Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.3%

Time: 13.2s

Alternatives: 13

Speedup: 1.2×

• 53.3% 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: 84.8% 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: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + (pow(r, -2.0) * 2.0)) + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (v + -1.0))))) - 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 + ((r ** (-2.0d0)) * 2.0d0)) + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (w * (r / (v + (-1.0d0)))))) - 4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. associate--l- 83.4%

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

   2. associate-*l* 76.4%

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

   3. sqr-neg 76.4%

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

   4. associate-*l* 83.4%

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

   5. associate-/l* 84.9%

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

   6. fma-define 84.9%

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

3. Simplified 84.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l* 84.9%

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

   2. *-commutative 84.9%

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

   3. associate-*r/ 84.9%

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

   4. associate-*l* 96.0%

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

   5. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. clear-num 99.7%

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

   2. associate-/r/ 99.7%

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

   3. pow2 99.7%

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

   4. pow-flip 99.8%

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

   5. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.65 \cdot 10^{-111}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \frac{r \cdot w}{\frac{v}{r \cdot w}}\right)\\ \mathbf{elif}\;r \leq 50:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - \frac{2}{r} \cdot \frac{-1}{r}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.65e-111)
   (- (+ 3.0 (/ 2.0 (* r r))) (+ 4.5 (* 0.375 (/ (* r w) (/ v (* r w))))))
   (if (<= r 50.0)
     (- (- -1.5 (* 0.375 (* r (* r (* w w))))) (* (/ 2.0 r) (/ -1.0 r)))
     (-
      3.0
      (+
       4.5
       (* (* 0.125 (+ 3.0 (* -2.0 v))) (/ (* w (* r w)) (/ (- 1.0 v) r))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.65e-111) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	} else if (r <= 50.0) {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((w * (r * w)) / ((1.0 - v) / r))));
	}
	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) :: tmp
    if (r <= 1.65d-111) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - (4.5d0 + (0.375d0 * ((r * w) / (v / (r * w)))))
    else if (r <= 50.0d0) then
        tmp = ((-1.5d0) - (0.375d0 * (r * (r * (w * w))))) - ((2.0d0 / r) * ((-1.0d0) / r))
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((w * (r * w)) / ((1.0d0 - v) / r))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.65e-111) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	} else if (r <= 50.0) {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((w * (r * w)) / ((1.0 - v) / r))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.65e-111:
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))))
	elif r <= 50.0:
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r))
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((w * (r * w)) / ((1.0 - v) / r))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.65e-111)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) / Float64(v / Float64(r * w))))));
	elseif (r <= 50.0)
		tmp = Float64(Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))) - Float64(Float64(2.0 / r) * Float64(-1.0 / r)));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(w * Float64(r * w)) / Float64(Float64(1.0 - v) / r)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.65e-111)
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	elseif (r <= 50.0)
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((w * (r * w)) / ((1.0 - v) / r))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.65e-111], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 50.0], N[(N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 / r), $MachinePrecision] * N[(-1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.65e-111

   1. Initial program 80.5%

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

      1. associate--l- 80.5%

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

      2. associate-*l* 74.4%

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

      3. sqr-neg 74.4%

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

      4. associate-*l* 80.5%

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

      5. associate-/l* 81.6%

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

      6. fma-define 81.6%

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

   3. Simplified 81.6%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r/ 81.6%

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

      4. associate-*l* 95.4%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. mul-1-neg 82.8%

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

      3. *-commutative 82.8%

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

      4. distribute-rgt-neg-in 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in v around 0 71.1%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \frac{w \cdot \left(-r\right)}{v}\right) + 4.5\right) \]
   11. Step-by-step derivation

