Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 99.7%

Time: 17.2s

Alternatives: 10

Speedup: 1.7×

• 53.2% 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: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. associate--l- 85.6%

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

   2. associate--l+ 85.6%

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

3. Simplified 88.5%

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

   1. associate-/r* 86.9%

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

   2. div-inv 86.9%

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

   3. associate-*r* 95.1%

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

   4. *-commutative 95.1%

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

   5. times-frac 99.8%

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

   6. *-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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 / ((((1.0d0 - v) / (r * w)) * (1.0d0 / (r * w))) / (3.0d0 + (v * (-2.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. associate--l- 85.6%

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

   2. associate--l+ 85.6%

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

3. Simplified 88.5%

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

   1. associate-*r* 84.8%

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

   2. *-commutative 84.8%

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

   3. div-inv 84.8%

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

   4. associate-*r/ 84.8%

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

   5. unswap-sqr 99.7%

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

   6. times-frac 99.7%

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

   7. *-commutative 99.7%

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

   8. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 93.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 6.9 \cdot 10^{-56}:\\ \;\;\;\;3 + \left(t_0 - \left(4.5 + \frac{0.125}{\frac{\frac{\frac{0.5}{r}}{r \cdot w}}{w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \frac{w \cdot \left(0.375 + v \cdot -0.25\right)}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 6.9e-56)
     (+ 3.0 (- t_0 (+ 4.5 (/ 0.125 (/ (/ (/ 0.5 r) (* r w)) w)))))
     (+
      t_0
      (- -1.5 (* (* r (* r w)) (/ (* w (+ 0.375 (* v -0.25))) (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.9e-56) {
		tmp = 3.0 + (t_0 - (4.5 + (0.125 / (((0.5 / r) / (r * w)) / w))));
	} else {
		tmp = t_0 + (-1.5 - ((r * (r * w)) * ((w * (0.375 + (v * -0.25))) / (1.0 - v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 6.9d-56) then
        tmp = 3.0d0 + (t_0 - (4.5d0 + (0.125d0 / (((0.5d0 / r) / (r * w)) / w))))
    else
        tmp = t_0 + ((-1.5d0) - ((r * (r * w)) * ((w * (0.375d0 + (v * (-0.25d0)))) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.9e-56) {
		tmp = 3.0 + (t_0 - (4.5 + (0.125 / (((0.5 / r) / (r * w)) / w))));
	} else {
		tmp = t_0 + (-1.5 - ((r * (r * w)) * ((w * (0.375 + (v * -0.25))) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 6.9e-56:
		tmp = 3.0 + (t_0 - (4.5 + (0.125 / (((0.5 / r) / (r * w)) / w))))
	else:
		tmp = t_0 + (-1.5 - ((r * (r * w)) * ((w * (0.375 + (v * -0.25))) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 6.9e-56)
		tmp = Float64(3.0 + Float64(t_0 - Float64(4.5 + Float64(0.125 / Float64(Float64(Float64(0.5 / r) / Float64(r * w)) / w)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(r * w)) * Float64(Float64(w * Float64(0.375 + Float64(v * -0.25))) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 6.9e-56)
		tmp = 3.0 + (t_0 - (4.5 + (0.125 / (((0.5 / r) / (r * w)) / w))));
	else
		tmp = t_0 + (-1.5 - ((r * (r * w)) * ((w * (0.375 + (v * -0.25))) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 6.9e-56], N[(3.0 + N[(t$95$0 - N[(4.5 + N[(0.125 / N[(N[(N[(0.5 / r), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 6.8999999999999996e-56

   1. Initial program 83.2%

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

      1. associate--l- 83.2%

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

      2. associate--l+ 83.2%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in v around inf 77.7%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{r}^{2} \cdot {w}^{2}}}} + 4.5\right)\right) \]
   5. Step-by-step derivation

      1. unpow2 77.7%

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

      2. unpow2 77.7%

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

      3. swap-sqr 92.9%

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

      4. unpow2 92.9%

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

   6. Simplified 92.9%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{\left(r \cdot w\right)}^{2}}}} + 4.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 92.9%

