Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 98.9%

Time: 15.5s

Alternatives: 7

Speedup: 1.5×

• 53.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -7000000000000 \lor \neg \left(v \leq 1.15\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{\frac{\frac{v}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{r}}{r}\right) + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= v -7000000000000.0) (not (<= v 1.15)))
   (+
    (/ 2.0 (* r r))
    (+ -1.5 (/ (+ 0.375 (* v -0.25)) (/ (/ (/ v r) w) (* r w)))))
   (-
    (+ (+ 3.0 (/ (/ 2.0 r) r)) (* (* 0.375 (* r w)) (/ w (/ (+ v -1.0) r))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((v <= -7000000000000.0) || !(v <= 1.15)) {
		tmp = (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / (((v / r) / w) / (r * w))));
	} else {
		tmp = ((3.0 + ((2.0 / r) / r)) + ((0.375 * (r * w)) * (w / ((v + -1.0) / r)))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((v <= (-7000000000000.0d0)) .or. (.not. (v <= 1.15d0))) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) / (((v / r) / w) / (r * w))))
    else
        tmp = ((3.0d0 + ((2.0d0 / r) / r)) + ((0.375d0 * (r * w)) * (w / ((v + (-1.0d0)) / r)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((v <= -7000000000000.0) || !(v <= 1.15)) {
		tmp = (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / (((v / r) / w) / (r * w))));
	} else {
		tmp = ((3.0 + ((2.0 / r) / r)) + ((0.375 * (r * w)) * (w / ((v + -1.0) / r)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (v <= -7000000000000.0) or not (v <= 1.15):
		tmp = (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / (((v / r) / w) / (r * w))))
	else:
		tmp = ((3.0 + ((2.0 / r) / r)) + ((0.375 * (r * w)) * (w / ((v + -1.0) / r)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((v <= -7000000000000.0) || !(v <= 1.15))
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(Float64(Float64(v / r) / w) / Float64(r * w)))));
	else
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / r) / r)) + Float64(Float64(0.375 * Float64(r * w)) * Float64(w / Float64(Float64(v + -1.0) / r)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((v <= -7000000000000.0) || ~((v <= 1.15)))
		tmp = (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / (((v / r) / w) / (r * w))));
	else
		tmp = ((3.0 + ((2.0 / r) / r)) + ((0.375 * (r * w)) * (w / ((v + -1.0) / r)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[v, -7000000000000.0], N[Not[LessEqual[v, 1.15]], $MachinePrecision]], 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[(v / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] + N[(N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w / N[(N[(v + -1.0), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -7e12 or 1.1499999999999999 < v

   1. Initial program 85.2%

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

   3. Simplified 87.2%

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

      1. *-commutative 87.2%

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

      2. fma-undefine 87.2%

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

      3. *-commutative 87.2%

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

      4. +-commutative 87.2%

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

      5. metadata-eval 87.2%

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

      6. cancel-sign-sub-inv 87.2%

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

      7. associate-*r/ 87.2%

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

      8. *-commutative 87.2%

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

      9. associate-/l* 88.3%

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

      10. *-commutative 88.3%

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

      11. clear-num 88.2%

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

      12. un-div-inv 88.2%

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

   6. Applied egg-rr 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.8%

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

   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. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in v around 0 94.9%

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

       1. associate-/r* 94.1%

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

       2. neg-mul-1 94.1%

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

       3. +-commutative 94.1%

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

       4. sub-neg 94.1%

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

       5. associate-/r* 95.8%

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

       6. div-sub 99.8%

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

       7. div-sub 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in v around inf 99.8%

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

       1. mul-1-neg 99.8%

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

       2. associate-/r* 99.8%

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

   16. Simplified 99.8%

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

   if -7e12 < v < 1.1499999999999999

   1. Initial program 90.7%

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

      1. associate-/r* 90.7%

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

      2. div-inv 90.6%

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

   4. Applied egg-rr 90.6%

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

      1. associate-*r/ 90.7%

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

      2. *-rgt-identity 90.7%

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

   6. Simplified 90.7%

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

      1. associate-/l* 90.7%

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

      2. cancel-sign-sub-inv 90.7%

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

      3. metadata-eval 90.7%

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

      4. +-commutative 90.7%

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

      5. *-commutative 90.7%

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

      6. fma-undefine 90.7%

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

      7. *-commutative 90.7%

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

      8. *-commutative 90.7%

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

      9. associate-/l* 90.7%

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

      10. *-commutative 90.7%

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

      11. associate-*r/ 90.7%

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

      12. associate-*r* 90.7%

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

      13. associate-*l* 98.4%

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

      14. associate-*r* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -7000000000000 \lor \neg \left(v \leq 1.15\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{\frac{\frac{v}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{r}}{r}\right) + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.2× speedup

Math

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

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(Float64(Float64(v + -1.0) / 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[(N[(N[(v + -1.0), $MachinePrecision] / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

