Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.7%

Time: 16.2s

Alternatives: 12

Speedup: 1.7×

• 52.3% 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: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ 3.0 (/ 2.0 (* r r)))
  (+ (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (/ (* r w) (- 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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (1.0 - v)))) + 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(Float64(r * w) / 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))) * ((r * w) * ((r * w) / (1.0 - v)))) + 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[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

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

   1. associate--l- 85.3%

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

   2. associate-*l* 78.4%

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

   3. sqr-neg 78.4%

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

   4. associate-*l* 85.3%

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

   5. associate-/l* 87.2%

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

   6. fma-define 87.2%

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

3. Simplified 87.2%

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

   1. *-un-lft-identity 87.2%

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

   2. add-sqr-sqrt 87.1%

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

   3. times-frac 87.1%

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

   4. *-commutative 87.1%

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

   5. sqrt-prod 41.4%

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

   6. *-commutative 41.4%

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

   7. sqrt-prod 41.4%

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

   8. sqrt-prod 18.7%

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

   9. add-sqr-sqrt 32.9%

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

   10. associate-*r* 32.9%

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

   11. add-sqr-sqrt 70.2%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -4.8 \cdot 10^{+97} \lor \neg \left(v \leq 8.4 \cdot 10^{-72}\right):\\ \;\;\;\;\left(3 + t\_0\right) + \left(\frac{v}{1 - v} \cdot \frac{w \cdot \left(r \cdot -0.25\right)}{\frac{\frac{-1}{w}}{r}} - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1}{w} \cdot \frac{1}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -4.8e+97) (not (<= v 8.4e-72)))
     (+
      (+ 3.0 t_0)
      (- (* (/ v (- 1.0 v)) (/ (* w (* r -0.25)) (/ (/ -1.0 w) r))) 4.5))
     (+ t_0 (- -1.5 (/ (* (* r w) 0.375) (* (/ 1.0 w) (/ 1.0 r))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -4.8e+97) || !(v <= 8.4e-72)) {
		tmp = (3.0 + t_0) + (((v / (1.0 - v)) * ((w * (r * -0.25)) / ((-1.0 / w) / r))) - 4.5);
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * 0.375) / ((1.0 / w) * (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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-4.8d+97)) .or. (.not. (v <= 8.4d-72))) then
        tmp = (3.0d0 + t_0) + (((v / (1.0d0 - v)) * ((w * (r * (-0.25d0))) / (((-1.0d0) / w) / r))) - 4.5d0)
    else
        tmp = t_0 + ((-1.5d0) - (((r * w) * 0.375d0) / ((1.0d0 / w) * (1.0d0 / r))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -4.8e+97) || !(v <= 8.4e-72)) {
		tmp = (3.0 + t_0) + (((v / (1.0 - v)) * ((w * (r * -0.25)) / ((-1.0 / w) / r))) - 4.5);
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * 0.375) / ((1.0 / w) * (1.0 / r))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -4.8e+97) or not (v <= 8.4e-72):
		tmp = (3.0 + t_0) + (((v / (1.0 - v)) * ((w * (r * -0.25)) / ((-1.0 / w) / r))) - 4.5)
	else:
		tmp = t_0 + (-1.5 - (((r * w) * 0.375) / ((1.0 / w) * (1.0 / r))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -4.8e+97) || !(v <= 8.4e-72))
		tmp = Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(v / Float64(1.0 - v)) * Float64(Float64(w * Float64(r * -0.25)) / Float64(Float64(-1.0 / w) / r))) - 4.5));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 / w) * Float64(1.0 / r)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -4.8e+97) || ~((v <= 8.4e-72)))
		tmp = (3.0 + t_0) + (((v / (1.0 - v)) * ((w * (r * -0.25)) / ((-1.0 / w) / r))) - 4.5);
	else
		tmp = t_0 + (-1.5 - (((r * w) * 0.375) / ((1.0 / w) * (1.0 / r))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -4.8e+97], N[Not[LessEqual[v, 8.4e-72]], $MachinePrecision]], N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(v / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r * -0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 / w), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 / w), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -4.8e97 or 8.4e-72 < v