       1. clear-num 71.1%

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

       2. un-div-inv 71.1%

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

       3. *-commutative 71.1%

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

       4. add-sqr-sqrt 46.9%

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

       5. sqrt-unprod 71.0%

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

       6. sqr-neg 71.0%

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

       7. sqrt-prod 24.8%

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

       8. add-sqr-sqrt 71.2%

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

   12. Applied egg-rr 71.2%

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

   if 1.65e-111 < r < 50

   1. Initial program 76.8%

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

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 57.9%

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

      1. associate-/r* 58.1%

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

      2. div-inv 57.9%

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

   7. Applied egg-rr 57.9%

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

   8. Taylor expanded in v around 0 69.9%

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

   if 50 < r

   1. Initial program 95.0%

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

      1. associate--l- 95.0%

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

      2. associate-*l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-*l* 95.0%

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

      5. associate-/l* 96.7%

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

      6. fma-define 96.7%

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

   3. Simplified 96.7%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-*r/ 96.6%

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

      4. associate-*l* 98.2%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 99.7%

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

      1. associate-*r* 99.9%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.65 \cdot 10^{-111}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \frac{r \cdot w}{\frac{v}{r \cdot w}}\right)\\ \mathbf{elif}\;r \leq 50:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - \frac{2}{r} \cdot \frac{-1}{r}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \frac{r \cdot w}{\frac{v}{r \cdot w}}\right)\\ \mathbf{elif}\;r \leq 6 \cdot 10^{+37}:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - \frac{2}{r} \cdot \frac{-1}{r}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e-113)
   (- (+ 3.0 (/ 2.0 (* r r))) (+ 4.5 (* 0.375 (/ (* r w) (/ v (* r w))))))
   (if (<= r 6e+37)
     (- (- -1.5 (* 0.375 (* r (* r (* w w))))) (* (/ 2.0 r) (/ -1.0 r)))
     (-
      3.0
      (+
       (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* w (/ r (- 1.0 v)))))
       4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e-113) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	} else if (r <= 6e+37) {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	} else {
		tmp = 3.0 - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))) + 4.5);
	}
	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
    real(8) :: tmp
    if (r <= 5d-113) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - (4.5d0 + (0.375d0 * ((r * w) / (v / (r * w)))))
    else if (r <= 6d+37) then
        tmp = ((-1.5d0) - (0.375d0 * (r * (r * (w * w))))) - ((2.0d0 / r) * ((-1.0d0) / r))
    else
        tmp = 3.0d0 - (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (w * (r / (1.0d0 - v))))) + 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e-113) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	} else if (r <= 6e+37) {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	} else {
		tmp = 3.0 - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))) + 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5e-113:
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))))
	elif r <= 6e+37:
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r))
	else:
		tmp = 3.0 - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))) + 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e-113)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) / Float64(v / Float64(r * w))))));
	elseif (r <= 6e+37)
		tmp = Float64(Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))) - Float64(Float64(2.0 / r) * Float64(-1.0 / r)));
	else
		tmp = Float64(3.0 - Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(w * Float64(r / Float64(1.0 - v))))) + 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5e-113)
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	elseif (r <= 6e+37)
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	else
		tmp = 3.0 - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))) + 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e-113], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 6e+37], N[(N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 / r), $MachinePrecision] * N[(-1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.9999999999999997e-113

   1. Initial program 80.5%

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

      1. associate--l- 80.5%

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

      2. associate-*l* 74.4%

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

      3. sqr-neg 74.4%

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

      4. associate-*l* 80.5%

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

      5. associate-/l* 81.6%

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

      6. fma-define 81.6%

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

   3. Simplified 81.6%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r/ 81.6%

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

      4. associate-*l* 95.4%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. mul-1-neg 82.8%

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

      3. *-commutative 82.8%

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

      4. distribute-rgt-neg-in 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in v around 0 71.1%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \frac{w \cdot \left(-r\right)}{v}\right) + 4.5\right) \]
   11. Step-by-step derivation

