Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.0%

Time: 11.8s

Alternatives: 8

Speedup: 1.7×

• 53.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r_m \leq 3.1 \cdot 10^{-65}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r_m}}{r_m} + -0.375 \cdot \left(\left(r_m \cdot w\right) \cdot \left(r_m \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r_m \cdot r_m} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r_m \cdot \left(w \cdot \left(r_m \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 3.1e-65)
   (+ -1.5 (+ (/ (/ 2.0 r_m) r_m) (* -0.375 (* (* r_m w) (* r_m w)))))
   (+
    -1.5
    (+
     (/ 2.0 (* r_m r_m))
     (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r_m (* w (* r_m w))))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 3.1e-65) {
		tmp = -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))));
	} else {
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r_m * (w * (r_m * w)))));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 3.1d-65) then
        tmp = (-1.5d0) + (((2.0d0 / r_m) / r_m) + ((-0.375d0) * ((r_m * w) * (r_m * w))))
    else
        tmp = (-1.5d0) + ((2.0d0 / (r_m * r_m)) + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r_m * (w * (r_m * w)))))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 3.1e-65) {
		tmp = -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))));
	} else {
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r_m * (w * (r_m * w)))));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 3.1e-65:
		tmp = -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))))
	else:
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r_m * (w * (r_m * w)))))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 3.1e-65)
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-0.375 * Float64(Float64(r_m * w) * Float64(r_m * w)))));
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r_m * Float64(w * Float64(r_m * w))))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 3.1e-65)
		tmp = -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))));
	else
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r_m * (w * (r_m * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 3.1e-65], N[(-1.5 + N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-0.375 * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.10000000000000016e-65

   1. Initial program 82.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in v around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

      4. swap-sqr 98.0%

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

      5. unpow2 98.0%

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

   6. Simplified 98.0%

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

      1. unpow2 98.0%

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

   8. Applied egg-rr 98.0%

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

      1. associate-/r* 98.0%

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

      2. div-inv 98.0%

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

      3. *-un-lft-identity 98.0%

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

      4. times-frac 98.0%

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

      5. metadata-eval 98.0%

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

   10. Applied egg-rr 98.0%

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

       1. associate-*r/ 98.0%

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

       2. div-inv 98.0%

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

   12. Applied egg-rr 98.0%

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

   if 3.10000000000000016e-65 < r

   1. Initial program 88.3%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in r around 0 81.4%

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

      1. unpow2 81.4%

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

      2. unpow2 81.4%

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

      3. swap-sqr 99.8%

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

      4. unpow2 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*r* 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.1 \cdot 10^{-65}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+
  (+
   (/ 2.0 (* r_m r_m))
   (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (pow (* r_m w) 2.0)))
  -1.5))

C

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

Fortran

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * Math.pow((r_m * w), 2.0))) + -1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * math.pow((r_m * w), 2.0))) + -1.5

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[Power[N[(r$95$m * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 87.0%

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

4. Taylor expanded in r around 0 80.1%

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

   1. unpow2 80.1%

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

   2. unpow2 80.1%

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

   3. swap-sqr 99.8%

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

   4. unpow2 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+
  (+
   3.0
   (-
    (/ 2.0 (* r_m r_m))
    (/
     (* 0.125 (+ 3.0 (* v -2.0)))
     (* (/ 1.0 (* r_m w)) (/ (- 1.0 v) (* r_m w))))))
  -4.5))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(3.0 + N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - v), $MachinePrecision] / N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 87.0%