   1. Initial program 81.4%

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

      1. associate--l- 81.4%

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

      2. associate-*l* 74.5%

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

      3. sqr-neg 74.5%

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

      4. associate-*l* 81.4%

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

      5. associate-/l* 85.7%

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

      6. fma-define 85.7%

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

   3. Simplified 85.7%

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

      1. *-un-lft-identity 85.7%

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

      2. add-sqr-sqrt 85.6%

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

      3. times-frac 85.6%

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

      4. *-commutative 85.6%

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

      5. sqrt-prod 42.2%

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

      6. *-commutative 42.2%

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

      7. sqrt-prod 42.2%

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

      8. sqrt-prod 18.1%

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

      9. add-sqr-sqrt 31.3%

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

      10. associate-*r* 31.3%

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

      11. add-sqr-sqrt 68.4%

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

   6. Applied egg-rr 99.7%

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

      1. /-rgt-identity 99.7%

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

      2. associate-*r* 95.4%

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

      3. clear-num 95.4%

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

      4. un-div-inv 95.4%

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

      5. distribute-lft-in 95.4%

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

      6. metadata-eval 95.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\color{blue}{0.375} + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      7. associate-*r* 95.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.375 + \color{blue}{\left(0.125 \cdot -2\right) \cdot v}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      8. metadata-eval 95.4%

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

   8. Applied egg-rr 95.4%

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

   9. Taylor expanded in v around inf 87.0%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)}}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]
   10. Step-by-step derivation

       1. *-commutative 87.0%

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

       2. *-commutative 87.0%

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

       3. associate-*l* 95.3%

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

       4. associate-*r* 95.3%

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

       5. associate-*l* 95.3%

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

   11. Simplified 95.3%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{v \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)}}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]
   12. Step-by-step derivation

       1. div-inv 95.2%

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

       2. times-frac 99.6%

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

       3. associate-/r* 99.6%

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

   13. Applied egg-rr 99.6%

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

   if -4.8e97 < v < 8.4e-72

   1. Initial program 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in v around 0 81.2%

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

      1. *-commutative 81.2%

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

      2. unpow2 81.2%

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

      3. unpow2 81.2%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

      6. *-commutative 99.3%

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

      7. *-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. pow2 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. pow2 99.3%

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

       2. metadata-eval 99.3%

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

       3. pow-div 99.3%

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

       4. pow1 99.3%

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

       5. inv-pow 99.3%

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

       6. associate-*l/ 99.3%

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

   11. Applied egg-rr 99.3%

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

       1. inv-pow 99.3%

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

       2. unpow-prod-down 99.4%

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

       3. inv-pow 99.4%

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

       4. inv-pow 99.4%

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

   13. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4.8 \cdot 10^{+97} \lor \neg \left(v \leq 8.4 \cdot 10^{-72}\right):\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) + \left(\frac{v}{1 - v} \cdot \frac{w \cdot \left(r \cdot -0.25\right)}{\frac{\frac{-1}{w}}{r}} - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1}{w} \cdot \frac{1}{r}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

