Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.4%

Time: 15.1s

Alternatives: 14

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: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ t_1 := t\_0 + \left(0.375 + v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right)\\ \mathbf{if}\;v \leq -2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5))
        (t_1 (+ t_0 (* (+ 0.375 (* v -0.25)) (* (* r w) (/ (* r w) v))))))
   (if (<= v -2.0)
     t_1
     (if (<= v 9.5e-6) (+ t_0 (* (* r w) (* w (* r -0.375)))) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double t_1 = t_0 + ((0.375 + (v * -0.25)) * ((r * w) * ((r * w) / v)));
	double tmp;
	if (v <= -2.0) {
		tmp = t_1;
	} else if (v <= 9.5e-6) {
		tmp = t_0 + ((r * w) * (w * (r * -0.375)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (2.0d0 / (r * r)) + (-1.5d0)
    t_1 = t_0 + ((0.375d0 + (v * (-0.25d0))) * ((r * w) * ((r * w) / v)))
    if (v <= (-2.0d0)) then
        tmp = t_1
    else if (v <= 9.5d-6) then
        tmp = t_0 + ((r * w) * (w * (r * (-0.375d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double t_1 = t_0 + ((0.375 + (v * -0.25)) * ((r * w) * ((r * w) / v)));
	double tmp;
	if (v <= -2.0) {
		tmp = t_1;
	} else if (v <= 9.5e-6) {
		tmp = t_0 + ((r * w) * (w * (r * -0.375)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	t_1 = t_0 + ((0.375 + (v * -0.25)) * ((r * w) * ((r * w) / v)))
	tmp = 0
	if v <= -2.0:
		tmp = t_1
	elif v <= 9.5e-6:
		tmp = t_0 + ((r * w) * (w * (r * -0.375)))
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	t_1 = Float64(t_0 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r * w) * Float64(Float64(r * w) / v))))
	tmp = 0.0
	if (v <= -2.0)
		tmp = t_1;
	elseif (v <= 9.5e-6)
		tmp = Float64(t_0 + Float64(Float64(r * w) * Float64(w * Float64(r * -0.375))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	t_1 = t_0 + ((0.375 + (v * -0.25)) * ((r * w) * ((r * w) / v)));
	tmp = 0.0;
	if (v <= -2.0)
		tmp = t_1;
	elseif (v <= 9.5e-6)
		tmp = t_0 + ((r * w) * (w * (r * -0.375)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.0], t$95$1, If[LessEqual[v, 9.5e-6], N[(t$95$0 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2 or 9.5000000000000005e-6 < v

   1. Initial program 85.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 89.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\frac{r \cdot \left(w \cdot w\right)}{v + -1} \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\frac{\left(r \cdot w\right) \cdot w}{v + -1} \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(r \cdot w\right) \cdot \frac{w}{v + -1}\right) \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{r}, r\right)\right), \frac{-3}{2}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(r \cdot w\right) \cdot \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{r}, r\right)\right), \frac{-3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\left(\frac{w}{v + -1}\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right), \frac{-3}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(v + -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      10. +-lowering-+.f64 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{+.f64}\left(v, -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \color{blue}{\left(\frac{r \cdot w}{v}\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{/.f64}\left(\left(r \cdot w\right), v\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right), \frac{-3}{2}\right)\right) \]

      2. *-lowering-*.f64 98.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), v\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   9. Simplified 98.6%

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

   if -2 < v < 9.5000000000000005e-6

   1. Initial program 86.8%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 86.8%

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

   5. Taylor expanded in v around 0

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

      1. associate--l+ N/A

         \[\leadsto \frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right), \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \left(2 \cdot \color{blue}{\frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right)\right) \]

      18. *-lowering-*.f64 83.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 83.3%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(r \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right), \frac{-3}{2}\right)\right) \]