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

      2. metadata-eval 92.9%

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

      3. unpow2 92.9%

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

      4. associate-*r* 92.9%

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

      5. *-commutative 92.9%

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

      6. associate-*l* 92.9%

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

      7. *-commutative 92.9%

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

      8. associate-*l* 92.9%

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

   8. Applied egg-rr 92.9%

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

      1. associate-*l* 92.2%

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

      2. *-commutative 92.2%

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

      3. associate-*l* 92.2%

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

      4. associate-*r* 92.2%

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

      5. metadata-eval 92.2%

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

      6. associate-/r/ 92.2%

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

      7. associate-/r* 92.2%

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

      8. associate-/r* 92.4%

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

   10. Applied egg-rr 92.4%

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

   if 6.8999999999999996e-56 < r

   1. Initial program 90.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in w around 0 94.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.9 \cdot 10^{-56}:\\ \;\;\;\;3 + \left(\frac{2}{r \cdot r} - \left(4.5 + \frac{0.125}{\frac{\frac{\frac{0.5}{r}}{r \cdot w}}{w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \frac{w \cdot \left(0.375 + v \cdot -0.25\right)}{1 - v}\right)\\ \end{array} \]

Alternative 4: 96.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.9 \cdot 10^{+181}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{\frac{1 - v}{w}} \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(t_0 - \left(4.5 + r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.9e+181)
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (/ (- 1.0 v) w)) (* r (* r w)))))
     (+ 3.0 (- t_0 (+ 4.5 (* r (* w (* w (* r 0.25))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.9e+181) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / ((1.0 - v) / w)) * (r * (r * w))));
	} else {
		tmp = 3.0 + (t_0 - (4.5 + (r * (w * (w * (r * 0.25))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2.9d+181) then
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / ((1.0d0 - v) / w)) * (r * (r * w))))
    else
        tmp = 3.0d0 + (t_0 - (4.5d0 + (r * (w * (w * (r * 0.25d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.9e+181) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / ((1.0 - v) / w)) * (r * (r * w))));
	} else {
		tmp = 3.0 + (t_0 - (4.5 + (r * (w * (w * (r * 0.25))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.9e+181:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / ((1.0 - v) / w)) * (r * (r * w))))
	else:
		tmp = 3.0 + (t_0 - (4.5 + (r * (w * (w * (r * 0.25))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.9e+181)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(Float64(1.0 - v) / w)) * Float64(r * Float64(r * w)))));
	else
		tmp = Float64(3.0 + Float64(t_0 - Float64(4.5 + Float64(r * Float64(w * Float64(w * Float64(r * 0.25)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.9e+181)
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / ((1.0 - v) / w)) * (r * (r * w))));
	else
		tmp = 3.0 + (t_0 - (4.5 + (r * (w * (w * (r * 0.25))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.9e+181], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(t$95$0 - N[(4.5 + N[(r * N[(w * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.9e181

   1. Initial program 85.6%

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

   3. Simplified 97.2%

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

   if 2.9e181 < r

   1. Initial program 86.0%

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

      1. associate--l- 86.0%

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

      2. associate--l+ 86.0%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in v around inf 74.8%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{r}^{2} \cdot {w}^{2}}}} + 4.5\right)\right) \]
   5. Step-by-step derivation

      1. unpow2 74.8%

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

      2. unpow2 74.8%

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

      3. swap-sqr 93.2%

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

      4. unpow2 93.2%

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

   6. Simplified 93.2%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{\left(r \cdot w\right)}^{2}}}} + 4.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 93.2%