   1. associate--l- 85.3%

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

   2. associate-*l* 78.4%

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

   3. sqr-neg 78.4%

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

   4. associate-*l* 85.3%

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

   5. associate-/l* 87.2%

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

   6. fma-define 87.2%

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

3. Simplified 87.2%

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

   1. associate-/l* 87.0%

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

   2. *-commutative 87.0%

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

   3. associate-*r/ 87.0%

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

   4. associate-*l* 97.7%

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

   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. add-sqr-sqrt 48.7%

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

   7. associate-*l* 48.7%

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

   8. add-sqr-sqrt 22.2%

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

   9. sqrt-prod 32.1%

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

   10. sqrt-prod 32.1%

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

   11. sqrt-prod 69.0%

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

   12. *-commutative 69.0%

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

   13. sqrt-prod 32.1%

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

   14. *-commutative 32.1%

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

   15. sqrt-prod 32.1%

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

   16. sqrt-prod 22.2%

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

   17. add-sqr-sqrt 48.7%

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

   18. associate-*r* 48.7%

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

   19. add-sqr-sqrt 99.7%

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

   20. clear-num 99.3%

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

   21. un-div-inv 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 8.4 \cdot 10^{-72}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1}{r} \cdot \frac{-1}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_0\right) + \left(\frac{v \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)}{\frac{v}{r \cdot w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 8.4e-72)
     (+ t_0 (+ -1.5 (/ (* (* r w) 0.375) (* (/ 1.0 r) (/ -1.0 w)))))
     (+ (+ 3.0 t_0) (- (/ (* v (* w (* r -0.25))) (/ v (* r w))) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 8.4e-72) {
		tmp = t_0 + (-1.5 + (((r * w) * 0.375) / ((1.0 / r) * (-1.0 / w))));
	} else {
		tmp = (3.0 + t_0) + (((v * (w * (r * -0.25))) / (v / (r * w))) - 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (v <= 8.4d-72) then
        tmp = t_0 + ((-1.5d0) + (((r * w) * 0.375d0) / ((1.0d0 / r) * ((-1.0d0) / w))))
    else
        tmp = (3.0d0 + t_0) + (((v * (w * (r * (-0.25d0)))) / (v / (r * w))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 8.4e-72) {
		tmp = t_0 + (-1.5 + (((r * w) * 0.375) / ((1.0 / r) * (-1.0 / w))));
	} else {
		tmp = (3.0 + t_0) + (((v * (w * (r * -0.25))) / (v / (r * w))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= 8.4e-72:
		tmp = t_0 + (-1.5 + (((r * w) * 0.375) / ((1.0 / r) * (-1.0 / w))))
	else:
		tmp = (3.0 + t_0) + (((v * (w * (r * -0.25))) / (v / (r * w))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 8.4e-72)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(Float64(r * w) * 0.375) / Float64(Float64(1.0 / r) * Float64(-1.0 / w)))));
	else
		tmp = Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(v * Float64(w * Float64(r * -0.25))) / Float64(v / Float64(r * w))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= 8.4e-72)
		tmp = t_0 + (-1.5 + (((r * w) * 0.375) / ((1.0 / r) * (-1.0 / w))));
	else
		tmp = (3.0 + t_0) + (((v * (w * (r * -0.25))) / (v / (r * w))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 8.4e-72], N[(t$95$0 + N[(-1.5 + N[(N[(N[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] / N[(N[(1.0 / r), $MachinePrecision] * N[(-1.0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(v * N[(w * N[(r * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 8.4e-72

   1. Initial program 87.2%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in v around 0 78.6%

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

      1. *-commutative 78.6%

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

      2. unpow2 78.6%

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

      3. unpow2 78.6%

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

      4. swap-sqr 96.3%

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

      5. unpow2 96.3%

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

      6. *-commutative 96.3%

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

      7. *-commutative 96.3%

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

   7. Simplified 96.3%

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

      1. *-commutative 96.3%

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

      2. pow2 96.3%

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

   9. Applied egg-rr 96.3%

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

       1. pow2 96.3%

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

       2. metadata-eval 96.3%

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

       3. pow-div 96.2%

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

       4. pow1 96.2%

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

       5. inv-pow 96.2%

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

       6. associate-*l/ 96.3%

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

   11. Applied egg-rr 96.3%

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

       1. inv-pow 96.3%

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

       2. unpow-prod-down 96.3%

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

       3. inv-pow 96.3%

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

       4. inv-pow 96.3%

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

   13. Applied egg-rr 96.3%

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

   if 8.4e-72 < v

   1. Initial program 80.0%

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

      1. associate--l- 80.0%

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

      2. associate-*l* 76.3%

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

      3. sqr-neg 76.3%

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

      4. associate-*l* 80.0%

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

      5. associate-/l* 82.9%

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

      6. fma-define 82.9%

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

   3. Simplified 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. add-sqr-sqrt 82.9%

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

      3. times-frac 82.8%

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

      4. *-commutative 82.8%

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

      5. sqrt-prod 45.6%

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

      6. *-commutative 45.6%

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

      7. sqrt-prod 45.5%

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

      8. sqrt-prod 20.2%

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

      9. add-sqr-sqrt 32.3%

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

      10. associate-*r* 32.3%

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

      11. add-sqr-sqrt 68.2%

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

   6. Applied egg-rr 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. associate-*r* 98.3%

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

      3. clear-num 98.3%

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

      4. un-div-inv 98.3%

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

      5. distribute-lft-in 98.3%

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

      6. metadata-eval 98.3%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\color{blue}{0.375} + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      7. associate-*r* 98.3%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.375 + \color{blue}{\left(0.125 \cdot -2\right) \cdot v}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

      8. metadata-eval 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in v around inf 89.3%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)}}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]
   10. Step-by-step derivation

       1. *-commutative 89.3%

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

       2. *-commutative 89.3%

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

       3. associate-*l* 98.1%

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

       4. associate-*r* 98.1%

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

       5. associate-*l* 98.1%

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

   11. Simplified 98.1%

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

   12. Taylor expanded in v around inf 98.2%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{v \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)}{\color{blue}{-1 \cdot \frac{v}{r \cdot w}}} + 4.5\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 98.2%