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + \left(0.375 + v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right)\\ \mathbf{elif}\;v \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + \left(0.375 + v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ t_1 := t\_0 + \left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \frac{w}{v + -1}\right)\right)\\ \mathbf{if}\;v \leq -3.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5))
        (t_1 (+ t_0 (* (* v -0.25) (* (* r w) (* r (/ w (+ v -1.0))))))))
   (if (<= v -3.2)
     t_1
     (if (<= v 9.5e-6) (+ t_0 (* (* r w) (* w (* r -0.375)))) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double t_1 = t_0 + ((v * -0.25) * ((r * w) * (r * (w / (v + -1.0)))));
	double tmp;
	if (v <= -3.2) {
		tmp = t_1;
	} else if (v <= 9.5e-6) {
		tmp = t_0 + ((r * w) * (w * (r * -0.375)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (2.0d0 / (r * r)) + (-1.5d0)
    t_1 = t_0 + ((v * (-0.25d0)) * ((r * w) * (r * (w / (v + (-1.0d0))))))
    if (v <= (-3.2d0)) then
        tmp = t_1
    else if (v <= 9.5d-6) then
        tmp = t_0 + ((r * w) * (w * (r * (-0.375d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double t_1 = t_0 + ((v * -0.25) * ((r * w) * (r * (w / (v + -1.0)))));
	double tmp;
	if (v <= -3.2) {
		tmp = t_1;
	} else if (v <= 9.5e-6) {
		tmp = t_0 + ((r * w) * (w * (r * -0.375)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / (r * r)) + -1.5
	t_1 = t_0 + ((v * -0.25) * ((r * w) * (r * (w / (v + -1.0)))))
	tmp = 0
	if v <= -3.2:
		tmp = t_1
	elif v <= 9.5e-6:
		tmp = t_0 + ((r * w) * (w * (r * -0.375)))
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	t_1 = Float64(t_0 + Float64(Float64(v * -0.25) * Float64(Float64(r * w) * Float64(r * Float64(w / Float64(v + -1.0))))))
	tmp = 0.0
	if (v <= -3.2)
		tmp = t_1;
	elseif (v <= 9.5e-6)
		tmp = Float64(t_0 + Float64(Float64(r * w) * Float64(w * Float64(r * -0.375))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / (r * r)) + -1.5;
	t_1 = t_0 + ((v * -0.25) * ((r * w) * (r * (w / (v + -1.0)))));
	tmp = 0.0;
	if (v <= -3.2)
		tmp = t_1;
	elseif (v <= 9.5e-6)
		tmp = t_0 + ((r * w) * (w * (r * -0.375)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * N[(w / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.2], t$95$1, If[LessEqual[v, 9.5e-6], N[(t$95$0 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.2000000000000002 or 9.5000000000000005e-6 < v

   1. Initial program 85.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 89.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\frac{r \cdot \left(w \cdot w\right)}{v + -1} \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\frac{\left(r \cdot w\right) \cdot w}{v + -1} \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(r \cdot w\right) \cdot \frac{w}{v + -1}\right) \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{r}, r\right)\right), \frac{-3}{2}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(r \cdot w\right) \cdot \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{r}, r\right)\right), \frac{-3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\left(\frac{w}{v + -1}\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right), \frac{-3}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(v + -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      10. +-lowering-+.f64 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{+.f64}\left(v, -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot v\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{+.f64}\left(v, -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(v \cdot \frac{-1}{4}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{+.f64}\left(v, -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      2. *-lowering-*.f64 97.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, \frac{-1}{4}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{+.f64}\left(v, -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   9. Simplified 97.2%

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

   if -3.2000000000000002 < v < 9.5000000000000005e-6

   1. Initial program 86.8%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 86.8%

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

   5. Taylor expanded in v around 0

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

      1. associate--l+ N/A

         \[\leadsto \frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right), \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \left(2 \cdot \color{blue}{\frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right)\right) \]

      18. *-lowering-*.f64 83.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 83.3%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(r \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right), \frac{-3}{2}\right)\right) \]

   9. Applied egg-rr 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.2:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + \left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \frac{w}{v + -1}\right)\right)\\ \mathbf{elif}\;v \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + \left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \frac{w}{v + -1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 10^{+143}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7e-5)
   (+ (/ 2.0 (* r r)) -1.5)
   (if (<= r 1e+143)
     (* (* r r) (+ (* -0.375 (* w w)) (/ -1.5 (* r r))))
     (* (* r w) (* (* r w) (/ (+ 0.375 (* v -0.25)) (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 1e+143) {
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	} else {
		tmp = (r * w) * ((r * w) * ((0.375 + (v * -0.25)) / (v + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 7d-5) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else if (r <= 1d+143) then
        tmp = (r * r) * (((-0.375d0) * (w * w)) + ((-1.5d0) / (r * r)))
    else
        tmp = (r * w) * ((r * w) * ((0.375d0 + (v * (-0.25d0))) / (v + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 1e+143) {
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	} else {
		tmp = (r * w) * ((r * w) * ((0.375 + (v * -0.25)) / (v + -1.0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7e-5:
		tmp = (2.0 / (r * r)) + -1.5
	elif r <= 1e+143:
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)))
	else:
		tmp = (r * w) * ((r * w) * ((0.375 + (v * -0.25)) / (v + -1.0)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7e-5)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	elseif (r <= 1e+143)
		tmp = Float64(Float64(r * r) * Float64(Float64(-0.375 * Float64(w * w)) + Float64(-1.5 / Float64(r * r))));
	else
		tmp = Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(v + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7e-5)
		tmp = (2.0 / (r * r)) + -1.5;
	elseif (r <= 1e+143)
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	else
		tmp = (r * w) * ((r * w) * ((0.375 + (v * -0.25)) / (v + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7e-5], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[r, 1e+143], N[(N[(r * r), $MachinePrecision] * N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 6.9999999999999994e-5

