Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.7%

Time: 13.4s

Alternatives: 7

Speedup: 1.7×

• 53.1% 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

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

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: 85.0% 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

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

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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{r}}{r} + \left(-1.5 + \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{-1}{1 - v}\right)\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ (/ 2.0 r) r)
  (+ -1.5 (* (fma v -0.25 0.375) (* (* r w) (* (* r w) (/ -1.0 (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 + (fma(v, -0.25, 0.375) * ((r * w) * ((r * w) * (-1.0 / (1.0 - v))))));
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 + Float64(fma(v, -0.25, 0.375) * Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(-1.0 / Float64(1.0 - v)))))))
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 + N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(-1.0 / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 96.4%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 96.4%

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

   2. clear-num 96.4%

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

   3. associate-*r* 99.8%

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

   4. pow2 99.8%

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

   5. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

   \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{{\left(w \cdot r\right)}^{2}}{1 - v}}\right) \]
7. Step-by-step derivation

   1. div-inv 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ (/ 2.0 r) r)
  (- -1.5 (* (* r w) (* (* r w) (/ (fma v -0.25 0.375) (- 1.0 v)))))))

C

double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 - ((r * w) * ((r * w) * (fma(v, -0.25, 0.375) / (1.0 - v)))));
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(fma(v, -0.25, 0.375) / Float64(1.0 - v))))))
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 96.4%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r/ 96.4%

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

   2. associate-*r* 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -1 \cdot 10^{+42}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;v \leq 0.2:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(\frac{0.125}{v} - 0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (<= v -1e+42)
     (+ t_0 (- -1.5 (* (* r w) (* w (* r 0.25)))))
     (if (<= v 0.2)
       (+ t_0 (- -1.5 (* (* r w) (* 0.375 (* r w)))))
       (+ t_0 (+ -1.5 (* (* r w) (* (* r w) (- (/ 0.125 v) 0.25)))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (v <= -1e+42) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	} else if (v <= 0.2) {
		tmp = t_0 + (-1.5 - ((r * w) * (0.375 * (r * w))));
	} else {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * ((0.125 / v) - 0.25))));
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / r) / r
    if (v <= (-1d+42)) then
        tmp = t_0 + ((-1.5d0) - ((r * w) * (w * (r * 0.25d0))))
    else if (v <= 0.2d0) then
        tmp = t_0 + ((-1.5d0) - ((r * w) * (0.375d0 * (r * w))))
    else
        tmp = t_0 + ((-1.5d0) + ((r * w) * ((r * w) * ((0.125d0 / v) - 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (v <= -1e+42) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	} else if (v <= 0.2) {
		tmp = t_0 + (-1.5 - ((r * w) * (0.375 * (r * w))));
	} else {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * ((0.125 / v) - 0.25))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if v <= -1e+42:
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))))
	elif v <= 0.2:
		tmp = t_0 + (-1.5 - ((r * w) * (0.375 * (r * w))))
	else:
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * ((0.125 / v) - 0.25))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (v <= -1e+42)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.25)))));
	elseif (v <= 0.2)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(0.375 * Float64(r * w)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(Float64(0.125 / v) - 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	tmp = 0.0;
	if (v <= -1e+42)
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	elseif (v <= 0.2)
		tmp = t_0 + (-1.5 - ((r * w) * (0.375 * (r * w))));
	else
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * ((0.125 / v) - 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[LessEqual[v, -1e+42], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 0.2], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(0.125 / v), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000004e42

   1. Initial program 87.6%

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

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 96.2%

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

      2. associate-*r* 99.7%

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

      3. associate-*r* 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 99.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \color{blue}{\left(0.25 \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot r\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* 99.7%

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

   9. Simplified 99.7%

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

   if -1.00000000000000004e42 < v < 0.20000000000000001

   1. Initial program 87.9%

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

   3. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 96.4%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

   if 0.20000000000000001 < v

   1. Initial program 72.5%

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

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 96.6%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\color{blue}{\left(0.25 - 0.125 \cdot \frac{1}{v}\right)} \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(0.25 - \color{blue}{\frac{0.125 \cdot 1}{v}}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\right) \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(0.25 - \frac{\color{blue}{0.125}}{v}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\right) \]

   9. Simplified 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\color{blue}{\left(0.25 - \frac{0.125}{v}\right)} \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;v \leq 0.2:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(\frac{0.125}{v} - 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;r \leq 9 \cdot 10^{-100}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (<= r 9e-100)
     (+ t_0 (- -1.5 (* (* r w) (* w (* r 0.25)))))
     (+
      t_0
      (- -1.5 (* (* r w) (/ (* r (* w (+ 0.375 (* v -0.25)))) (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (r <= 9e-100) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * ((r * (w * (0.375 + (v * -0.25)))) / (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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / r) / r
    if (r <= 9d-100) then
        tmp = t_0 + ((-1.5d0) - ((r * w) * (w * (r * 0.25d0))))
    else
        tmp = t_0 + ((-1.5d0) - ((r * w) * ((r * (w * (0.375d0 + (v * (-0.25d0))))) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (r <= 9e-100) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * ((r * (w * (0.375 + (v * -0.25)))) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if r <= 9e-100:
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))))
	else:
		tmp = t_0 + (-1.5 - ((r * w) * ((r * (w * (0.375 + (v * -0.25)))) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (r <= 9e-100)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(Float64(r * Float64(w * Float64(0.375 + Float64(v * -0.25)))) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	tmp = 0.0;
	if (r <= 9e-100)
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	else
		tmp = t_0 + (-1.5 - ((r * w) * ((r * (w * (0.375 + (v * -0.25)))) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[LessEqual[r, 9e-100], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(N[(r * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 9.0000000000000002e-100