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

       2. neg-mul-1 98.2%

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

   14. Simplified 98.2%

       \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{v \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)}{\color{blue}{\frac{-v}{r \cdot w}}} + 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 8.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{\left(r \cdot w\right) \cdot 0.375}{\frac{1}{r} \cdot \frac{-1}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) + \left(\frac{v \cdot \left(w \cdot \left(r \cdot -0.25\right)\right)}{\frac{v}{r \cdot w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (+ 3.0 (/ 2.0 (* r r)))
  (- (/ (* (* r w) (+ 0.375 (* v -0.25))) (/ (+ v -1.0) (* r w))) 4.5)))

C

double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 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))) + ((((r * w) * (0.375d0 + (v * (-0.25d0)))) / ((v + (-1.0d0)) / (r * w))) - 4.5d0)
end function

Java

public static double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5);
}

Python

def code(v, w, r):
	return (3.0 + (2.0 / (r * r))) + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5)

Julia

function code(v, w, r)
	return Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(Float64(Float64(r * w) * Float64(0.375 + Float64(v * -0.25))) / Float64(Float64(v + -1.0) / Float64(r * w))) - 4.5))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + (2.0 / (r * r))) + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5);
end

Wolfram

code[v_, w_, r_] := N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

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

   1. associate--l- 85.3%

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

   2. associate-*l* 78.4%

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

   3. sqr-neg 78.4%

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

   4. associate-*l* 85.3%

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

   5. associate-/l* 87.2%

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

   6. fma-define 87.2%

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

3. Simplified 87.2%

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

   1. *-un-lft-identity 87.2%

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

   2. add-sqr-sqrt 87.1%

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

   3. times-frac 87.1%

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

   4. *-commutative 87.1%

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

   5. sqrt-prod 41.4%

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

   6. *-commutative 41.4%

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

   7. sqrt-prod 41.4%

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

   8. sqrt-prod 18.7%

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

   9. add-sqr-sqrt 32.9%

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

   10. associate-*r* 32.9%

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

   11. add-sqr-sqrt 70.2%

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

6. Applied egg-rr 99.7%

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

   1. /-rgt-identity 99.7%

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

   2. associate-*r* 97.9%

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

   3. clear-num 97.9%

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

   4. un-div-inv 97.9%

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

   5. distribute-lft-in 97.9%

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

   6. metadata-eval 97.9%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\color{blue}{0.375} + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

   7. associate-*r* 97.9%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.375 + \color{blue}{\left(0.125 \cdot -2\right) \cdot v}\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{w \cdot r}} + 4.5\right) \]