       1. clear-num 71.1%

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

       2. un-div-inv 71.1%

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

       3. *-commutative 71.1%

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

       4. add-sqr-sqrt 46.9%

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

       5. sqrt-unprod 71.0%

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

       6. sqr-neg 71.0%

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

       7. sqrt-prod 24.8%

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

       8. add-sqr-sqrt 71.2%

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

   12. Applied egg-rr 71.2%

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

   if 4.9999999999999997e-113 < r < 6.00000000000000043e37

   1. Initial program 83.1%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 69.4%

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

      1. associate-/r* 69.5%

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

      2. div-inv 69.4%

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

   7. Applied egg-rr 69.4%

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

   8. Taylor expanded in v around 0 78.1%

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

   if 6.00000000000000043e37 < r

   1. Initial program 94.2%

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

      1. associate--l- 94.2%

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

      2. associate-*l* 79.8%

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

      3. sqr-neg 79.8%

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

      4. associate-*l* 94.2%

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

      5. associate-/l* 96.2%

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

      6. fma-define 96.2%

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

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 96.2%

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

      2. *-commutative 96.2%

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

      3. associate-*r/ 96.2%

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

      4. associate-*l* 98.0%

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

      5. associate-*r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in r around inf 99.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \frac{r \cdot w}{\frac{v}{r \cdot w}}\right)\\ \mathbf{elif}\;r \leq 6 \cdot 10^{+37}:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - \frac{2}{r} \cdot \frac{-1}{r}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \frac{r \cdot w}{\frac{v}{r \cdot w}}\right)\\ \mathbf{elif}\;r \leq 52:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - \frac{2}{r} \cdot \frac{-1}{r}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(w \cdot \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.2e-113)
   (- (+ 3.0 (/ 2.0 (* r r))) (+ 4.5 (* 0.375 (/ (* r w) (/ v (* r w))))))
   (if (<= r 52.0)
     (- (- -1.5 (* 0.375 (* r (* r (* w w))))) (* (/ 2.0 r) (/ -1.0 r)))
     (-
      3.0
      (+
       4.5
       (* (* 0.125 (+ 3.0 (* -2.0 v))) (* w (* r (* w (/ r (- 1.0 v)))))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.2e-113) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	} else if (r <= 52.0) {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * (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
    real(8) :: tmp
    if (r <= 5.2d-113) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - (4.5d0 + (0.375d0 * ((r * w) / (v / (r * w)))))
    else if (r <= 52.0d0) then
        tmp = ((-1.5d0) - (0.375d0 * (r * (r * (w * w))))) - ((2.0d0 / r) * ((-1.0d0) / r))
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * (w * (r * (w * (r / (1.0d0 - v)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.2e-113) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	} else if (r <= 52.0) {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * (w * (r * (w * (r / (1.0 - v)))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.2e-113:
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))))
	elif r <= 52.0:
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r))
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * (w * (r * (w * (r / (1.0 - v)))))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.2e-113)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) / Float64(v / Float64(r * w))))));
	elseif (r <= 52.0)
		tmp = Float64(Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))) - Float64(Float64(2.0 / r) * Float64(-1.0 / r)));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(w * Float64(r * Float64(w * Float64(r / Float64(1.0 - v))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.2e-113)
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	elseif (r <= 52.0)
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * (w * (r * (w * (r / (1.0 - v)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.2e-113], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 52.0], N[(N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 / r), $MachinePrecision] * N[(-1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.1999999999999998e-113

   1. Initial program 80.5%

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

      1. associate--l- 80.5%

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

      2. associate-*l* 74.4%

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

      3. sqr-neg 74.4%

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

      4. associate-*l* 80.5%

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

      5. associate-/l* 81.6%

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

      6. fma-define 81.6%

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

   3. Simplified 81.6%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r/ 81.6%

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

      4. associate-*l* 95.4%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. mul-1-neg 82.8%

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

      3. *-commutative 82.8%

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

      4. distribute-rgt-neg-in 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in v around 0 71.1%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \frac{w \cdot \left(-r\right)}{v}\right) + 4.5\right) \]
   11. Step-by-step derivation