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

   1. associate-*r* 97.4%

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

   2. *-commutative 97.4%

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

   3. *-un-lft-identity 97.4%

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

   4. associate-*r* 99.8%

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

   5. times-frac 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r_m \leq 2 \cdot 10^{-57}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r_m}}{r_m} + -0.375 \cdot \left(\left(r_m \cdot w\right) \cdot \left(r_m \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r_m \cdot r_m} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r_m \cdot \left(r_m \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 2e-57)
   (+ -1.5 (+ (/ (/ 2.0 r_m) r_m) (* -0.375 (* (* r_m w) (* r_m w)))))
   (+
    -1.5
    (+
     (/ 2.0 (* r_m r_m))
     (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r_m (* r_m (* w w))))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2e-57) {
		tmp = -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))));
	} else {
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r_m * (r_m * (w * w)))));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 2d-57) then
        tmp = (-1.5d0) + (((2.0d0 / r_m) / r_m) + ((-0.375d0) * ((r_m * w) * (r_m * w))))
    else
        tmp = (-1.5d0) + ((2.0d0 / (r_m * r_m)) + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r_m * (r_m * (w * w)))))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2e-57) {
		tmp = -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))));
	} else {
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r_m * (r_m * (w * w)))));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 2e-57:
		tmp = -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))))
	else:
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r_m * (r_m * (w * w)))))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 2e-57)
		tmp = Float64(-1.5 + Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-0.375 * Float64(Float64(r_m * w) * Float64(r_m * w)))));
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r_m * Float64(r_m * Float64(w * w))))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 2e-57)
		tmp = -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))));
	else
		tmp = -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r_m * (r_m * (w * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 2e-57], N[(-1.5 + N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-0.375 * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(r$95$m * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.99999999999999991e-57

   1. Initial program 82.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in v around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. unpow2 77.6%

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

      4. swap-sqr 98.0%

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

      5. unpow2 98.0%

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

   6. Simplified 98.0%

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

      1. unpow2 98.0%

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

   8. Applied egg-rr 98.0%

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

      1. associate-/r* 98.0%

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

      2. div-inv 98.0%

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

      3. *-un-lft-identity 98.0%

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

      4. times-frac 98.0%

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

      5. metadata-eval 98.0%

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

   10. Applied egg-rr 98.0%

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

       1. associate-*r/ 98.0%

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

       2. div-inv 98.0%

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

   12. Applied egg-rr 98.0%

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

   if 1.99999999999999991e-57 < r

   1. Initial program 88.3%

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

   3. Simplified 95.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-57}:\\ \;\;\;\;-1.5 + \left(\frac{\frac{2}{r}}{r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.2× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+
  -1.5
  (+
   (/ 2.0 (* r_m r_m))
   (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* w (* r_m (* r_m w)))))))

C

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

Fortran

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

Java

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

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -1.5 + ((2.0 / (r_m * r_m)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (w * (r_m * (r_m * w)))))

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(-1.5 + N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r$95$m * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 87.0%

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

4. Taylor expanded in r around 0 80.1%

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

   1. unpow2 80.1%

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

   2. unpow2 80.1%

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

   3. swap-sqr 99.8%

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

   4. unpow2 99.8%

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

6. Simplified 99.8%

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

   1. unpow2 99.8%

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

   2. associate-*r* 98.0%

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

8. Applied egg-rr 98.0%

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

9. Final simplification 98.0%

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

Alternative 6: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r_m \cdot r_m}\\ \mathbf{if}\;v \leq -5 \lor \neg \left(v \leq 0.108\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r_m \cdot w\right) \cdot \left(w \cdot \left(r_m \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_0 + -0.375 \cdot \left(\left(r_m \cdot w\right) \cdot \left(r_m \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (or (<= v -5.0) (not (<= v 0.108)))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r_m w) (* w (* r_m 0.25))))))
     (+ -1.5 (+ t_0 (* -0.375 (* (* r_m w) (* r_m w))))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if ((v <= -5.0) || !(v <= 0.108)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r_m * w) * (w * (r_m * 0.25)))));
	} else {
		tmp = -1.5 + (t_0 + (-0.375 * ((r_m * w) * (r_m * w))));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    if ((v <= (-5.0d0)) .or. (.not. (v <= 0.108d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r_m * w) * (w * (r_m * 0.25d0)))))
    else
        tmp = (-1.5d0) + (t_0 + ((-0.375d0) * ((r_m * w) * (r_m * w))))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if ((v <= -5.0) || !(v <= 0.108)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r_m * w) * (w * (r_m * 0.25)))));
	} else {
		tmp = -1.5 + (t_0 + (-0.375 * ((r_m * w) * (r_m * w))));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	tmp = 0
	if (v <= -5.0) or not (v <= 0.108):
		tmp = -4.5 + (3.0 + (t_0 - ((r_m * w) * (w * (r_m * 0.25)))))
	else:
		tmp = -1.5 + (t_0 + (-0.375 * ((r_m * w) * (r_m * w))))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if ((v <= -5.0) || !(v <= 0.108))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r_m * w) * Float64(w * Float64(r_m * 0.25))))));
	else
		tmp = Float64(-1.5 + Float64(t_0 + Float64(-0.375 * Float64(Float64(r_m * w) * Float64(r_m * w)))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 2.0 / (r_m * r_m);
	tmp = 0.0;
	if ((v <= -5.0) || ~((v <= 0.108)))
		tmp = -4.5 + (3.0 + (t_0 - ((r_m * w) * (w * (r_m * 0.25)))));
	else
		tmp = -1.5 + (t_0 + (-0.375 * ((r_m * w) * (r_m * w))));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -5.0], N[Not[LessEqual[v, 0.108]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r$95$m * w), $MachinePrecision] * N[(w * N[(r$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$0 + N[(-0.375 * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -5 or 0.107999999999999999 < v

