N[(r * r), $MachinePrecision] * 3.0), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision] / N[(v * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(r * r), $MachinePrecision] * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Power[N[(w * w), $MachinePrecision], -1.0], $MachinePrecision] * 3.0 + N[(N[Power[N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.0999999999999999e-121

   1. Initial program 85.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   2. Taylor expanded in v around -inf

      \[\leadsto \color{blue}{\left(\left(3 + \left(\frac{1}{8} \cdot \frac{3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - 2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} - \frac{9}{2} \]

   4. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\left(\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(r \cdot r\right), w \cdot w, \left(\left(r \cdot r\right) \cdot 3\right) \cdot \left(w \cdot w\right)\right) \cdot 0.125, -r \cdot r, -2 \cdot v\right)}{v \cdot \left(-r \cdot r\right)} + 3\right) - \left(0.25 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\right)} - 4.5 \]

   if 2.0999999999999999e-121 < w

   1. Initial program 83.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   2. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)} - \frac{9}{2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

   4. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left({\left(r \cdot r\right)}^{-1}, 2, 3\right)}{w \cdot w} - \frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(-2, v, 3\right)}{1 - v} \cdot 0.125\right) \cdot \left(w \cdot w\right)} - 4.5 \]

   6. Applied rewrites 91.5%

      \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot \color{blue}{w} - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (*
   (*
    (fma
     (/ (* (* r r) (fma v -2.0 3.0)) (- 1.0 v))
     -0.125
     (fma (pow (* w w) -1.0) 3.0 (* (pow (pow (* w r) 2.0) -1.0) 2.0)))
    w)
   w)
  4.5))

C

double code(double v, double w, double r) {
	return ((fma((((r * r) * fma(v, -2.0, 3.0)) / (1.0 - v)), -0.125, fma(pow((w * w), -1.0), 3.0, (pow(pow((w * r), 2.0), -1.0) * 2.0))) * w) * w) - 4.5;
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(fma(Float64(Float64(Float64(r * r) * fma(v, -2.0, 3.0)) / Float64(1.0 - v)), -0.125, fma((Float64(w * w) ^ -1.0), 3.0, Float64(((Float64(w * r) ^ 2.0) ^ -1.0) * 2.0))) * w) * w) - 4.5)
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(N[(N[(N[(N[(r * r), $MachinePrecision] * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Power[N[(w * w), $MachinePrecision], -1.0], $MachinePrecision] * 3.0 + N[(N[Power[N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

2. Taylor expanded in w around inf

   \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)} - \frac{9}{2} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

4. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left({\left(r \cdot r\right)}^{-1}, 2, 3\right)}{w \cdot w} - \frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(-2, v, 3\right)}{1 - v} \cdot 0.125\right) \cdot \left(w \cdot w\right)} - 4.5 \]

6. Applied rewrites 73.2%

   \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot \color{blue}{w} - 4.5 \]
7. Add Preprocessing

Alternative 6: 63.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (*
   (*
    (fma
     (/ (* r (* r (fma v -2.0 3.0))) (- 1.0 v))
     -0.125
     (fma (pow (* w w) -1.0) 3.0 (* (pow (pow (* w r) 2.0) -1.0) 2.0)))
    w)
   w)
  4.5))

C

double code(double v, double w, double r) {
	return ((fma(((r * (r * fma(v, -2.0, 3.0))) / (1.0 - v)), -0.125, fma(pow((w * w), -1.0), 3.0, (pow(pow((w * r), 2.0), -1.0) * 2.0))) * w) * w) - 4.5;
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(fma(Float64(Float64(r * Float64(r * fma(v, -2.0, 3.0))) / Float64(1.0 - v)), -0.125, fma((Float64(w * w) ^ -1.0), 3.0, Float64(((Float64(w * r) ^ 2.0) ^ -1.0) * 2.0))) * w) * w) - 4.5)
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(N[(N[(N[(r * N[(r * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Power[N[(w * w), $MachinePrecision], -1.0], $MachinePrecision] * 3.0 + N[(N[Power[N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

2. Taylor expanded in w around inf

   \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)} - \frac{9}{2} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(3 \cdot \frac{1}{{w}^{2}} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{1}{8} \cdot \frac{{r}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \cdot \color{blue}{{w}^{2}} - \frac{9}{2} \]

4. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left({\left(r \cdot r\right)}^{-1}, 2, 3\right)}{w \cdot w} - \frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(-2, v, 3\right)}{1 - v} \cdot 0.125\right) \cdot \left(w \cdot w\right)} - 4.5 \]

6. Applied rewrites 73.2%

   \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot \color{blue}{w} - 4.5 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \left(v \cdot -2 + 3\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   4. associate-*l* N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   7. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + -2 \cdot v\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + -2 \cdot v\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(3 + v \cdot -2\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   10. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, \frac{-1}{8}, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - \frac{9}{2} \]

   11. lift-fma.f64 73.2

       \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5 \]

8. Applied rewrites 73.2%

   \[\leadsto \left(\mathsf{fma}\left(\frac{r \cdot \left(r \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, -0.125, \mathsf{fma}\left({\left(w \cdot w\right)}^{-1}, 3, {\left({\left(w \cdot r\right)}^{2}\right)}^{-1} \cdot 2\right)\right) \cdot w\right) \cdot w - 4.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))