Rosa's TurbineBenchmark

Average Error: 12.8 → 0.4

Time: 22.2s

Precision: binary64

Cost: 14404

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 3 + t_0\\ t_2 := {\left(w \cdot r\right)}^{2}\\ \mathbf{if}\;v \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\left(t_2 \cdot 0.25 + \left(t_1 + \frac{t_2}{-2}\right)\right) - 4.5\\ \mathbf{elif}\;v \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(t_1 - \frac{\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(-0.25 \cdot {\left(r \cdot w\right)}^{2} + t_0\right) + -1.5\\ \end{array} \]

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))

↓

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ 3.0 t_0)) (t_2 (pow (* w r) 2.0)))
   (if (<= v -1e+31)
     (- (+ (* t_2 0.25) (+ t_1 (/ t_2 -2.0))) 4.5)
     (if (<= v 5e+15)
       (-
        (- t_1 (/ (* (- 3.0 (+ v v)) (* 0.125 (* (* w r) (* w r)))) (- 1.0 v)))
        4.5)
       (+ (+ (* -0.25 (pow (* r w) 2.0)) t_0) -1.5)))))

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;
}

↓

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double t_2 = pow((w * r), 2.0);
	double tmp;
	if (v <= -1e+31) {
		tmp = ((t_2 * 0.25) + (t_1 + (t_2 / -2.0))) - 4.5;
	} else if (v <= 5e+15) {
		tmp = (t_1 - (((3.0 - (v + v)) * (0.125 * ((w * r) * (w * r)))) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((-0.25 * pow((r * w), 2.0)) + t_0) + -1.5;
	}
	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) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = 3.0d0 + t_0
    t_2 = (w * r) ** 2.0d0
    if (v <= (-1d+31)) then
        tmp = ((t_2 * 0.25d0) + (t_1 + (t_2 / (-2.0d0)))) - 4.5d0
    else if (v <= 5d+15) then
        tmp = (t_1 - (((3.0d0 - (v + v)) * (0.125d0 * ((w * r) * (w * r)))) / (1.0d0 - v))) - 4.5d0
    else
        tmp = (((-0.25d0) * ((r * w) ** 2.0d0)) + t_0) + (-1.5d0)
    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 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double t_2 = Math.pow((w * r), 2.0);
	double tmp;
	if (v <= -1e+31) {
		tmp = ((t_2 * 0.25) + (t_1 + (t_2 / -2.0))) - 4.5;
	} else if (v <= 5e+15) {
		tmp = (t_1 - (((3.0 - (v + v)) * (0.125 * ((w * r) * (w * r)))) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((-0.25 * Math.pow((r * w), 2.0)) + t_0) + -1.5;
	}
	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 = 2.0 / (r * r)
	t_1 = 3.0 + t_0
	t_2 = math.pow((w * r), 2.0)
	tmp = 0
	if v <= -1e+31:
		tmp = ((t_2 * 0.25) + (t_1 + (t_2 / -2.0))) - 4.5
	elif v <= 5e+15:
		tmp = (t_1 - (((3.0 - (v + v)) * (0.125 * ((w * r) * (w * r)))) / (1.0 - v))) - 4.5
	else:
		tmp = ((-0.25 * math.pow((r * w), 2.0)) + t_0) + -1.5
	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(2.0 / Float64(r * r))
	t_1 = Float64(3.0 + t_0)
	t_2 = Float64(w * r) ^ 2.0
	tmp = 0.0
	if (v <= -1e+31)
		tmp = Float64(Float64(Float64(t_2 * 0.25) + Float64(t_1 + Float64(t_2 / -2.0))) - 4.5);
	elseif (v <= 5e+15)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(3.0 - Float64(v + v)) * Float64(0.125 * Float64(Float64(w * r) * Float64(w * r)))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(Float64(-0.25 * (Float64(r * w) ^ 2.0)) + t_0) + -1.5);
	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 = 2.0 / (r * r);
	t_1 = 3.0 + t_0;
	t_2 = (w * r) ^ 2.0;
	tmp = 0.0;
	if (v <= -1e+31)
		tmp = ((t_2 * 0.25) + (t_1 + (t_2 / -2.0))) - 4.5;
	elseif (v <= 5e+15)
		tmp = (t_1 - (((3.0 - (v + v)) * (0.125 * ((w * r) * (w * r)))) / (1.0 - v))) - 4.5;
	else
		tmp = ((-0.25 * ((r * w) ^ 2.0)) + t_0) + -1.5;
	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[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[v, -1e+31], N[(N[(N[(t$95$2 * 0.25), $MachinePrecision] + N[(t$95$1 + N[(t$95$2 / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 5e+15], N[(N[(t$95$1 - N[(N[(N[(3.0 - N[(v + v), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(-0.25 * N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + -1.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -9.9999999999999996e30