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

      2. metadata-eval 93.2%

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

      3. unpow2 93.2%

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

      4. associate-*r* 93.2%

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

      5. *-commutative 93.2%

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

      6. associate-*l* 93.2%

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

      7. *-commutative 93.2%

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

      8. *-commutative 93.2%

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

      9. associate-*r* 93.2%

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

      10. associate-*l* 93.2%

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

   8. Applied egg-rr 93.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.9 \cdot 10^{+181}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{\frac{1 - v}{w}} \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{2}{r \cdot r} - \left(4.5 + r \cdot \left(w \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 96.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -100000000000 \lor \neg \left(v \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot \left(w \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot \left(w \cdot 0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -100000000000.0) (not (<= v 2e-11)))
     (+ t_1 (- -1.5 (* t_0 (* w 0.25))))
     (+ t_1 (- -1.5 (* t_0 (* w 0.375)))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -100000000000.0) || !(v <= 2e-11)) {
		tmp = t_1 + (-1.5 - (t_0 * (w * 0.25)));
	} else {
		tmp = t_1 + (-1.5 - (t_0 * (w * 0.375)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = r * (r * w)
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-100000000000.0d0)) .or. (.not. (v <= 2d-11))) then
        tmp = t_1 + ((-1.5d0) - (t_0 * (w * 0.25d0)))
    else
        tmp = t_1 + ((-1.5d0) - (t_0 * (w * 0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -100000000000.0) || !(v <= 2e-11)) {
		tmp = t_1 + (-1.5 - (t_0 * (w * 0.25)));
	} else {
		tmp = t_1 + (-1.5 - (t_0 * (w * 0.375)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -100000000000.0) or not (v <= 2e-11):
		tmp = t_1 + (-1.5 - (t_0 * (w * 0.25)))
	else:
		tmp = t_1 + (-1.5 - (t_0 * (w * 0.375)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -100000000000.0) || !(v <= 2e-11))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * Float64(w * 0.25))));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * Float64(w * 0.375))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -100000000000.0) || ~((v <= 2e-11)))
		tmp = t_1 + (-1.5 - (t_0 * (w * 0.25)));
	else
		tmp = t_1 + (-1.5 - (t_0 * (w * 0.375)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -100000000000.0], N[Not[LessEqual[v, 2e-11]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1e11 or 1.99999999999999988e-11 < v

   1. Initial program 79.3%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in v around inf 97.2%

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

      1. *-commutative 97.2%

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

   6. Simplified 97.2%

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

   if -1e11 < v < 1.99999999999999988e-11

   1. Initial program 91.6%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in v around 0 95.5%

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

      1. *-commutative 95.5%

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

   6. Simplified 95.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -100000000000 \lor \neg \left(v \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.375\right)\right)\\ \end{array} \]

Alternative 6: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := r \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -1.25:\\ \;\;\;\;3 + \left(t_0 - \left(4.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-11}:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot \left(w \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot \left(w \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* r (* r w))))
   (if (<= v -1.25)
     (+ 3.0 (- t_0 (+ 4.5 (* (* r w) (* w (* r 0.25))))))
     (if (<= v 2e-11)
       (+ t_0 (- -1.5 (* t_1 (* w 0.375))))
       (+ t_0 (- -1.5 (* t_1 (* w 0.25))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r * (r * w);
	double tmp;
	if (v <= -1.25) {
		tmp = 3.0 + (t_0 - (4.5 + ((r * w) * (w * (r * 0.25)))));
	} else if (v <= 2e-11) {
		tmp = t_0 + (-1.5 - (t_1 * (w * 0.375)));
	} else {
		tmp = t_0 + (-1.5 - (t_1 * (w * 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = r * (r * w)
    if (v <= (-1.25d0)) then
        tmp = 3.0d0 + (t_0 - (4.5d0 + ((r * w) * (w * (r * 0.25d0)))))
    else if (v <= 2d-11) then
        tmp = t_0 + ((-1.5d0) - (t_1 * (w * 0.375d0)))
    else
        tmp = t_0 + ((-1.5d0) - (t_1 * (w * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r * (r * w);
	double tmp;
	if (v <= -1.25) {
		tmp = 3.0 + (t_0 - (4.5 + ((r * w) * (w * (r * 0.25)))));
	} else if (v <= 2e-11) {
		tmp = t_0 + (-1.5 - (t_1 * (w * 0.375)));
	} else {
		tmp = t_0 + (-1.5 - (t_1 * (w * 0.25)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = r * (r * w)
	tmp = 0
	if v <= -1.25:
		tmp = 3.0 + (t_0 - (4.5 + ((r * w) * (w * (r * 0.25)))))
	elif v <= 2e-11:
		tmp = t_0 + (-1.5 - (t_1 * (w * 0.375)))
	else:
		tmp = t_0 + (-1.5 - (t_1 * (w * 0.25)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(r * Float64(r * w))
	tmp = 0.0
	if (v <= -1.25)
		tmp = Float64(3.0 + Float64(t_0 - Float64(4.5 + Float64(Float64(r * w) * Float64(w * Float64(r * 0.25))))));
	elseif (v <= 2e-11)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * Float64(w * 0.375))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * Float64(w * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = r * (r * w);
	tmp = 0.0;
	if (v <= -1.25)
		tmp = 3.0 + (t_0 - (4.5 + ((r * w) * (w * (r * 0.25)))));
	elseif (v <= 2e-11)
		tmp = t_0 + (-1.5 - (t_1 * (w * 0.375)));
	else
		tmp = t_0 + (-1.5 - (t_1 * (w * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1.25], N[(3.0 + N[(t$95$0 - N[(4.5 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2e-11], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.25