   1. Initial program 80.0%

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

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 94.8%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 96.6%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \color{blue}{\left(0.25 \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot r\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* 96.6%

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

   9. Simplified 96.6%

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

   if 9.0000000000000002e-100 < r

   1. Initial program 93.0%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 99.8%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in w around 0 98.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)}{1 - v}} \cdot \left(w \cdot r\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{1 - v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -1.5 \cdot 10^{+41} \lor \neg \left(v \leq 8.5 \cdot 10^{-62}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (or (<= v -1.5e+41) (not (<= v 8.5e-62)))
     (+ t_0 (- -1.5 (* (* r w) (* w (* r 0.25)))))
     (+ t_0 (- -1.5 (* (* r w) (* 0.375 (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if ((v <= -1.5e+41) || !(v <= 8.5e-62)) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * (0.375 * (r * w))));
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / r) / r
    if ((v <= (-1.5d+41)) .or. (.not. (v <= 8.5d-62))) then
        tmp = t_0 + ((-1.5d0) - ((r * w) * (w * (r * 0.25d0))))
    else
        tmp = t_0 + ((-1.5d0) - ((r * w) * (0.375d0 * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if ((v <= -1.5e+41) || !(v <= 8.5e-62)) {
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	} else {
		tmp = t_0 + (-1.5 - ((r * w) * (0.375 * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if (v <= -1.5e+41) or not (v <= 8.5e-62):
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))))
	else:
		tmp = t_0 + (-1.5 - ((r * w) * (0.375 * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -1.5e+41) || !(v <= 8.5e-62))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.25)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * w) * Float64(0.375 * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	tmp = 0.0;
	if ((v <= -1.5e+41) || ~((v <= 8.5e-62)))
		tmp = t_0 + (-1.5 - ((r * w) * (w * (r * 0.25))));
	else
		tmp = t_0 + (-1.5 - ((r * w) * (0.375 * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -1.5e+41], N[Not[LessEqual[v, 8.5e-62]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.4999999999999999e41 or 8.4999999999999995e-62 < v

   1. Initial program 81.0%

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

   3. Simplified 96.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 96.8%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf 99.7%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \color{blue}{\left(0.25 \cdot \left(r \cdot w\right)\right)} \cdot \left(w \cdot r\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* 99.7%

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

   9. Simplified 99.7%

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

   if -1.4999999999999999e41 < v < 8.4999999999999995e-62

   1. Initial program 87.4%

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

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 96.0%

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

      2. associate-*r* 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

      \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\color{blue}{0.375} \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.5 \cdot 10^{+41} \lor \neg \left(v \leq 8.5 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ (/ 2.0 r) r)
  (+ -1.5 (* (* r w) (/ (- (* v -0.25) -0.375) (/ (/ (+ v -1.0) r) w))))))

C

double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 + ((r * w) * (((v * -0.25) - -0.375) / (((v + -1.0) / r) / w))));
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / r) / r) + ((-1.5d0) + ((r * w) * (((v * (-0.25d0)) - (-0.375d0)) / (((v + (-1.0d0)) / r) / w))))
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 + ((r * w) * (((v * -0.25) - -0.375) / (((v + -1.0) / r) / w))));
}

Python

def code(v, w, r):
	return ((2.0 / r) / r) + (-1.5 + ((r * w) * (((v * -0.25) - -0.375) / (((v + -1.0) / r) / w))))

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(Float64(v * -0.25) - -0.375) / Float64(Float64(Float64(v + -1.0) / r) / w)))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / r) / r) + (-1.5 + ((r * w) * (((v * -0.25) - -0.375) / (((v + -1.0) / r) / w))));
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(N[(v * -0.25), $MachinePrecision] - -0.375), $MachinePrecision] / N[(N[(N[(v + -1.0), $MachinePrecision] / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 96.4%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r/ 96.4%

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

   2. associate-*r* 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.1%

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

   2. frac-2neg 99.1%

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

8. Applied egg-rr 99.1%

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

   1. distribute-lft-neg-in 99.1%

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

   2. associate-/l* 99.8%

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

   3. neg-sub0 99.8%

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

   4. fma-udef 99.8%

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

   5. *-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. *-commutative 99.8%

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

   8. associate--r+ 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-/r* 99.4%

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

   12. neg-sub0 99.4%

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

   13. associate--r- 99.4%

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

   14. metadata-eval 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 7: 93.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+ (/ (/ 2.0 r) r) (- -1.5 (* (* r w) (* 0.375 (* r w))))))

C

double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 - ((r * w) * (0.375 * (r * w))));
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / r) / r) + ((-1.5d0) - ((r * w) * (0.375d0 * (r * w))))
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 - ((r * w) * (0.375 * (r * w))));
}

Python

def code(v, w, r):
	return ((2.0 / r) / r) + (-1.5 - ((r * w) * (0.375 * (r * w))))

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(Float64(r * w) * Float64(0.375 * Float64(r * w)))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / r) / r) + (-1.5 - ((r * w) * (0.375 * (r * w))));
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(N[(r * w), $MachinePrecision] * N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 96.4%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r/ 96.4%

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

   2. associate-*r* 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in v around 0 94.3%

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

8. Final simplification 94.3%

   \[\leadsto \frac{\frac{2}{r}}{r} + \left(-1.5 - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))