Rosa's TurbineBenchmark

Average Error: 13.1 → 7.8

Time: 18.8s

Precision: binary64

Cost: 14024

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ \mathbf{if}\;r \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;t_1 - \frac{r \cdot \left(w \cdot \left(r \cdot \left(w \cdot \left(0.375 - 0.25 \cdot v\right)\right)\right)\right)}{1 - v}\\ \mathbf{elif}\;r \leq -1 \cdot 10^{-50}:\\ \;\;\;\;t_1 - {w}^{2} \cdot \left(0.25 \cdot {r}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right)\\ \end{array} \]

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 (/ 2.0 (* r r))) (t_1 (+ t_0 -1.5)))
   (if (<= r -1.3e+154)
     (- t_1 (/ (* r (* w (* r (* w (- 0.375 (* 0.25 v)))))) (- 1.0 v)))
     (if (<= r -1e-50)
       (- t_1 (* (pow w 2.0) (* 0.25 (pow r 2.0))))
       (+
        t_0
        (-
         -1.5
         (/ (* (- 0.375 (* v 0.25)) (* (* w r) (* w r))) (- 1.0 v))))))))

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 = t_0 + -1.5;
	double tmp;
	if (r <= -1.3e+154) {
		tmp = t_1 - ((r * (w * (r * (w * (0.375 - (0.25 * v)))))) / (1.0 - v));
	} else if (r <= -1e-50) {
		tmp = t_1 - (pow(w, 2.0) * (0.25 * pow(r, 2.0)));
	} else {
		tmp = t_0 + (-1.5 - (((0.375 - (v * 0.25)) * ((w * r) * (w * r))) / (1.0 - v)));
	}
	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) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = t_0 + (-1.5d0)
    if (r <= (-1.3d+154)) then
        tmp = t_1 - ((r * (w * (r * (w * (0.375d0 - (0.25d0 * v)))))) / (1.0d0 - v))
    else if (r <= (-1d-50)) then
        tmp = t_1 - ((w ** 2.0d0) * (0.25d0 * (r ** 2.0d0)))
    else
        tmp = t_0 + ((-1.5d0) - (((0.375d0 - (v * 0.25d0)) * ((w * r) * (w * r))) / (1.0d0 - v)))
    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 = t_0 + -1.5;
	double tmp;
	if (r <= -1.3e+154) {
		tmp = t_1 - ((r * (w * (r * (w * (0.375 - (0.25 * v)))))) / (1.0 - v));
	} else if (r <= -1e-50) {
		tmp = t_1 - (Math.pow(w, 2.0) * (0.25 * Math.pow(r, 2.0)));
	} else {
		tmp = t_0 + (-1.5 - (((0.375 - (v * 0.25)) * ((w * r) * (w * r))) / (1.0 - v)));
	}
	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 = t_0 + -1.5
	tmp = 0
	if r <= -1.3e+154:
		tmp = t_1 - ((r * (w * (r * (w * (0.375 - (0.25 * v)))))) / (1.0 - v))
	elif r <= -1e-50:
		tmp = t_1 - (math.pow(w, 2.0) * (0.25 * math.pow(r, 2.0)))
	else:
		tmp = t_0 + (-1.5 - (((0.375 - (v * 0.25)) * ((w * r) * (w * r))) / (1.0 - v)))
	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(t_0 + -1.5)
	tmp = 0.0
	if (r <= -1.3e+154)
		tmp = Float64(t_1 - Float64(Float64(r * Float64(w * Float64(r * Float64(w * Float64(0.375 - Float64(0.25 * v)))))) / Float64(1.0 - v)));
	elseif (r <= -1e-50)
		tmp = Float64(t_1 - Float64((w ^ 2.0) * Float64(0.25 * (r ^ 2.0))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 - Float64(v * 0.25)) * Float64(Float64(w * r) * Float64(w * r))) / Float64(1.0 - v))));
	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 = t_0 + -1.5;
	tmp = 0.0;
	if (r <= -1.3e+154)
		tmp = t_1 - ((r * (w * (r * (w * (0.375 - (0.25 * v)))))) / (1.0 - v));
	elseif (r <= -1e-50)
		tmp = t_1 - ((w ^ 2.0) * (0.25 * (r ^ 2.0)));
	else
		tmp = t_0 + (-1.5 - (((0.375 - (v * 0.25)) * ((w * r) * (w * r))) / (1.0 - v)));
	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[(t$95$0 + -1.5), $MachinePrecision]}, If[LessEqual[r, -1.3e+154], N[(t$95$1 - N[(N[(r * N[(w * N[(r * N[(w * N[(0.375 - N[(0.25 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, -1e-50], N[(t$95$1 - N[(N[Power[w, 2.0], $MachinePrecision] * N[(0.25 * N[Power[r, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 - N[(v * 0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < -1.29999999999999994e154

