Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 99.7%

Time: 11.1s

Alternatives: 9

Speedup: 1.5×

• 52.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

3. Simplified 88.4%

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

   1. fma-undefine 88.4%

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

   2. *-commutative 88.4%

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

   3. +-commutative 88.4%

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

   4. metadata-eval 88.4%

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

   5. cancel-sign-sub-inv 88.4%

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

   6. associate-*r/ 88.4%

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

   7. *-commutative 88.4%

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

   8. associate-/l* 89.4%

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

   9. clear-num 89.4%

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

   10. un-div-inv 89.4%

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

   11. cancel-sign-sub-inv 89.4%

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

   12. metadata-eval 89.4%

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

   13. distribute-rgt-in 89.4%

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

   14. metadata-eval 89.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   15. *-commutative 89.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   16. associate-*l* 89.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   17. metadata-eval 89.8%

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

6. Applied egg-rr 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. *-commutative 99.4%

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

   3. unpow2 99.4%

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

   4. times-frac 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -18.5 \lor \neg \left(v \leq 300000\right):\\ \;\;\;\;3 + \left(\left(v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -18.5) (not (<= v 300000.0)))
   (+ 3.0 (- (* (* v -0.25) (/ (* r (* r (* w w))) (+ v -1.0))) 4.5))
   (- (+ 3.0 (* (* 0.375 (* r w)) (* w (/ r (+ v -1.0))))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -18.5) || !(v <= 300000.0)) {
		tmp = 3.0 + (((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))) - 4.5);
	} else {
		tmp = (3.0 + ((0.375 * (r * w)) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((v <= (-18.5d0)) .or. (.not. (v <= 300000.0d0))) then
        tmp = 3.0d0 + (((v * (-0.25d0)) * ((r * (r * (w * w))) / (v + (-1.0d0)))) - 4.5d0)
    else
        tmp = (3.0d0 + ((0.375d0 * (r * w)) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -18.5) || !(v <= 300000.0)) {
		tmp = 3.0 + (((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))) - 4.5);
	} else {
		tmp = (3.0 + ((0.375 * (r * w)) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -18.5) or not (v <= 300000.0):
		tmp = 3.0 + (((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))) - 4.5)
	else:
		tmp = (3.0 + ((0.375 * (r * w)) * (w * (r / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -18.5) || !(v <= 300000.0))
		tmp = Float64(3.0 + Float64(Float64(Float64(v * -0.25) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0))) - 4.5));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 * Float64(r * w)) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -18.5) || ~((v <= 300000.0)))
		tmp = 3.0 + (((v * -0.25) * ((r * (r * (w * w))) / (v + -1.0))) - 4.5);
	else
		tmp = (3.0 + ((0.375 * (r * w)) * (w * (r / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -18.5], N[Not[LessEqual[v, 300000.0]], $MachinePrecision]], N[(3.0 + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -18.5 or 3e5 < v

   1. Initial program 82.5%

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

      1. associate--l- 82.5%

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

      2. associate-*l* 76.2%

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

      3. sqr-neg 76.2%

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

      4. associate-*l* 82.5%

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

      5. associate-/l* 89.3%

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

      6. fma-define 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in r around inf 50.6%

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

   6. Taylor expanded in v around inf 50.3%

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

      1. *-commutative 50.3%

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

   8. Simplified 50.3%

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

   if -18.5 < v < 3e5

   1. Initial program 89.5%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in v around 0 99.1%

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

   6. Taylor expanded in r around inf 52.3%

      \[\leadsto \left(\color{blue}{3} - \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -18.5 \lor \neg \left(v \leq 300000\right):\\ \;\;\;\;3 + \left(\left(v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 15:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 15.0)
   (- (- (+ (/ 2.0 (* r r)) 3.0) (* (* r w) (* 0.375 (* r w)))) 4.5)
   (+
    3.0
    (- (* (+ 0.375 (* v -0.25)) (* (* r w) (* w (/ r (+ v -1.0))))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 15.0) {
		tmp = (((2.0 / (r * r)) + 3.0) - ((r * w) * (0.375 * (r * w)))) - 4.5;
	} else {
		tmp = 3.0 + (((0.375 + (v * -0.25)) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 15.0d0) then
        tmp = (((2.0d0 / (r * r)) + 3.0d0) - ((r * w) * (0.375d0 * (r * w)))) - 4.5d0
    else
        tmp = 3.0d0 + (((0.375d0 + (v * (-0.25d0))) * ((r * w) * (w * (r / (v + (-1.0d0)))))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 15.0) {
		tmp = (((2.0 / (r * r)) + 3.0) - ((r * w) * (0.375 * (r * w)))) - 4.5;
	} else {
		tmp = 3.0 + (((0.375 + (v * -0.25)) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 15.0:
		tmp = (((2.0 / (r * r)) + 3.0) - ((r * w) * (0.375 * (r * w)))) - 4.5
	else:
		tmp = 3.0 + (((0.375 + (v * -0.25)) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 15.0)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(r * w) * Float64(0.375 * Float64(r * w)))) - 4.5);
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r * w) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 15.0)
		tmp = (((2.0 / (r * r)) + 3.0) - ((r * w) * (0.375 * (r * w)))) - 4.5;
	else
		tmp = 3.0 + (((0.375 + (v * -0.25)) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 15.0], N[(N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 + N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 15