   1. Initial program 84.8%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 86.1%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right) \]

      8. *-lowering-*.f64 64.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right) \]

   7. Simplified 64.8%

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

   if 6.9999999999999994e-5 < r < 1e143

   1. Initial program 89.6%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in v around 0

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

      1. associate--l+ N/A

         \[\leadsto \frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right), \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \left(2 \cdot \color{blue}{\frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right)\right) \]

      18. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 91.0%

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

   8. Taylor expanded in r around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot {w}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \left({w}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \left(w \cdot w\right)\right), \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{\frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{\frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2} \cdot 1}{{r}^{2}}\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2}}{{r}^{2}}\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{\mathsf{neg}\left(\frac{3}{2}\right)}{\color{blue}{{r}^{2}}}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{\frac{-3}{2}}{{\color{blue}{r}}^{2}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \color{blue}{\left({r}^{2}\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{r}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right)\right)\right) \]

   10. Simplified 91.0%

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

   if 1e143 < r

   1. Initial program 90.4%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 93.2%

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

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right) \]

      15. +-lowering-+.f64 77.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 77.6%

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

      1. associate-/l* N/A

         \[\leadsto \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \color{blue}{\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{r}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \color{blue}{w}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot w\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(r \cdot w\right)\right), \color{blue}{\left(r \cdot w\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right), \left(r \cdot w\right)\right), \left(\color{blue}{r} \cdot w\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(v + -1\right)\right), \left(r \cdot w\right)\right), \left(r \cdot w\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(v + -1\right)\right), \left(r \cdot w\right)\right), \left(r \cdot w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(v + -1\right)\right), \left(r \cdot w\right)\right), \left(r \cdot w\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \left(r \cdot w\right)\right), \left(r \cdot w\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \left(r \cdot w\right)\right) \]

      15. *-lowering-*.f64 85.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right) \]

   9. Applied egg-rr 85.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 10^{+143}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 2.7 \cdot 10^{+143}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(w \cdot \frac{\left(0.375 + v \cdot -0.25\right) \cdot r}{v + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7.8e-5)
   (+ (/ 2.0 (* r r)) -1.5)
   (if (<= r 2.7e+143)
     (* (* r r) (+ (* -0.375 (* w w)) (/ -1.5 (* r r))))
     (* (* r w) (* w (/ (* (+ 0.375 (* v -0.25)) r) (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.8e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 2.7e+143) {
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	} else {
		tmp = (r * w) * (w * (((0.375 + (v * -0.25)) * r) / (v + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 7.8d-5) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else if (r <= 2.7d+143) then
        tmp = (r * r) * (((-0.375d0) * (w * w)) + ((-1.5d0) / (r * r)))
    else
        tmp = (r * w) * (w * (((0.375d0 + (v * (-0.25d0))) * r) / (v + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.8e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 2.7e+143) {
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	} else {
		tmp = (r * w) * (w * (((0.375 + (v * -0.25)) * r) / (v + -1.0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7.8e-5:
		tmp = (2.0 / (r * r)) + -1.5
	elif r <= 2.7e+143:
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)))
	else:
		tmp = (r * w) * (w * (((0.375 + (v * -0.25)) * r) / (v + -1.0)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7.8e-5)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	elseif (r <= 2.7e+143)
		tmp = Float64(Float64(r * r) * Float64(Float64(-0.375 * Float64(w * w)) + Float64(-1.5 / Float64(r * r))));
	else
		tmp = Float64(Float64(r * w) * Float64(w * Float64(Float64(Float64(0.375 + Float64(v * -0.25)) * r) / Float64(v + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7.8e-5)
		tmp = (2.0 / (r * r)) + -1.5;
	elseif (r <= 2.7e+143)
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	else
		tmp = (r * w) * (w * (((0.375 + (v * -0.25)) * r) / (v + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7.8e-5], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[r, 2.7e+143], N[(N[(r * r), $MachinePrecision] * N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(w * N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.7999999999999999e-5