   1. Initial program 27.6

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

   2. Simplified 27.7

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

      [Start] 27.6	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
      rational_best_oopsla_all_46_json_45_simplify-105 [=>] 27.6	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 27.6	\[ \color{blue}{\left(\frac{2}{r \cdot r} + \left(3 - 4.5\right)\right)} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \]
      metadata-eval [=>] 27.6	\[ \left(\frac{2}{r \cdot r} + \color{blue}{-1.5}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.6	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\color{blue}{\left(3 \cdot 0.125 - 0.125 \cdot \left(2 \cdot v\right)\right)} \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      metadata-eval [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(\color{blue}{0.375} - 0.125 \cdot \left(2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - 0.125 \cdot \color{blue}{\left(v \cdot 2\right)}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - \color{blue}{v \cdot \left(0.125 \cdot 2\right)}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      metadata-eval [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot \color{blue}{0.25}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right)}{1 - v} \]

   3. Applied egg-rr 27.7

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right) + 0\right)}\right)}{1 - v} \]

   4. Simplified 11.3

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

      [Start] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \left(r \cdot \left(w \cdot w\right) + 0\right)\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 11.3	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 11.3	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \left(w \cdot \color{blue}{\left(w \cdot r\right)}\right)\right)}{1 - v} \]

   5. Applied egg-rr 27.7

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \color{blue}{\left(\left(0.375 - v \cdot 0.25\right) \cdot \left(r \cdot \left(w \cdot w\right)\right) + 0\right)}}{1 - v} \]

   6. Simplified 11.3

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

      [Start] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \left(r \cdot \left(w \cdot w\right)\right) + 0\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 27.7	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \color{blue}{\left(\left(0.375 - v \cdot 0.25\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 11.3	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 11.3	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \color{blue}{\left(w \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \left(r \cdot w\right)\right)\right)}}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 11.3	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(w \cdot \color{blue}{\left(r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot w\right)\right)}\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 11.3	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(w \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(0.375 - v \cdot 0.25\right)\right)}\right)\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 11.3	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(w \cdot \left(r \cdot \left(w \cdot \left(0.375 - \color{blue}{0.25 \cdot v}\right)\right)\right)\right)}{1 - v} \]

   if -1.29999999999999994e154 < r < -1.00000000000000001e-50

   1. Initial program 7.2

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

   2. Simplified 7.5

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

      [Start] 7.2	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
      rational_best_oopsla_all_46_json_45_simplify-105 [=>] 7.2	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 7.2	\[ \color{blue}{\left(\frac{2}{r \cdot r} + \left(3 - 4.5\right)\right)} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \]
      metadata-eval [=>] 7.2	\[ \left(\frac{2}{r \cdot r} + \color{blue}{-1.5}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 7.2	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 7.5	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{\color{blue}{r \cdot \left(\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 7.5	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\color{blue}{\left(3 \cdot 0.125 - 0.125 \cdot \left(2 \cdot v\right)\right)} \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      metadata-eval [=>] 7.5	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(\color{blue}{0.375} - 0.125 \cdot \left(2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 7.5	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - 0.125 \cdot \color{blue}{\left(v \cdot 2\right)}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 7.5	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - \color{blue}{v \cdot \left(0.125 \cdot 2\right)}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      metadata-eval [=>] 7.5	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot \color{blue}{0.25}\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{1 - v} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 7.5	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right)}{1 - v} \]

   3. Taylor expanded in v around inf 9.9

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

   4. Simplified 9.9

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

      [Start] 9.9	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - 0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 9.9	\[ \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{{w}^{2} \cdot \left(0.25 \cdot {r}^{2}\right)} \]

   if -1.00000000000000001e-50 < r

   1. Initial program 12.8

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

   2. Simplified 12.8

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

      [Start] 12.8	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 12.8	\[ \color{blue}{\left(\frac{2}{r \cdot r} + \left(3 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} - 4.5 \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 12.8	\[ \color{blue}{\left(\left(3 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \frac{2}{r \cdot r}\right)} - 4.5 \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 12.8	\[ \color{blue}{\frac{2}{r \cdot r} + \left(\left(3 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-105 [=>] 12.8	\[ \frac{2}{r \cdot r} + \color{blue}{\left(\left(3 - 4.5\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} \]
      metadata-eval [=>] 12.8	\[ \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 12.8	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(3 \cdot 0.125 - 0.125 \cdot \left(2 \cdot v\right)\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
      metadata-eval [=>] 12.8	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(\color{blue}{0.375} - 0.125 \cdot \left(2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 12.8	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - 0.125 \cdot \color{blue}{\left(v \cdot 2\right)}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 12.8	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - \color{blue}{v \cdot \left(0.125 \cdot 2\right)}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
      metadata-eval [=>] 12.8	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - v \cdot \color{blue}{0.25}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 12.8	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}}{1 - v}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 12.8	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right)}{1 - v}\right) \]

   3. Applied egg-rr 6.5

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(0 - \left(r \cdot w\right) \cdot \left(r \cdot \left(-w\right)\right)\right)}}{1 - v}\right) \]

   4. Applied egg-rr 8.7

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(0 + \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)}{1 - v}\right)}\right) \]