   1. Initial program 81.3%

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

   3. Simplified 87.2%

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

      1. associate-/r/ 87.2%

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

      2. associate-*r* 80.9%

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

      3. swap-sqr 99.7%

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

      4. associate-*r* 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-rgt-in 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*l* 99.7%

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

      9. metadata-eval 99.7%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.7%

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

      11. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in v around inf 99.0%

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

      1. associate-*r* 99.0%

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

   8. Simplified 99.0%

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

   if -5 < v < 0.107999999999999999

   1. Initial program 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in v around 0 79.3%

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

      1. *-commutative 79.3%

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

      2. unpow2 79.3%

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

      3. unpow2 79.3%

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

      4. swap-sqr 99.4%

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

      5. unpow2 99.4%

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

   6. Simplified 99.4%

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

      1. unpow2 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5 \lor \neg \left(v \leq 0.108\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(\frac{2}{r \cdot r} + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 7: 93.3% accurate, 1.7× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+ -1.5 (+ (/ 2.0 (* r_m r_m)) (* -0.375 (* (* r_m w) (* r_m w))))))

C

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

Fortran

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

Java

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

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -1.5 + ((2.0 / (r_m * r_m)) + (-0.375 * ((r_m * w) * (r_m * w))))

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(-1.5 + N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 87.0%

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

4. Taylor expanded in v around 0 76.7%

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

   1. *-commutative 76.7%

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

   2. unpow2 76.7%

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

   3. unpow2 76.7%

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

   4. swap-sqr 94.6%

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

   5. unpow2 94.6%

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

6. Simplified 94.6%

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

   1. unpow2 94.6%

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

8. Applied egg-rr 94.6%

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

9. Final simplification 94.6%

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

Alternative 8: 93.3% accurate, 1.7× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+ -1.5 (+ (/ (/ 2.0 r_m) r_m) (* -0.375 (* (* r_m w) (* r_m w))))))

C

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

Fortran

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

Java

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

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -1.5 + (((2.0 / r_m) / r_m) + (-0.375 * ((r_m * w) * (r_m * w))))

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(-1.5 + N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-0.375 * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

3. Simplified 87.0%

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

4. Taylor expanded in v around 0 76.7%

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

   1. *-commutative 76.7%

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

   2. unpow2 76.7%

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

   3. unpow2 76.7%

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

   4. swap-sqr 94.6%

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

   5. unpow2 94.6%

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

6. Simplified 94.6%

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

   1. unpow2 94.6%

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

8. Applied egg-rr 94.6%

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

   1. associate-/r* 94.7%

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

   2. div-inv 94.7%

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

   3. *-un-lft-identity 94.7%

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

   4. times-frac 94.7%

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

   5. metadata-eval 94.7%

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

10. Applied egg-rr 94.7%

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

    1. associate-*r/ 94.7%

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

    2. div-inv 94.7%

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

12. Applied egg-rr 94.7%

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

13. Final simplification 94.7%

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

Reproduce

herbie shell --seed 2023339 
(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))