   1. Initial program 84.8%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 86.1%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right) \]

      8. *-lowering-*.f64 64.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right) \]

   7. Simplified 64.8%

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

   if 7.7999999999999999e-5 < r < 2.7000000000000002e143

   1. Initial program 89.6%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in v around 0

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

      1. associate--l+ N/A

         \[\leadsto \frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right), \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \left(2 \cdot \color{blue}{\frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right)\right) \]

      18. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 91.0%

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

   8. Taylor expanded in r around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot {w}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \left({w}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \left(w \cdot w\right)\right), \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{\frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{\frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2} \cdot 1}{{r}^{2}}\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2}}{{r}^{2}}\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{\mathsf{neg}\left(\frac{3}{2}\right)}{\color{blue}{{r}^{2}}}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{\frac{-3}{2}}{{\color{blue}{r}}^{2}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \color{blue}{\left({r}^{2}\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{r}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right)\right)\right) \]

   10. Simplified 91.0%

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

   if 2.7000000000000002e143 < r

   1. Initial program 90.4%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 93.2%

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

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right) \]

      15. +-lowering-+.f64 77.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 77.6%

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

      1. associate-/l* N/A

         \[\leadsto \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \color{blue}{\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \frac{\color{blue}{\frac{3}{8} + v \cdot \frac{-1}{4}}}{v + -1} \]

      3. associate-*l* N/A

         \[\leadsto \left(r \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\color{blue}{r} \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(w \cdot \left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(r \cdot w\right), \color{blue}{\left(w \cdot \left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\color{blue}{w} \cdot \left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \color{blue}{\left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \left(\frac{r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v + -1}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right), \color{blue}{\left(v + -1\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right), \left(\color{blue}{v} + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 82.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right)\right) \]

   9. Applied egg-rr 82.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 2.7 \cdot 10^{+143}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(w \cdot \frac{\left(0.375 + v \cdot -0.25\right) \cdot r}{v + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 7.4 \cdot 10^{+152}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7.4e-5)
   (+ (/ 2.0 (* r r)) -1.5)
   (if (<= r 7.4e+152)
     (* (* r r) (+ (* -0.375 (* w w)) (/ -1.5 (* r r))))
     (* (* r w) (* r (* -0.25 w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.4e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 7.4e+152) {
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	} else {
		tmp = (r * w) * (r * (-0.25 * w));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 7.4d-5) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else if (r <= 7.4d+152) then
        tmp = (r * r) * (((-0.375d0) * (w * w)) + ((-1.5d0) / (r * r)))
    else
        tmp = (r * w) * (r * ((-0.25d0) * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.4e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 7.4e+152) {
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	} else {
		tmp = (r * w) * (r * (-0.25 * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7.4e-5:
		tmp = (2.0 / (r * r)) + -1.5
	elif r <= 7.4e+152:
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)))
	else:
		tmp = (r * w) * (r * (-0.25 * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7.4e-5)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	elseif (r <= 7.4e+152)
		tmp = Float64(Float64(r * r) * Float64(Float64(-0.375 * Float64(w * w)) + Float64(-1.5 / Float64(r * r))));
	else
		tmp = Float64(Float64(r * w) * Float64(r * Float64(-0.25 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7.4e-5)
		tmp = (2.0 / (r * r)) + -1.5;
	elseif (r <= 7.4e+152)
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	else
		tmp = (r * w) * (r * (-0.25 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7.4e-5], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[r, 7.4e+152], N[(N[(r * r), $MachinePrecision] * N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(r * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.39999999999999962e-5

   1. Initial program 84.8%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 86.1%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right) \]

      8. *-lowering-*.f64 64.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right) \]

   7. Simplified 64.8%

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

   if 7.39999999999999962e-5 < r < 7.39999999999999992e152

   1. Initial program 89.6%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in v around 0

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

      1. associate--l+ N/A

         \[\leadsto \frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right), \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \left(2 \cdot \color{blue}{\frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right)\right) \]

      18. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 91.0%

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

   8. Taylor expanded in r around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot {w}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \left({w}^{2}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \left(w \cdot w\right)\right), \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{\frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{\frac{1}{{r}^{2}}}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2} \cdot 1}{{r}^{2}}\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2}}{{r}^{2}}\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{\mathsf{neg}\left(\frac{3}{2}\right)}{\color{blue}{{r}^{2}}}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{\frac{-3}{2}}{{\color{blue}{r}}^{2}}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \color{blue}{\left({r}^{2}\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{r}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, w\right)\right), \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right)\right)\right) \]

