Rosa's TurbineBenchmark

Average Error: 12.2 → 0.4

Time: 14.5s

Precision: binary64

Cost: 2632

Math

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

↓

\[\begin{array}{l} t_0 := r \cdot \left(w \cdot \left(r \cdot w\right)\right)\\ t_1 := \left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{w}}\\ t_2 := \frac{2}{r \cdot r}\\ t_3 := 3 + t_2\\ \mathbf{if}\;r \leq -1 \cdot 10^{+54}:\\ \;\;\;\;-4.5 + \left(t_3 + t_0 \cdot \frac{-0.375 - v \cdot -0.25}{1 - v}\right)\\ \mathbf{elif}\;r \leq 1.26 \cdot 10^{+129}:\\ \;\;\;\;t_2 + \left(-1.5 - \left(0.375 \cdot t_1 + \left(v \cdot -0.25\right) \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(t_3 + \frac{0.125 \cdot \left(-3 + 2 \cdot v\right)}{\frac{1 - v}{t_0}}\right)\\ \end{array} \]

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))

↓

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* w (* r w))))
        (t_1 (* (* r w) (/ r (/ (- 1.0 v) w))))
        (t_2 (/ 2.0 (* r r)))
        (t_3 (+ 3.0 t_2)))
   (if (<= r -1e+54)
     (+ -4.5 (+ t_3 (* t_0 (/ (- -0.375 (* v -0.25)) (- 1.0 v)))))
     (if (<= r 1.26e+129)
       (+ t_2 (- -1.5 (+ (* 0.375 t_1) (* (* v -0.25) t_1))))
       (+ -4.5 (+ t_3 (/ (* 0.125 (+ -3.0 (* 2.0 v))) (/ (- 1.0 v) t_0))))))))

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;
}

↓

double code(double v, double w, double r) {
	double t_0 = r * (w * (r * w));
	double t_1 = (r * w) * (r / ((1.0 - v) / w));
	double t_2 = 2.0 / (r * r);
	double t_3 = 3.0 + t_2;
	double tmp;
	if (r <= -1e+54) {
		tmp = -4.5 + (t_3 + (t_0 * ((-0.375 - (v * -0.25)) / (1.0 - v))));
	} else if (r <= 1.26e+129) {
		tmp = t_2 + (-1.5 - ((0.375 * t_1) + ((v * -0.25) * t_1)));
	} else {
		tmp = -4.5 + (t_3 + ((0.125 * (-3.0 + (2.0 * v))) / ((1.0 - v) / t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    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

↓

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = r * (w * (r * w))
    t_1 = (r * w) * (r / ((1.0d0 - v) / w))
    t_2 = 2.0d0 / (r * r)
    t_3 = 3.0d0 + t_2
    if (r <= (-1d+54)) then
        tmp = (-4.5d0) + (t_3 + (t_0 * (((-0.375d0) - (v * (-0.25d0))) / (1.0d0 - v))))
    else if (r <= 1.26d+129) then
        tmp = t_2 + ((-1.5d0) - ((0.375d0 * t_1) + ((v * (-0.25d0)) * t_1)))
    else
        tmp = (-4.5d0) + (t_3 + ((0.125d0 * ((-3.0d0) + (2.0d0 * v))) / ((1.0d0 - v) / t_0)))
    end if
    code = tmp
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;
}

↓

public static double code(double v, double w, double r) {
	double t_0 = r * (w * (r * w));
	double t_1 = (r * w) * (r / ((1.0 - v) / w));
	double t_2 = 2.0 / (r * r);
	double t_3 = 3.0 + t_2;
	double tmp;
	if (r <= -1e+54) {
		tmp = -4.5 + (t_3 + (t_0 * ((-0.375 - (v * -0.25)) / (1.0 - v))));
	} else if (r <= 1.26e+129) {
		tmp = t_2 + (-1.5 - ((0.375 * t_1) + ((v * -0.25) * t_1)));
	} else {
		tmp = -4.5 + (t_3 + ((0.125 * (-3.0 + (2.0 * v))) / ((1.0 - v) / t_0)));
	}
	return tmp;
}

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

↓

def code(v, w, r):
	t_0 = r * (w * (r * w))
	t_1 = (r * w) * (r / ((1.0 - v) / w))
	t_2 = 2.0 / (r * r)
	t_3 = 3.0 + t_2
	tmp = 0
	if r <= -1e+54:
		tmp = -4.5 + (t_3 + (t_0 * ((-0.375 - (v * -0.25)) / (1.0 - v))))
	elif r <= 1.26e+129:
		tmp = t_2 + (-1.5 - ((0.375 * t_1) + ((v * -0.25) * t_1)))
	else:
		tmp = -4.5 + (t_3 + ((0.125 * (-3.0 + (2.0 * v))) / ((1.0 - v) / t_0)))
	return tmp