   1. Initial program 18.7

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

   2. Simplified 21.6

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

      [Start] 18.7	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 18.7	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 19.7	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 19.7	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 21.6	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{r \cdot \color{blue}{\left(r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(w \cdot w\right)\right)\right)}}{1 - v}\right) - 4.5 \]

   3. Taylor expanded in v around inf 17.5

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

   4. Simplified 0.4

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

      [Start] 17.5	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)\right) - 4.5 \]
      rational_best-simplify-2 [=>] 17.5	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left({w}^{2} \cdot {r}^{2}\right) \cdot 0.25}\right) - 4.5 \]
      exponential-simplify-27 [=>] 0.4	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot 0.25\right) - 4.5 \]
      rational_best-simplify-2 [=>] 0.4	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - {\color{blue}{\left(w \cdot r\right)}}^{2} \cdot 0.25\right) - 4.5 \]

   5. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\left(\left(3 + \left(\frac{2}{r \cdot r} + {\left(w \cdot r\right)}^{2} \cdot 0.25\right)\right) + \left(-{\left(w \cdot r\right)}^{2} \cdot 0.5\right)\right)} - 4.5 \]

   6. Simplified 0.4

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

      [Start] 0.4	\[ \left(\left(3 + \left(\frac{2}{r \cdot r} + {\left(w \cdot r\right)}^{2} \cdot 0.25\right)\right) + \left(-{\left(w \cdot r\right)}^{2} \cdot 0.5\right)\right) - 4.5 \]
      rational_best-simplify-1 [=>] 0.4	\[ \color{blue}{\left(\left(-{\left(w \cdot r\right)}^{2} \cdot 0.5\right) + \left(3 + \left(\frac{2}{r \cdot r} + {\left(w \cdot r\right)}^{2} \cdot 0.25\right)\right)\right)} - 4.5 \]
      rational_best-simplify-43 [=>] 0.4	\[ \left(\left(-{\left(w \cdot r\right)}^{2} \cdot 0.5\right) + \color{blue}{\left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\frac{2}{r \cdot r} + 3\right)\right)}\right) - 4.5 \]
      rational_best-simplify-1 [<=] 0.4	\[ \left(\left(-{\left(w \cdot r\right)}^{2} \cdot 0.5\right) + \left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \color{blue}{\left(3 + \frac{2}{r \cdot r}\right)}\right)\right) - 4.5 \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(\left(-{\left(w \cdot r\right)}^{2} \cdot 0.5\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + {\left(w \cdot r\right)}^{2} \cdot 0.25\right)}\right) - 4.5 \]
      rational_best-simplify-43 [=>] 0.4	\[ \color{blue}{\left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-{\left(w \cdot r\right)}^{2} \cdot 0.5\right)\right)\right)} - 4.5 \]
      rational_best-simplify-11 [=>] 0.4	\[ \left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(0 - {\left(w \cdot r\right)}^{2} \cdot 0.5\right)}\right)\right) - 4.5 \]
      rational_best-simplify-4 [<=] 0.4	\[ \left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(0 - \color{blue}{\left({\left(w \cdot r\right)}^{2} \cdot 0.5 + 0\right)}\right)\right)\right) - 4.5 \]
      rational_best-simplify-60 [=>] 0.4	\[ \left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \color{blue}{\frac{{\left(w \cdot r\right)}^{2} \cdot 0.5 + {\left(w \cdot r\right)}^{2} \cdot 0.5}{-2}}\right)\right) - 4.5 \]
      rational_best-simplify-47 [=>] 0.4	\[ \left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{\color{blue}{{\left(w \cdot r\right)}^{2} \cdot \left(0.5 + 0.5\right)}}{-2}\right)\right) - 4.5 \]
      metadata-eval [=>] 0.4	\[ \left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{{\left(w \cdot r\right)}^{2} \cdot \color{blue}{1}}{-2}\right)\right) - 4.5 \]
      rational_best-simplify-5 [=>] 0.4	\[ \left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{\color{blue}{{\left(w \cdot r\right)}^{2}}}{-2}\right)\right) - 4.5 \]

   if -9.9999999999999996e30 < v < 5e15

   1. Initial program 8.2

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

   2. Simplified 8.2

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

      [Start] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{r \cdot \color{blue}{\left(r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(w \cdot w\right)\right)\right)}}{1 - v}\right) - 4.5 \]