   1. Initial program 74.3%

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

      1. associate--l- 74.3%

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

      2. associate--l+ 74.3%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in v around inf 75.4%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{r}^{2} \cdot {w}^{2}}}} + 4.5\right)\right) \]
   5. Step-by-step derivation

      1. unpow2 75.4%

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

      2. unpow2 75.4%

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

      3. swap-sqr 98.3%

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

      4. unpow2 98.3%

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

   6. Simplified 98.3%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{\left(r \cdot w\right)}^{2}}}} + 4.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 98.3%

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

      2. metadata-eval 98.3%

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

      3. unpow2 98.3%

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

      4. associate-*r* 98.3%

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

      5. *-commutative 98.3%

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

      6. associate-*l* 98.3%

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

      7. *-commutative 98.3%

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

      8. associate-*l* 98.3%

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

   8. Applied egg-rr 98.3%

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

   if -1.25 < v < 1.99999999999999988e-11

   1. Initial program 91.2%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in v around 0 96.0%

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

      1. *-commutative 96.0%

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

   6. Simplified 96.0%

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

   if 1.99999999999999988e-11 < v

   1. Initial program 85.8%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in v around inf 99.5%

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

      1. *-commutative 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.25:\\ \;\;\;\;3 + \left(\frac{2}{r \cdot r} - \left(4.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 7: 91.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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 * (r * w)) * (w * 0.25d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Simplified 95.2%

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

4. Taylor expanded in v around inf 90.3%

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

   1. *-commutative 90.3%

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

6. Simplified 90.3%

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

7. Final simplification 90.3%

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

Alternative 8: 63.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4.1e+136) (+ (/ 2.0 (* r r)) -1.5) (* (* w w) (* (* r r) -0.25))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.1e+136) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (w * w) * ((r * r) * -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 4.1d+136) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (w * w) * ((r * r) * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.1e+136) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (w * w) * ((r * r) * -0.25);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 4.1e+136:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (w * w) * ((r * r) * -0.25)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4.1e+136)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(w * w) * Float64(Float64(r * r) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 4.1e+136)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (w * w) * ((r * r) * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4.1e+136], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(w * w), $MachinePrecision] * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.0999999999999998e136

   1. Initial program 85.7%

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

      1. associate--l- 85.7%

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

      2. associate--l+ 85.7%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in v around inf 79.7%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{r}^{2} \cdot {w}^{2}}}} + 4.5\right)\right) \]
   5. Step-by-step derivation

      1. unpow2 79.7%

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

      2. unpow2 79.7%

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

      3. swap-sqr 91.9%

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

      4. unpow2 91.9%

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

   6. Simplified 91.9%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{\left(r \cdot w\right)}^{2}}}} + 4.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 91.8%