3. Simplified 89.0%

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

   1. *-commutative 89.0%

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

   2. fma-undefine 89.0%

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

   3. *-commutative 89.0%

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

   4. +-commutative 89.0%

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

   5. metadata-eval 89.0%

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

   6. cancel-sign-sub-inv 89.0%

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

   7. associate-*r/ 89.0%

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

   8. *-commutative 89.0%

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

   9. associate-/l* 89.5%

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

   10. *-commutative 89.5%

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

   11. clear-num 89.5%

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

   12. un-div-inv 89.5%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

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. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in v around 0 94.7%

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

    1. associate-/r* 94.3%

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

    2. neg-mul-1 94.3%

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

    3. +-commutative 94.3%

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

    4. sub-neg 94.3%

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

    5. associate-/r* 96.7%

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

    6. div-sub 99.8%

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

    7. div-sub 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 3: 99.8% 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 88.0%

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

3. Simplified 89.0%

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

   1. *-commutative 89.0%

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

   2. fma-undefine 89.0%

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

   3. *-commutative 89.0%

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

   4. +-commutative 89.0%

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

   5. metadata-eval 89.0%

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

   6. cancel-sign-sub-inv 89.0%

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

   7. associate-*r/ 89.0%

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

   8. *-commutative 89.0%

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

   9. associate-/l* 89.5%

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

   10. *-commutative 89.5%

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

   11. clear-num 89.5%

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

   12. un-div-inv 89.5%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

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(Float64(v + -1.0) / 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[(N[(v + -1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

3. Simplified 89.0%

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

   1. *-commutative 89.0%

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

   2. fma-undefine 89.0%

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

   3. *-commutative 89.0%

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

   4. +-commutative 89.0%

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

   5. metadata-eval 89.0%

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

   6. cancel-sign-sub-inv 89.0%

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

   7. associate-*r/ 89.0%

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

   8. *-commutative 89.0%

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

   9. associate-/l* 89.5%

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

   10. *-commutative 89.5%

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

   11. clear-num 89.5%

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

   12. un-div-inv 89.5%

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

6. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. unpow2 99.8%

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

   3. times-frac 99.8%

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

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. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 5: 92.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + ((2.0d0 / r) / r)) - ((r * w) * (w * (r * 0.375d0)))) - 4.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. associate-/r* 88.0%

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

   2. div-inv 88.0%

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

4. Applied egg-rr 88.0%

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

   1. associate-*r/ 88.0%

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

   2. *-rgt-identity 88.0%

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

6. Simplified 88.0%

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

   1. associate-/l* 89.5%

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

   2. cancel-sign-sub-inv 89.5%

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

   3. metadata-eval 89.5%

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

   4. +-commutative 89.5%

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

   5. *-commutative 89.5%

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

   6. fma-undefine 89.5%

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

   7. *-commutative 89.5%

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

   8. *-commutative 89.5%

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

   9. associate-/l* 89.0%

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

   10. *-commutative 89.0%

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

   11. associate-*r/ 89.0%

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

   12. associate-*r* 83.9%

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

   13. associate-*l* 92.2%

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

   14. associate-*r* 93.0%

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

8. Applied egg-rr 93.0%

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

9. Taylor expanded in v around 0 86.2%

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

    1. associate-*r* 86.2%

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

11. Simplified 86.2%

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

12. Taylor expanded in v around 0 94.7%

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

13. Final simplification 94.7%

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

Alternative 6: 55.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Simplified 82.6%

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

5. Taylor expanded in r around 0 61.7%

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

   1. associate--l+ 61.7%

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

   2. div-inv 61.7%

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

   3. fma-neg 61.7%

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

   4. pow2 61.7%

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

   5. pow-flip 61.8%

      \[\leadsto 3 + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -4.5\right) \]

   6. metadata-eval 61.8%

      \[\leadsto 3 + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -4.5\right) \]

   7. metadata-eval 61.8%

      \[\leadsto 3 + \mathsf{fma}\left(2, {r}^{-2}, \color{blue}{-4.5}\right) \]

7. Applied egg-rr 61.8%

   \[\leadsto \color{blue}{3 + \mathsf{fma}\left(2, {r}^{-2}, -4.5\right)} \]
8. Step-by-step derivation

   1. +-commutative 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, {r}^{-2}, -4.5\right) + 3} \]

   2. fma-undefine 61.8%

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

   3. associate-+l+ 61.8%

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

   4. metadata-eval 61.8%

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

9. Simplified 61.8%

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

    1. sqr-pow 61.7%

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

    2. metadata-eval 61.7%

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

    3. inv-pow 61.7%

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

    4. metadata-eval 61.7%

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

    5. inv-pow 61.7%

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

    6. associate-*r* 61.7%

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

    7. div-inv 61.7%

       \[\leadsto \color{blue}{\frac{2}{r}} \cdot \frac{1}{r} + -1.5 \]

    8. div-inv 61.7%

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

11. Applied egg-rr 61.7%

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

12. Final simplification 61.7%

    \[\leadsto -1.5 + \frac{\frac{2}{r}}{r} \]
13. Add Preprocessing

Alternative 7: 13.6% 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 88.0%

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

3. Simplified 82.6%

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

5. Taylor expanded in r around 0 61.7%

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

6. Taylor expanded in r around inf 17.8%

   \[\leadsto \color{blue}{-1.5} \]

7. Final simplification 17.8%

   \[\leadsto -1.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))