   5. Simplified 6.5

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

      [Start] 8.7	\[ \frac{2}{r \cdot r} + \left(-1.5 - \left(0 + \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)}{1 - v}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 8.7	\[ \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)}{1 - v} + 0\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 8.7	\[ \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)}{1 - v}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 8.7	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(w \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot r\right)}\right)}{1 - v}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 6.5	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 6.5	\[ \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r \cdot \left(w \cdot \left(r \cdot \left(w \cdot \left(0.375 - 0.25 \cdot v\right)\right)\right)\right)}{1 - v}\\ \mathbf{elif}\;r \leq -1 \cdot 10^{-50}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - {w}^{2} \cdot \left(0.25 \cdot {r}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	9.0
Cost	2128

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ t_2 := t_1 - \frac{0 - \left(r \cdot \left(w \cdot -0.375\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\\ t_3 := -1.5 + \left(t_0 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)}{1 - v}\right)\\ \mathbf{if}\;r \leq -3.9 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq -1.4 \cdot 10^{-16}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;r \leq 2 \cdot 10^{-68}:\\ \;\;\;\;t_1 - \frac{w \cdot \left(w \cdot \left(r \cdot \left(0.375 \cdot r\right)\right)\right)}{1 - v}\\ \mathbf{elif}\;r \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	9.2
Cost	2128

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ t_2 := t_0 + \left(-1.5 - \frac{\left(0.375 - v \cdot 0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{1 - v}\right)\\ \mathbf{if}\;w \leq -6.4 \cdot 10^{+61}:\\ \;\;\;\;t_1 - \frac{w \cdot \left(\left(0.375 \cdot w\right) \cdot \left(r \cdot r\right)\right)}{1 - v}\\ \mathbf{elif}\;w \leq -1.9 \cdot 10^{-96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;w \leq 10^{-145}:\\ \;\;\;\;t_1 - \frac{0 - \left(r \cdot \left(w \cdot -0.375\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\\ \mathbf{elif}\;w \leq 1.1 \cdot 10^{+26}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{w \cdot \left(w \cdot \left(r \cdot \left(0.375 \cdot r\right)\right)\right)}{1 - v}\\ \end{array} \]

Alternative 3
Error	11.4
Cost	1608

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ t_1 := t_0 - \frac{r \cdot \left(0.375 \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\\ \mathbf{if}\;r \leq -1 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 2 \cdot 10^{-68}:\\ \;\;\;\;t_0 - \frac{0.375 \cdot \left(w \cdot \left(r \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	11.3
Cost	1608

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;r \leq -500000:\\ \;\;\;\;t_0 - \frac{r \cdot \left(w \cdot \left(0.375 \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\\ \mathbf{elif}\;r \leq 5 \cdot 10^{-66}:\\ \;\;\;\;t_0 - \frac{0.375 \cdot \left(w \cdot \left(r \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{r \cdot \left(0.375 \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\\ \end{array} \]

Alternative 5
Error	11.3
Cost	1608

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;r \leq -4000000000:\\ \;\;\;\;t_0 - \frac{r \cdot \left(w \cdot \left(r \cdot \left(0.375 \cdot w\right)\right)\right)}{1 - v}\\ \mathbf{elif}\;r \leq 5 \cdot 10^{-66}:\\ \;\;\;\;t_0 - \frac{0.375 \cdot \left(w \cdot \left(r \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{r \cdot \left(0.375 \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\\ \end{array} \]

Alternative 6
Error	11.3
Cost	1608

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;r \leq -1 \cdot 10^{-36}:\\ \;\;\;\;t_0 - \frac{r \cdot \left(w \cdot \left(r \cdot \left(0.375 \cdot w\right)\right)\right)}{1 - v}\\ \mathbf{elif}\;r \leq 4 \cdot 10^{-18}:\\ \;\;\;\;t_0 - \frac{w \cdot \left(w \cdot \left(r \cdot \left(0.375 \cdot r\right)\right)\right)}{1 - v}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{r \cdot \left(0.375 \cdot \left(w \cdot \left(w \cdot r\right)\right)\right)}{1 - v}\\ \end{array} \]

Alternative 7
Error	9.8
Cost	1604

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq 420000:\\ \;\;\;\;t_0 - \frac{-\left(\left(r \cdot w\right) \cdot -0.375\right) \cdot \left(r \cdot w\right)}{1 - v}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{r \cdot \left(w \cdot \left(v \cdot \left(-0.25 \cdot \left(w \cdot r\right)\right)\right)\right)}{1 - v}\\ \end{array} \]

Alternative 8
Error	9.8
Cost	1604

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq 260000:\\ \;\;\;\;t_0 - \frac{0 - \left(r \cdot \left(w \cdot -0.375\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{r \cdot \left(w \cdot \left(v \cdot \left(-0.25 \cdot \left(w \cdot r\right)\right)\right)\right)}{1 - v}\\ \end{array} \]

Alternative 9
Error	7.1
Cost	1600

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

Alternative 10
Error	11.2
Cost	1408

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

Alternative 11
Error	13.5
Cost	1344

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

Reproduce

herbie shell --seed 2023090 
(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))