   3. Applied egg-rr 8.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{r \cdot \left(\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)}{1 - v} + 0\right)}\right) - 4.5 \]

   4. Simplified 0.4

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

      [Start] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{r \cdot \left(\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)}{1 - v} + 0\right)\right) - 4.5 \]
      rational_best-simplify-4 [=>] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{r \cdot \left(\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)}{1 - v}}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(3 - \left(v + v\right)\right) \cdot \left(r \cdot \left(0.125 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)}}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 8.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(3 - \left(v + v\right)\right) \cdot \color{blue}{\left(0.125 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)}}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 2.3	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}\right)\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 2.3	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right)\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 0.4	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 0.4	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot \left(r \cdot w\right)\right)\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 0.4	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)\right)}{1 - v}\right) - 4.5 \]

   if 5e15 < v

   1. Initial program 17.8

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

   2. Simplified 21.0

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

      [Start] 17.8	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 17.8	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 18.9	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) - 4.5 \]
      rational_best-simplify-2 [=>] 18.9	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right)}{1 - v}\right) - 4.5 \]
      rational_best-simplify-44 [=>] 21.0	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{r \cdot \color{blue}{\left(r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(w \cdot w\right)\right)\right)}}{1 - v}\right) - 4.5 \]

   3. Taylor expanded in v around inf 17.0

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

   4. Simplified 0.4

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

      [Start] 17.0	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)\right) - 4.5 \]
      rational_best-simplify-2 [=>] 17.0	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left({w}^{2} \cdot {r}^{2}\right) \cdot 0.25}\right) - 4.5 \]
      exponential-simplify-27 [=>] 0.4	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot 0.25\right) - 4.5 \]
      rational_best-simplify-2 [=>] 0.4	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - {\color{blue}{\left(w \cdot r\right)}}^{2} \cdot 0.25\right) - 4.5 \]

   5. Applied egg-rr 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5 - {\left(w \cdot r\right)}^{2} \cdot 0.25\right) \]
      rational_best-simplify-6 [<=] 0.4	\[ \left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5 - \color{blue}{\left({\left(w \cdot r\right)}^{2} \cdot 0.25 - 0\right)}\right) \]
      rational_best-simplify-45 [=>] 0.4	\[ \left(3 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(0 - \left({\left(w \cdot r\right)}^{2} \cdot 0.25 - -4.5\right)\right)} \]
      rational_best-simplify-46 [=>] 0.4	\[ \left(3 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(-4.5 + \left(0 - {\left(w \cdot r\right)}^{2} \cdot 0.25\right)\right)} \]
      rational_best-simplify-11 [<=] 0.4	\[ \left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5 + \color{blue}{\left(-{\left(w \cdot r\right)}^{2} \cdot 0.25\right)}\right) \]
      rational_best-simplify-1 [<=] 0.4	\[ \left(3 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(\left(-{\left(w \cdot r\right)}^{2} \cdot 0.25\right) + -4.5\right)} \]
      rational_best-simplify-43 [<=] 0.4	\[ \color{blue}{-4.5 + \left(\left(-{\left(w \cdot r\right)}^{2} \cdot 0.25\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} \]
      rational_best-simplify-1 [=>] 0.4	\[ -4.5 + \left(\left(-{\left(w \cdot r\right)}^{2} \cdot 0.25\right) + \color{blue}{\left(\frac{2}{r \cdot r} + 3\right)}\right) \]
      rational_best-simplify-43 [=>] 0.4	\[ -4.5 + \color{blue}{\left(3 + \left(\frac{2}{r \cdot r} + \left(-{\left(w \cdot r\right)}^{2} \cdot 0.25\right)\right)\right)} \]
      rational_best-simplify-43 [=>] 0.4	\[ \color{blue}{\left(\frac{2}{r \cdot r} + \left(-{\left(w \cdot r\right)}^{2} \cdot 0.25\right)\right) + \left(3 + -4.5\right)} \]
      metadata-eval [=>] 0.4	\[ \left(\frac{2}{r \cdot r} + \left(-{\left(w \cdot r\right)}^{2} \cdot 0.25\right)\right) + \color{blue}{-1.5} \]

   7. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\left(\left(\frac{2}{r \cdot r} + {\left(r \cdot w\right)}^{2} \cdot -0.25\right) - 0\right)} + -1.5 \]