   8. metadata-eval 97.9%

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

8. Applied egg-rr 97.9%

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

9. Final simplification 97.9%

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

Alternative 6: 93.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

3. Simplified 87.0%

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

5. Taylor expanded in v around 0 76.2%

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

   1. *-commutative 76.2%

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

   2. unpow2 76.2%

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

   3. unpow2 76.2%

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

   4. swap-sqr 93.6%

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

   5. unpow2 93.6%

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

   6. *-commutative 93.6%

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

   7. *-commutative 93.6%

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

7. Simplified 93.6%

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

   1. *-commutative 93.6%

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

   2. pow2 93.6%

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

9. Applied egg-rr 93.6%

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

    1. pow2 93.6%

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

    2. metadata-eval 93.6%

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

    3. pow-div 93.6%

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

    4. pow1 93.6%

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

    5. inv-pow 93.6%

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

    6. associate-*l/ 93.6%

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

11. Applied egg-rr 93.6%

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

    1. inv-pow 93.6%

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

    2. unpow-prod-down 93.6%

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

    3. inv-pow 93.6%

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

    4. inv-pow 93.6%

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

13. Applied egg-rr 93.6%

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

14. Final simplification 93.6%

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

Alternative 7: 93.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

3. Simplified 87.0%

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

5. Taylor expanded in v around 0 76.2%

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

   1. *-commutative 76.2%

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

   2. unpow2 76.2%

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

   3. unpow2 76.2%

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

   4. swap-sqr 93.6%

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

   5. unpow2 93.6%

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

   6. *-commutative 93.6%

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

   7. *-commutative 93.6%

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

7. Simplified 93.6%

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

   1. *-commutative 93.6%

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

   2. pow2 93.6%

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

9. Applied egg-rr 93.6%

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

    1. pow2 93.6%

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

    2. metadata-eval 93.6%

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

    3. pow-div 93.6%

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

    4. pow1 93.6%

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

    5. inv-pow 93.6%

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

    6. associate-*l/ 93.6%

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

11. Applied egg-rr 93.6%

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

    1. *-commutative 93.6%

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

    2. *-un-lft-identity 93.6%

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

    3. times-frac 93.6%

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

    4. metadata-eval 93.6%

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

    5. associate-/r* 93.6%

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

13. Applied egg-rr 93.6%

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

14. Final simplification 93.6%

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

Alternative 8: 91.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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) - (w * ((r * w) * (r * 0.375d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

3. Simplified 87.0%

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

5. Taylor expanded in v around 0 76.2%

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

   1. *-commutative 76.2%

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

   2. unpow2 76.2%

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

   3. unpow2 76.2%

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

   4. swap-sqr 93.6%

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

   5. unpow2 93.6%

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

   6. *-commutative 93.6%

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

   7. *-commutative 93.6%

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

7. Simplified 93.6%

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

   1. *-commutative 93.6%

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

   2. pow2 93.6%

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

9. Applied egg-rr 93.6%

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

    1. pow2 93.6%

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

    2. metadata-eval 93.6%

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

    3. pow-div 93.6%

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

    4. pow1 93.6%

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

    5. inv-pow 93.6%

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

    6. associate-*l/ 93.6%

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

11. Applied egg-rr 93.6%

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

    1. *-un-lft-identity 93.6%

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

    2. div-inv 93.6%

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

    3. associate-*l* 93.6%

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

    4. inv-pow 93.6%

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

    5. pow-flip 93.6%

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

    6. metadata-eval 93.6%

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

    7. pow1 93.6%

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

13. Applied egg-rr 93.6%

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

    1. *-lft-identity 93.6%

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

    2. associate-*l* 91.9%

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

    3. *-commutative 91.9%

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

15. Simplified 91.9%

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

16. Final simplification 91.9%

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

Alternative 9: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

3. Simplified 87.0%

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

5. Taylor expanded in v around 0 76.2%

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

   1. *-commutative 76.2%

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

   2. unpow2 76.2%

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

   3. unpow2 76.2%

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

   4. swap-sqr 93.6%

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

   5. unpow2 93.6%

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

   6. *-commutative 93.6%

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

   7. *-commutative 93.6%

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

7. Simplified 93.6%

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

   1. *-commutative 93.6%

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

   2. pow2 93.6%

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

9. Applied egg-rr 93.6%

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

    1. pow2 93.6%

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

    2. metadata-eval 93.6%

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

    3. pow-div 93.6%

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

    4. pow1 93.6%

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

    5. inv-pow 93.6%

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

    6. associate-*l/ 93.6%

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

11. Applied egg-rr 93.6%

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

    1. div-inv 93.6%

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

    2. associate-*l* 93.6%

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

    3. inv-pow 93.6%

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

    4. pow-flip 93.6%

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

    5. metadata-eval 93.6%

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

    6. pow1 93.6%

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

13. Applied egg-rr 93.6%

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

14. Final simplification 93.6%

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

Alternative 10: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

3. Simplified 87.0%

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

5. Taylor expanded in v around 0 76.2%

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

   1. *-commutative 76.2%

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

   2. unpow2 76.2%

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

   3. unpow2 76.2%

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

   4. swap-sqr 93.6%

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

   5. unpow2 93.6%

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

   6. *-commutative 93.6%

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

   7. *-commutative 93.6%

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

7. Simplified 93.6%

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

   1. *-commutative 93.6%

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

   2. pow2 93.6%

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

9. Applied egg-rr 93.6%

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

10. Final simplification 93.6%

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

Alternative 11: 57.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

3. Simplified 78.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 58.1%

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

6. Final simplification 58.1%

   \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - 4.5 \]
7. Add Preprocessing

Alternative 12: 57.0% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (v w r) :precision binary64 (- (+ 3.0 (/ (/ 2.0 r) r)) 4.5))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 85.4%

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

3. Simplified 78.9%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 58.1%

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

   1. associate-/r* 58.2%

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

   2. div-inv 58.1%

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

7. Applied egg-rr 58.1%

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

   1. associate-*r/ 58.2%

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

   2. *-rgt-identity 58.2%

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

9. Simplified 58.2%

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

10. Final simplification 58.2%

    \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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))