       1. clear-num 71.1%

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

       2. un-div-inv 71.1%

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

       3. *-commutative 71.1%

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

       4. add-sqr-sqrt 46.9%

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

       5. sqrt-unprod 71.0%

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

       6. sqr-neg 71.0%

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

       7. sqrt-prod 24.8%

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

       8. add-sqr-sqrt 71.2%

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

   12. Applied egg-rr 71.2%

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

   if 5.1999999999999998e-113 < r < 52

   1. Initial program 76.8%

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

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 57.9%

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

      1. associate-/r* 58.1%

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

      2. div-inv 57.9%

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

   7. Applied egg-rr 57.9%

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

   8. Taylor expanded in v around 0 69.9%

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

   if 52 < r

   1. Initial program 95.0%

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

      1. associate--l- 95.0%

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

      2. associate-*l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-*l* 95.0%

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

      5. associate-/l* 96.7%

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

      6. fma-define 96.7%

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

   3. Simplified 96.7%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-*r/ 96.6%

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

      4. *-commutative 96.6%

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

      5. associate-*l* 98.2%

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

      6. associate-*l* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.2 \cdot 10^{-113}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \frac{r \cdot w}{\frac{v}{r \cdot w}}\right)\\ \mathbf{elif}\;r \leq 52:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - \frac{2}{r} \cdot \frac{-1}{r}\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(w \cdot \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(3 - \frac{\frac{-1}{r}}{r \cdot 0.5}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (- 3.0 (/ (/ -1.0 r) (* r 0.5)))
  (+ (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* w (/ r (- 1.0 v))))) 4.5)))

C

double code(double v, double w, double r) {
	return (3.0 - ((-1.0 / r) / (r * 0.5))) - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (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 - (((-1.0d0) / r) / (r * 0.5d0))) - (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (w * (r / (1.0d0 - v))))) + 4.5d0)
end function

Java

public static double code(double v, double w, double r) {
	return (3.0 - ((-1.0 / r) / (r * 0.5))) - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))) + 4.5);
}

Python

def code(v, w, r):
	return (3.0 - ((-1.0 / r) / (r * 0.5))) - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))) + 4.5)

Julia

function code(v, w, r)
	return Float64(Float64(3.0 - Float64(Float64(-1.0 / r) / Float64(r * 0.5))) - Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(w * Float64(r / Float64(1.0 - v))))) + 4.5))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 - ((-1.0 / r) / (r * 0.5))) - (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))) + 4.5);
end

Wolfram

code[v_, w_, r_] := N[(N[(3.0 - N[(N[(-1.0 / r), $MachinePrecision] / N[(r * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

   1. associate--l- 83.4%

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

   2. associate-*l* 76.4%

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

   3. sqr-neg 76.4%

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

   4. associate-*l* 83.4%

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

   5. associate-/l* 84.9%

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

   6. fma-define 84.9%

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

3. Simplified 84.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l* 84.9%

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

   2. *-commutative 84.9%

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

   3. associate-*r/ 84.9%

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

   4. associate-*l* 96.0%

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

   5. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. clear-num 99.7%

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

   2. associate-/r/ 99.7%

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

   3. pow2 99.7%

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

   4. pow-flip 99.8%

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

   5. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. metadata-eval 99.8%

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

   3. metadata-eval 99.8%

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

   4. pow-sqr 99.7%

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

   5. inv-pow 99.7%

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

   6. inv-pow 99.7%

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

   7. un-div-inv 99.8%

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

   8. times-frac 99.8%

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

   9. *-un-lft-identity 99.8%

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

   10. associate-*l/ 99.7%

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

   11. *-commutative 99.7%

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

   12. clear-num 99.7%

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

   13. un-div-inv 99.8%

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

   14. div-inv 99.8%

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

   15. metadata-eval 99.8%

       \[\leadsto \left(3 + \frac{\frac{1}{r}}{r \cdot \color{blue}{0.5}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]

10. Applied egg-rr 99.8%

    \[\leadsto \left(3 + \color{blue}{\frac{\frac{1}{r}}{r \cdot 0.5}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]

11. Final simplification 99.8%

    \[\leadsto \left(3 - \frac{\frac{-1}{r}}{r \cdot 0.5}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
12. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (+ 3.0 (/ 2.0 (* r r)))
  (- (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* w (/ r (+ v -1.0))))) 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))) * ((r * w) * (w * (r / (v + -1.0))))) - 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))) * ((r * w) * (w * (r / (v + (-1.0d0)))))) - 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))) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. associate--l- 83.4%