   8. Simplified 0.4

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

      [Start] 0.4	\[ \left(\left(\frac{2}{r \cdot r} + {\left(r \cdot w\right)}^{2} \cdot -0.25\right) - 0\right) + -1.5 \]
      rational_best-simplify-6 [=>] 0.4	\[ \color{blue}{\left(\frac{2}{r \cdot r} + {\left(r \cdot w\right)}^{2} \cdot -0.25\right)} + -1.5 \]
      rational_best-simplify-1 [=>] 0.4	\[ \color{blue}{\left({\left(r \cdot w\right)}^{2} \cdot -0.25 + \frac{2}{r \cdot r}\right)} + -1.5 \]
      rational_best-simplify-2 [=>] 0.4	\[ \left(\color{blue}{-0.25 \cdot {\left(r \cdot w\right)}^{2}} + \frac{2}{r \cdot r}\right) + -1.5 \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\left({\left(w \cdot r\right)}^{2} \cdot 0.25 + \left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{{\left(w \cdot r\right)}^{2}}{-2}\right)\right) - 4.5\\ \mathbf{elif}\;v \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(-0.25 \cdot {\left(r \cdot w\right)}^{2} + \frac{2}{r \cdot r}\right) + -1.5\\ \end{array} \]

Alternatives

Alternative 1
Error	4.0
Cost	8708

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ \mathbf{if}\;t_0 - \frac{t_1 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -\infty:\\ \;\;\;\;-0.25 \cdot {\left(w \cdot r\right)}^{2} - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - \frac{t_1 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \end{array} \]

Alternative 2
Error	0.4
Cost	7560

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(-0.25 \cdot {\left(r \cdot w\right)}^{2} + t_0\right) + -1.5\\ \mathbf{if}\;v \leq -1 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 2.9 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(3 + t_0\right) - \frac{\left(3 - \left(v + v\right)\right) \cdot \left(0.125 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	6.2
Cost	3652

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ \mathbf{if}\;t_0 - \frac{t_1 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(t_0 - \frac{\left(r \cdot \left(0.375 \cdot w\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - \frac{t_1 \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \end{array} \]

Alternative 4
Error	9.8
Cost	2128

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{\left(r \cdot \left(0.375 \cdot w\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;v \leq -1.28 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq -1500000:\\ \;\;\;\;\left(t_0 - \frac{-0.25 \cdot \left(r \cdot \left(v \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{elif}\;v \leq 3.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+103}:\\ \;\;\;\;\left(t_0 - \frac{-0.25 \cdot \left(\left(v \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - \frac{r \cdot \left(\left(0.375 \cdot r\right) \cdot \left(w \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \]

Alternative 5
Error	6.9
Cost	2120

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{r \cdot \left(0.125 \cdot \left(w \cdot \left(r \cdot \left(\left(3 - \left(v + v\right)\right) \cdot w\right)\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;r \leq -4000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 360000000:\\ \;\;\;\;\left(t_0 - \frac{\left(w \cdot r\right) \cdot \left(w \cdot \left(0.375 \cdot r\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	7.8
Cost	1996

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{\left(r \cdot \left(0.375 \cdot w\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ t_2 := \left(t_0 - \frac{v \cdot \left(\left(w \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;v \leq -5.8 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq -1500000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	7.8
Cost	1996

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{\left(r \cdot \left(0.375 \cdot w\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;v \leq -4.3 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq -1500000:\\ \;\;\;\;\left(t_0 - \frac{v \cdot \left(\left(w \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - \frac{\left(w \cdot r\right) \cdot \left(-0.25 \cdot \left(v \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \]

Alternative 8
Error	7.9
Cost	1996

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{\left(r \cdot \left(0.375 \cdot w\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;v \leq -5.5 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq -1550000:\\ \;\;\;\;\left(t_0 - \frac{r \cdot \left(0.125 \cdot \left(w \cdot \left(r \cdot \left(-2 \cdot \left(v \cdot w\right)\right)\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 - \frac{\left(w \cdot r\right) \cdot \left(-0.25 \cdot \left(v \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \]

Alternative 9
Error	10.5
Cost	1864

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{\left(r \cdot \left(0.375 \cdot w\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;v \leq -1.15 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq -35000000:\\ \;\;\;\;\left(t_0 - \frac{-0.25 \cdot \left(r \cdot \left(v \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	12.3
Cost	1472

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

Alternative 11
Error	11.0
Cost	1472

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

Reproduce

herbie shell --seed 2023092 
(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))