   1. Initial program 85.0%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in v around 0 87.4%

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

   6. Taylor expanded in v around 0 94.9%

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

   if 15 < r

   1. Initial program 90.0%

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

      1. associate--l- 90.0%

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

      2. associate-*l* 73.6%

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

      3. sqr-neg 73.6%

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

      4. associate-*l* 90.0%

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

      5. associate-/l* 93.2%

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

      6. fma-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in r around inf 93.1%

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

   6. Taylor expanded in v around 0 93.1%

      \[\leadsto 3 - \left(\color{blue}{\left(0.375 + -0.25 \cdot v\right)} \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   7. Step-by-step derivation

      1. associate-/l* 91.5%

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

      2. *-commutative 91.5%

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

      3. associate-*r/ 91.5%

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

      4. associate-*l* 96.1%

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

      5. associate-*r* 99.7%

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

      6. *-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 15:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

3. Simplified 88.4%

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

   1. fma-undefine 88.4%

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

   2. *-commutative 88.4%

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

   3. +-commutative 88.4%

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

   4. metadata-eval 88.4%

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

   5. cancel-sign-sub-inv 88.4%

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

   6. associate-*r/ 88.4%

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

   7. *-commutative 88.4%

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

   8. associate-/l* 89.4%

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

   9. clear-num 89.4%

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

   10. un-div-inv 89.4%

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

   11. cancel-sign-sub-inv 89.4%

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

   12. metadata-eval 89.4%

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

   13. distribute-rgt-in 89.4%

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

   14. metadata-eval 89.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   15. *-commutative 89.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   16. associate-*l* 89.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   17. metadata-eval 89.8%

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

6. Applied egg-rr 99.4%

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

   1. unpow2 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 5: 93.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

4. Applied egg-rr 93.0%

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

5. Taylor expanded in v around 0 83.8%

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

6. Taylor expanded in v around 0 93.7%

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

7. Final simplification 93.7%

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

Alternative 6: 40.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (3.0d0 + ((0.375d0 * (r * w)) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

4. Applied egg-rr 93.0%

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

5. Taylor expanded in v around 0 83.8%

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

6. Taylor expanded in r around inf 37.4%

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

7. Final simplification 37.4%

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

Alternative 7: 42.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = 3.0d0 - (4.5d0 + (0.375d0 * ((r * (r * (w * w))) / (1.0d0 - v))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. associate--l- 86.2%

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

   2. associate-*l* 79.7%

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

   3. sqr-neg 79.7%

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

   4. associate-*l* 86.2%

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

   5. associate-/l* 89.4%

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

   6. fma-define 89.4%

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

3. Simplified 89.4%

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

5. Taylor expanded in r around inf 50.0%

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

6. Taylor expanded in v around 0 37.1%

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

7. Final simplification 37.1%

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

Alternative 8: 39.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = 3.0d0 + ((0.375d0 * (w * ((r * w) * (r / (v + (-1.0d0)))))) - 4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. associate--l- 86.2%

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

   2. associate-*l* 79.7%

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

   3. sqr-neg 79.7%

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

   4. associate-*l* 86.2%

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

   5. associate-/l* 89.4%

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

   6. fma-define 89.4%

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

3. Simplified 89.4%

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

5. Taylor expanded in r around inf 50.0%

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

6. Taylor expanded in v around 0 37.1%

   \[\leadsto 3 - \left(\color{blue}{0.375} \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
7. Step-by-step derivation

   1. associate-/l* 36.4%

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

   2. *-commutative 36.4%

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

   3. associate-*r/ 36.4%

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

   4. associate-*l* 37.9%

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

   5. associate-*r* 37.1%

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

   6. add-sqr-sqrt 19.4%

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

   7. sqrt-prod 22.3%

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

   8. add-sqr-sqrt 9.3%

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

   9. sqrt-prod 23.3%

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

   10. sqrt-prod 23.3%

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

   11. associate-*r* 30.1%

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

   12. associate-*l* 30.1%

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

   13. *-commutative 30.1%

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

   14. associate-*l* 29.7%

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

8. Applied egg-rr 36.0%

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

9. Final simplification 36.0%

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

Alternative 9: 15.1% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. associate--l- 86.2%

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

   2. associate-*l* 79.7%

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

   3. sqr-neg 79.7%

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

   4. associate-*l* 86.2%

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

   5. associate-/l* 89.4%

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

   6. fma-define 89.4%

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

3. Simplified 89.4%

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

5. Taylor expanded in r around inf 50.0%

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

6. Taylor expanded in v around 0 37.1%

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

7. Taylor expanded in r around 0 15.0%

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

Reproduce

herbie shell --seed 2024123 
(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))