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

↓

function code(v, w, r)
	t_0 = Float64(r * Float64(w * Float64(r * w)))
	t_1 = Float64(Float64(r * w) * Float64(r / Float64(Float64(1.0 - v) / w)))
	t_2 = Float64(2.0 / Float64(r * r))
	t_3 = Float64(3.0 + t_2)
	tmp = 0.0
	if (r <= -1e+54)
		tmp = Float64(-4.5 + Float64(t_3 + Float64(t_0 * Float64(Float64(-0.375 - Float64(v * -0.25)) / Float64(1.0 - v)))));
	elseif (r <= 1.26e+129)
		tmp = Float64(t_2 + Float64(-1.5 - Float64(Float64(0.375 * t_1) + Float64(Float64(v * -0.25) * t_1))));
	else
		tmp = Float64(-4.5 + Float64(t_3 + Float64(Float64(0.125 * Float64(-3.0 + Float64(2.0 * v))) / Float64(Float64(1.0 - v) / t_0))));
	end
	return tmp
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

↓

function tmp_2 = code(v, w, r)
	t_0 = r * (w * (r * w));
	t_1 = (r * w) * (r / ((1.0 - v) / w));
	t_2 = 2.0 / (r * r);
	t_3 = 3.0 + t_2;
	tmp = 0.0;
	if (r <= -1e+54)
		tmp = -4.5 + (t_3 + (t_0 * ((-0.375 - (v * -0.25)) / (1.0 - v))));
	elseif (r <= 1.26e+129)
		tmp = t_2 + (-1.5 - ((0.375 * t_1) + ((v * -0.25) * t_1)));
	else
		tmp = -4.5 + (t_3 + ((0.125 * (-3.0 + (2.0 * v))) / ((1.0 - v) / t_0)));
	end
	tmp_2 = tmp;
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]

↓

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * N[(r / N[(N[(1.0 - v), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 + t$95$2), $MachinePrecision]}, If[LessEqual[r, -1e+54], N[(-4.5 + N[(t$95$3 + N[(t$95$0 * N[(N[(-0.375 - N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.26e+129], N[(t$95$2 + N[(-1.5 - N[(N[(0.375 * t$95$1), $MachinePrecision] + N[(N[(v * -0.25), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(t$95$3 + N[(N[(0.125 * N[(-3.0 + N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if r < -1.0000000000000001e54

   1. Initial program 16.2

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

   2. Simplified 9.6

      \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -4.5} \]
      Proof
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (*.f64 (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (-.f64 1 v)) (*.f64 r (*.f64 r (*.f64 w w))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (*.f64 (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (-.f64 1 v)) (*.f64 r (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 w w) r))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (*.f64 (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (-.f64 1 v)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 w w) r) r)))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))) -9/2): 16 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) (Rewrite<= metadata-eval (neg.f64 9/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) 9/2)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in v around 0 9.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{-0.25 \cdot v + 0.375}}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -4.5 \]

   4. Taylor expanded in r around 0 9.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{-0.25 \cdot v + 0.375}{1 - v} \cdot \left(r \cdot \color{blue}{\left({w}^{2} \cdot r\right)}\right)\right) + -4.5 \]

   5. Simplified 0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{-0.25 \cdot v + 0.375}{1 - v} \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right)\right) + -4.5 \]
      Proof
      (*.f64 w (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w w) r)): 42 points increase in error, 43 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) r): 0 points increase in error, 0 points decrease in error