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

   2. associate-*l* 76.4%

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

   3. sqr-neg 76.4%

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

   4. associate-*l* 83.4%

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

   5. associate-/l* 84.9%

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

   6. fma-define 84.9%

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

3. Simplified 84.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l* 84.9%

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

   2. *-commutative 84.9%

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

   3. associate-*r/ 84.9%

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

   4. associate-*l* 96.0%

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

   5. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 41:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{\frac{v + -1}{r}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 41.0)
   (+
    (+ 3.0 (/ 2.0 (* r r)))
    (- (* (* v -0.25) (* (* r w) (/ (* r w) v))) 4.5))
   (+
    3.0
    (-
     (* (* 0.125 (+ 3.0 (* -2.0 v))) (/ (* w (* r w)) (/ (+ v -1.0) r)))
     4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 41.0) {
		tmp = (3.0 + (2.0 / (r * r))) + (((v * -0.25) * ((r * w) * ((r * w) / v))) - 4.5);
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((w * (r * w)) / ((v + -1.0) / r))) - 4.5);
	}
	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
    real(8) :: tmp
    if (r <= 41.0d0) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) + (((v * (-0.25d0)) * ((r * w) * ((r * w) / v))) - 4.5d0)
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((w * (r * w)) / ((v + (-1.0d0)) / r))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 41.0) {
		tmp = (3.0 + (2.0 / (r * r))) + (((v * -0.25) * ((r * w) * ((r * w) / v))) - 4.5);
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((w * (r * w)) / ((v + -1.0) / r))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 41.0:
		tmp = (3.0 + (2.0 / (r * r))) + (((v * -0.25) * ((r * w) * ((r * w) / v))) - 4.5)
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((w * (r * w)) / ((v + -1.0) / r))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 41.0)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(Float64(v * -0.25) * Float64(Float64(r * w) * Float64(Float64(r * w) / v))) - 4.5));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(w * Float64(r * w)) / Float64(Float64(v + -1.0) / r))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 41.0)
		tmp = (3.0 + (2.0 / (r * r))) + (((v * -0.25) * ((r * w) * ((r * w) / v))) - 4.5);
	else
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((w * (r * w)) / ((v + -1.0) / r))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 41.0], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 41

   1. Initial program 80.1%

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

      1. associate--l- 80.1%

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

      2. associate-*l* 74.7%

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

      3. sqr-neg 74.7%

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

      4. associate-*l* 80.1%

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

      5. associate-/l* 81.6%

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

      6. fma-define 81.5%

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

   3. Simplified 81.6%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r/ 81.6%

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

      4. associate-*l* 95.4%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 81.2%

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

      1. associate-*r/ 81.2%

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

      2. mul-1-neg 81.2%

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

      3. *-commutative 81.2%

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

      4. distribute-rgt-neg-in 81.2%

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

   9. Simplified 81.2%

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

   10. Taylor expanded in v around inf 89.2%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(\left(r \cdot w\right) \cdot \frac{w \cdot \left(-r\right)}{v}\right) + 4.5\right) \]
   11. Step-by-step derivation

       1. *-commutative 89.2%

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

   12. Simplified 89.2%

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

   if 41 < r

   1. Initial program 95.0%

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

      1. associate--l- 95.0%

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

      2. associate-*l* 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-*l* 95.0%

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

      5. associate-/l* 96.7%

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

      6. fma-define 96.7%

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

   3. Simplified 96.7%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-*r/ 96.6%

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

      4. associate-*l* 98.2%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 99.7%

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

      1. associate-*r* 99.9%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 41:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{\frac{v + -1}{r}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.9 \cdot 10^{-113}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \frac{r \cdot w}{\frac{v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - \frac{2}{r} \cdot \frac{-1}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.9e-113)
   (- (+ 3.0 (/ 2.0 (* r r))) (+ 4.5 (* 0.375 (/ (* r w) (/ v (* r w))))))
   (- (- -1.5 (* 0.375 (* r (* r (* w w))))) (* (/ 2.0 r) (/ -1.0 r)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.9e-113) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	} else {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	}
	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) :: tmp
    if (r <= 2.9d-113) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - (4.5d0 + (0.375d0 * ((r * w) / (v / (r * w)))))
    else
        tmp = ((-1.5d0) - (0.375d0 * (r * (r * (w * w))))) - ((2.0d0 / r) * ((-1.0d0) / r))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.9e-113) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	} else {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.9e-113:
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))))
	else:
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.9e-113)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) / Float64(v / Float64(r * w))))));
	else
		tmp = Float64(Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))) - Float64(Float64(2.0 / r) * Float64(-1.0 / r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.9e-113)
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) / (v / (r * w)))));
	else
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) - ((2.0 / r) * (-1.0 / r));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.9e-113], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] / N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 / r), $MachinePrecision] * N[(-1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.90000000000000004e-113