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

      2. metadata-eval 91.8%

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

      3. unpow2 91.8%

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

      4. associate-*r* 91.8%

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

      5. *-commutative 91.8%

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

      6. associate-*l* 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*l* 91.8%

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

   8. Applied egg-rr 91.8%

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

   9. Taylor expanded in r around 0 65.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   10. Step-by-step derivation

       1. sub-neg 65.4%

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

       2. associate-*r/ 65.4%

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

       3. metadata-eval 65.4%

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

       4. unpow2 65.4%

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

       5. metadata-eval 65.4%

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   11. Simplified 65.4%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r} + -1.5} \]

   if 4.0999999999999998e136 < r

   1. Initial program 84.9%

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

      1. associate--l- 84.9%

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

      2. associate--l+ 84.9%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in v around inf 74.7%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{r}^{2} \cdot {w}^{2}}}} + 4.5\right)\right) \]
   5. Step-by-step derivation

      1. unpow2 74.7%

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

      2. unpow2 74.7%

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

      3. swap-sqr 90.3%

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

      4. unpow2 90.3%

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

   6. Simplified 90.3%

      \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{\left(r \cdot w\right)}^{2}}}} + 4.5\right)\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 90.3%

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

      2. metadata-eval 90.3%

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

      3. unpow2 90.3%

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

      4. associate-*r* 90.3%

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

      5. *-commutative 90.3%

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

      6. associate-*l* 90.3%

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

      7. *-commutative 90.3%

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

      8. associate-*l* 90.3%

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

   8. Applied egg-rr 90.3%

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

   9. Taylor expanded in r around inf 72.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 72.1%

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

       2. *-commutative 72.1%

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

       3. unpow2 72.1%

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

       4. *-commutative 72.1%

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

       5. unpow2 72.1%

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

   11. Simplified 72.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 9: 57.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. associate--l- 85.6%

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

   2. associate--l+ 85.6%

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

3. Simplified 88.5%

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

4. Taylor expanded in v around inf 79.0%

   \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{r}^{2} \cdot {w}^{2}}}} + 4.5\right)\right) \]
5. Step-by-step derivation

   1. unpow2 79.0%

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

   2. unpow2 79.0%

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

   3. swap-sqr 91.6%

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

   4. unpow2 91.6%

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

6. Simplified 91.6%

   \[\leadsto 3 + \left(\frac{2}{r \cdot r} - \left(\frac{0.125}{\color{blue}{\frac{0.5}{{\left(r \cdot w\right)}^{2}}}} + 4.5\right)\right) \]
7. Step-by-step derivation

   1. associate-/r/ 91.6%

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

   2. metadata-eval 91.6%

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

   3. unpow2 91.6%

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

   4. associate-*r* 91.6%

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

   5. *-commutative 91.6%

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

   6. associate-*l* 91.6%

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

   7. *-commutative 91.6%

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

   8. associate-*l* 91.6%

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

8. Applied egg-rr 91.6%

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

9. Taylor expanded in r around 0 57.2%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
10. Step-by-step derivation

    1. sub-neg 57.2%

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

    2. associate-*r/ 57.2%

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

    3. metadata-eval 57.2%

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

    4. unpow2 57.2%

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

    5. metadata-eval 57.2%

       \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

11. Simplified 57.2%

    \[\leadsto \color{blue}{\frac{2}{r \cdot r} + -1.5} \]

12. Final simplification 57.2%

    \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

Alternative 10: 44.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (/ 2.0 (* r r)))

C

double code(double v, double w, double r) {
	return 2.0 / (r * 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)
end function

Java

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

Python

def code(v, w, r):
	return 2.0 / (r * r)

Julia

function code(v, w, r)
	return Float64(2.0 / Float64(r * r))
end

MATLAB

function tmp = code(v, w, r)
	tmp = 2.0 / (r * r);
end

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Simplified 95.2%

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

4. Taylor expanded in v around inf 90.3%

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

   1. *-commutative 90.3%

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

6. Simplified 90.3%

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

7. Taylor expanded in r around 0 43.2%

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 43.2%

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

   2. associate-/r* 43.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

9. Simplified 43.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

10. Taylor expanded in r around 0 43.2%

    \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 43.2%

       \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

12. Simplified 43.2%

    \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]

13. Final simplification 43.2%

    \[\leadsto \frac{2}{r \cdot r} \]

Reproduce

herbie shell --seed 2023297 
(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))