   if -1.0000000000000001e54 < r < 1.26e129

   1. Initial program 9.0

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

   2. Simplified 6.6

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{r}{\frac{\frac{1 - v}{r}}{w \cdot w}}\right)} \]
      Proof
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 -3/2 (*.f64 (fma.f64 v -1/4 3/8) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (Rewrite<= metadata-eval (-.f64 3 9/2)) (*.f64 (fma.f64 v -1/4 3/8) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (fma.f64 v (Rewrite<= metadata-eval (*.f64 -2 1/8)) 3/8) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (fma.f64 v (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) 1/8) 3/8) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (fma.f64 v (*.f64 (neg.f64 2) 1/8) (Rewrite<= metadata-eval (*.f64 3 1/8))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 v (*.f64 (neg.f64 2) 1/8)) (*.f64 3 1/8))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 v (neg.f64 2)) 1/8)) (*.f64 3 1/8)) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 2 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (+.f64 (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 v 2))) 1/8) (*.f64 3 1/8)) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (+.f64 (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 v))) 1/8) (*.f64 3 1/8)) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/8 (+.f64 (neg.f64 (*.f64 2 v)) 3))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (Rewrite<= +-commutative_binary64 (+.f64 3 (neg.f64 (*.f64 2 v))))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (Rewrite<= sub-neg_binary64 (-.f64 3 (*.f64 2 v)))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 r (Rewrite=> associate-/l/_binary64 (/.f64 (-.f64 1 v) (*.f64 (*.f64 w w) r))))))): 7 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 r (*.f64 (*.f64 w w) r)) (-.f64 1 v)))))): 4 points increase in error, 9 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (-.f64 1 v)) (*.f64 1/8 (-.f64 3 (*.f64 2 v))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (*.f64 1/8 (-.f64 3 (*.f64 2 v)))) (-.f64 1 v))))): 17 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r))) (-.f64 1 v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (Rewrite<= associate--r+_binary64 (-.f64 3 (+.f64 9/2 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 3 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)) 9/2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 2 (*.f64 r r)) 3) (+.f64 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)) 9/2))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 3 (/.f64 2 (*.f64 r r)))) (+.f64 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)) 9/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l-_binary64 (-.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) 9/2)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.3

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \color{blue}{\left(\frac{r}{\frac{1 - v}{w \cdot r}} \cdot w\right)}\right) \]

   4. Applied egg-rr 0.5

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{w}}\right) \cdot \left(v \cdot -0.25\right) + \left(\left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{w}}\right) \cdot 0.375\right)}\right) \]

   if 1.26e129 < r

   1. Initial program 23.9

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

   2. Simplified 16.2

      \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) + -4.5} \]
      Proof
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (+.f64 3 (*.f64 -2 v))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (+.f64 3 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) v))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 3 (*.f64 2 v)))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (-.f64 1 v) (*.f64 r (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 w w) r)))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (-.f64 1 v) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 w w) r) r))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))) -9/2): 19 points increase in error, 10 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) (Rewrite<= metadata-eval (neg.f64 9/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) 9/2)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in r around 0 16.2

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

   4. Simplified 0.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}}}\right) + -4.5 \]
      Proof
      (*.f64 w (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w w) r)): 42 points increase in error, 43 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) r): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -1 \cdot 10^{+54}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{-0.375 - v \cdot -0.25}{1 - v}\right)\\ \mathbf{elif}\;r \leq 1.26 \cdot 10^{+129}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{w}}\right) + \left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{w}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{0.125 \cdot \left(-3 + 2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	8000

\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

Alternative 2
Error	0.3
Cost	2368

\[\frac{2}{r \cdot r} + \left(-1.5 + \left(\frac{r \cdot w}{\frac{1 - v}{r \cdot w}} \cdot \left(v \cdot 0.25\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{w}}\right)\right)\right) \]

Alternative 3
Error	0.5
Cost	1992

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + \left(-1.5 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{if}\;v \leq -4 \cdot 10^{+199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 2900000000:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) + \left(r \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(-0.375 - v \cdot -0.25\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	1.3
Cost	1736

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + \left(-1.5 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{if}\;v \leq -2800000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 6 \cdot 10^{-46}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.375 + v \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.8
Cost	1736

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := -4.5 + \left(t_0 - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(0.25 - \frac{0.125}{v}\right)\right)\right)\\ \mathbf{if}\;v \leq -2800000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;-4.5 + \left(t_0 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.375 + v \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	2.3
Cost	1728

\[-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{-0.375 - v \cdot -0.25}{1 - v}\right) \]

Alternative 7
Error	15.8
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ t_2 := t_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -7 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 2 \cdot 10^{-148}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 8.5 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.8
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -7.2 \cdot 10^{-39}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{elif}\;r \leq 5 \cdot 10^{-148}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 8.5 \cdot 10^{+153}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	8.9
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + \left(-1.5 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\\ t_2 := t_0 + \left(-1.5 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{if}\;r \leq -5.4 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq -2.6 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 3 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	18.2
Cost	1480

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ \mathbf{if}\;w \cdot w \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \cdot w \leq \infty:\\ \;\;\;\;t_0 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	1.3
Cost	1480

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + \left(-1.5 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{if}\;v \leq -2800000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 6 \cdot 10^{-46}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	1.3
Cost	1352

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := t_1 + \left(-1.5 + -0.25 \cdot t_0\right)\\ \mathbf{if}\;v \leq -2800000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 6 \cdot 10^{-46}:\\ \;\;\;\;t_1 + \left(-1.5 - 0.375 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	20.8
Cost	448

\[\frac{2}{r \cdot r} + -1.5 \]

Alternative 14
Error	20.8
Cost	448

\[-1.5 + \frac{\frac{2}{r}}{r} \]

Reproduce

herbie shell --seed 2022329 
(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))