   1. Initial program 80.5%

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

      1. associate--l- 80.5%

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

      2. associate-*l* 74.4%

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

      3. sqr-neg 74.4%

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

      4. associate-*l* 80.5%

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

      5. associate-/l* 81.6%

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

      6. fma-define 81.6%

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

   3. Simplified 81.6%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r/ 81.6%

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

      4. associate-*l* 95.4%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. mul-1-neg 82.8%

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

      3. *-commutative 82.8%

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

      4. distribute-rgt-neg-in 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in v around 0 71.1%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \frac{w \cdot \left(-r\right)}{v}\right) + 4.5\right) \]
   11. Step-by-step derivation

       1. clear-num 71.1%

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

       2. un-div-inv 71.1%

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

       3. *-commutative 71.1%

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

       4. add-sqr-sqrt 46.9%

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

       5. sqrt-unprod 71.0%

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

       6. sqr-neg 71.0%

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

       7. sqrt-prod 24.8%

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

       8. add-sqr-sqrt 71.2%

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

   12. Applied egg-rr 71.2%

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

   if 2.90000000000000004e-113 < r

   1. Initial program 90.1%

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

   3. Simplified 92.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 73.7%

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

      1. associate-/r* 73.7%

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

      2. div-inv 73.7%

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

   7. Applied egg-rr 73.7%

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

   8. Taylor expanded in v around 0 87.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.9 \cdot 10^{-113}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \frac{r \cdot w}{\frac{v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) - \frac{2}{r} \cdot \frac{-1}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / ((v + -1.0) / ((r * w) * (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) + (((v * (-0.25d0)) + 0.375d0) / ((v + (-1.0d0)) / ((r * w) * (r * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 84.9%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 84.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   2. *-commutative 84.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   3. +-commutative 84.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   4. associate-*r/ 84.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

   5. *-commutative 84.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

   6. associate-/l* 84.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

   7. clear-num 84.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   8. un-div-inv 85.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   9. +-commutative 85.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   10. distribute-rgt-in 85.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 85.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 85.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 85.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   14. metadata-eval 85.0%

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

   15. associate-*r* 77.6%

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

   16. pow2 77.6%

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

   17. pow2 77.6%

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

   18. pow-prod-down 99.7%

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

6. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

   2. *-commutative 99.7%

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

   3. *-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 10: 46.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.5 \cdot 10^{-124}:\\ \;\;\;\;3 - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \left(w + v \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3.5e-124)
   (- 3.0 (+ 4.5 (* 0.375 (* (* r w) (* r (+ w (* v w)))))))
   (- 3.0 (+ 4.5 (* 0.375 (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.5e-124) {
		tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (r * (w + (v * w))))));
	} else {
		tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (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) :: tmp
    if (r <= 3.5d-124) then
        tmp = 3.0d0 - (4.5d0 + (0.375d0 * ((r * w) * (r * (w + (v * w))))))
    else
        tmp = 3.0d0 - (4.5d0 + (0.375d0 * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.5e-124) {
		tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (r * (w + (v * w))))));
	} else {
		tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3.5e-124:
		tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (r * (w + (v * w))))))
	else:
		tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3.5e-124)
		tmp = Float64(3.0 - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) * Float64(r * Float64(w + Float64(v * w)))))));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3.5e-124)
		tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (r * (w + (v * w))))));
	else
		tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3.5e-124], N[(3.0 - N[(4.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * N[(w + N[(v * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.4999999999999999e-124

   1. Initial program 80.4%

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

      1. associate--l- 80.4%

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

      2. associate-*l* 74.3%

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

      3. sqr-neg 74.3%

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

      4. associate-*l* 80.4%

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

      5. associate-/l* 81.5%

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

      6. fma-define 81.5%

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

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 81.5%

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

      2. *-commutative 81.5%

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

      3. associate-*r/ 81.5%

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

      4. associate-*l* 95.4%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 40.6%

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

   8. Taylor expanded in v around 0 29.2%

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

   9. Taylor expanded in v around 0 27.4%

      \[\leadsto 3 - \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w + r \cdot \left(v \cdot w\right)\right)}\right) + 4.5\right) \]
   10. Step-by-step derivation

       1. distribute-lft-out 28.0%

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

       2. *-commutative 28.0%

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

   11. Simplified 28.0%

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

   if 3.4999999999999999e-124 < r

   1. Initial program 90.2%

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

      1. associate--l- 90.2%

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

      2. associate-*l* 81.3%

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

      3. sqr-neg 81.3%

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

      4. associate-*l* 90.2%

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

      5. associate-/l* 92.6%

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

      6. fma-define 92.6%

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

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 92.6%

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

      2. *-commutative 92.6%

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

      3. associate-*r/ 92.6%

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

      4. associate-*l* 97.5%

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

      5. associate-*r* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 86.4%

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

   8. Taylor expanded in v around 0 62.1%

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

   9. Taylor expanded in v around 0 79.4%

      \[\leadsto 3 - \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) + 4.5\right) \]
   10. Step-by-step derivation

       1. *-commutative 79.4%

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

   11. Simplified 79.4%

       \[\leadsto 3 - \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) + 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.5 \cdot 10^{-124}:\\ \;\;\;\;3 - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \left(w + v \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 84.9%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 73.2%

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

   1. associate-/r* 73.2%

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

   2. div-inv 73.2%

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

7. Applied egg-rr 73.2%

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

8. Taylor expanded in v around 0 81.2%

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

9. Final simplification 81.2%

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

Alternative 12: 49.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ 3 - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (- 3.0 (+ 4.5 (* 0.375 (* (* r w) (* r w))))))

C

double code(double v, double w, double r) {
	return 3.0 - (4.5 + (0.375 * ((r * w) * (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 = 3.0d0 - (4.5d0 + (0.375d0 * ((r * w) * (r * w))))
end function

Java

public static double code(double v, double w, double r) {
	return 3.0 - (4.5 + (0.375 * ((r * w) * (r * w))));
}

Python

def code(v, w, r):
	return 3.0 - (4.5 + (0.375 * ((r * w) * (r * w))))

Julia

function code(v, w, r)
	return Float64(3.0 - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = 3.0 - (4.5 + (0.375 * ((r * w) * (r * w))));
end

Wolfram

code[v_, w_, r_] := N[(3.0 - N[(4.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

   1. associate--l- 83.4%

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

   2. associate-*l* 76.4%

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

   3. sqr-neg 76.4%

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

   4. associate-*l* 83.4%

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

   5. associate-/l* 84.9%

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

   6. fma-define 84.9%

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

3. Simplified 84.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l* 84.9%

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

   2. *-commutative 84.9%

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

   3. associate-*r/ 84.9%

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

   4. associate-*l* 96.0%

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

   5. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in r around inf 54.7%

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

8. Taylor expanded in v around 0 39.3%

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

9. Taylor expanded in v around 0 48.8%

   \[\leadsto 3 - \left(0.375 \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) + 4.5\right) \]
10. Step-by-step derivation

    1. *-commutative 48.8%

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

11. Simplified 48.8%

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

12. Final simplification 48.8%

    \[\leadsto 3 - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \]
13. Add Preprocessing

Alternative 13: 14.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 83.4%

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

   1. associate--l- 83.4%

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

   2. associate-*l* 76.4%

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

   3. sqr-neg 76.4%

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

   4. associate-*l* 83.4%

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

   5. associate-/l* 84.9%

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

   6. fma-define 84.9%

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

3. Simplified 84.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l* 84.9%

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

   2. *-commutative 84.9%

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

   3. associate-*r/ 84.9%

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

   4. associate-*l* 96.0%

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

   5. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in r around inf 54.7%

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

8. Taylor expanded in v around 0 39.3%

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

9. Taylor expanded in r around 0 15.7%

   \[\leadsto \color{blue}{-1.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024176 
(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))