   10. Simplified 91.0%

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

   if 7.39999999999999992e152 < r

   1. Initial program 90.4%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 93.2%

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

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right) \]

      15. +-lowering-+.f64 77.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 77.6%

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

   8. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-1}{4}} \]

      2. associate-*l* N/A

         \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right) \]

      4. associate-*l* N/A

         \[\leadsto r \cdot \color{blue}{\left(r \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right)\right) \]

      9. *-lowering-*.f64 77.8%

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right)\right) \]

   10. Simplified 77.8%

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

       1. *-commutative N/A

          \[\leadsto r \cdot \left(\left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right) \cdot \color{blue}{r}\right) \]

       2. associate-*l* N/A

          \[\leadsto r \cdot \left(\left(w \cdot \left(w \cdot \frac{-1}{4}\right)\right) \cdot r\right) \]

       3. associate-*l* N/A

          \[\leadsto r \cdot \left(w \cdot \color{blue}{\left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right)}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot w\right), \color{blue}{\left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot r\right), \left(\color{blue}{\left(w \cdot \frac{-1}{4}\right)} \cdot r\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, r\right), \left(\color{blue}{\left(w \cdot \frac{-1}{4}\right)} \cdot r\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, r\right), \left(r \cdot \color{blue}{\left(w \cdot \frac{-1}{4}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, r\right), \mathsf{*.f64}\left(r, \color{blue}{\left(w \cdot \frac{-1}{4}\right)}\right)\right) \]

       10. *-lowering-*.f64 82.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, r\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

   12. Applied egg-rr 82.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 7.4 \cdot 10^{+152}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. sub-neg N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

   3. associate--l+ N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

3. Simplified 88.5%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\frac{r \cdot \left(w \cdot w\right)}{v + -1} \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\frac{\left(r \cdot w\right) \cdot w}{v + -1} \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(r \cdot w\right) \cdot \frac{w}{v + -1}\right) \cdot r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{r}, r\right)\right), \frac{-3}{2}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(r \cdot w\right) \cdot \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\frac{w}{v + -1} \cdot r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{r}, r\right)\right), \frac{-3}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\left(\frac{w}{v + -1}\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right), \frac{-3}{2}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \left(v + -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   10. +-lowering-+.f64 99.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(w, \mathsf{+.f64}\left(v, -1\right)\right), r\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 7: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.25 \cdot 10^{+222}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + w \cdot \left(-0.375 \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.25e+222)
   (+ (+ (/ 2.0 (* r r)) -1.5) (* w (* -0.375 (* r (* r w)))))
   (* (* r w) (* (* r w) (/ (+ 0.375 (* v -0.25)) (+ v -1.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.25e+222) {
		tmp = ((2.0 / (r * r)) + -1.5) + (w * (-0.375 * (r * (r * w))));
	} else {
		tmp = (r * w) * ((r * w) * ((0.375 + (v * -0.25)) / (v + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.25d+222) then
        tmp = ((2.0d0 / (r * r)) + (-1.5d0)) + (w * ((-0.375d0) * (r * (r * w))))
    else
        tmp = (r * w) * ((r * w) * ((0.375d0 + (v * (-0.25d0))) / (v + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.25e+222) {
		tmp = ((2.0 / (r * r)) + -1.5) + (w * (-0.375 * (r * (r * w))));
	} else {
		tmp = (r * w) * ((r * w) * ((0.375 + (v * -0.25)) / (v + -1.0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.25e+222:
		tmp = ((2.0 / (r * r)) + -1.5) + (w * (-0.375 * (r * (r * w))))
	else:
		tmp = (r * w) * ((r * w) * ((0.375 + (v * -0.25)) / (v + -1.0)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.25e+222)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + -1.5) + Float64(w * Float64(-0.375 * Float64(r * Float64(r * w)))));
	else
		tmp = Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(v + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.25e+222)
		tmp = ((2.0 / (r * r)) + -1.5) + (w * (-0.375 * (r * (r * w))));
	else
		tmp = (r * w) * ((r * w) * ((0.375 + (v * -0.25)) / (v + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.25e+222], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] + N[(w * N[(-0.375 * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.25000000000000006e222

   1. Initial program 85.8%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 88.1%

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

   5. Taylor expanded in v around 0

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

      1. associate--l+ N/A

         \[\leadsto \frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right), \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \left(2 \cdot \color{blue}{\frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right)\right) \]

      18. *-lowering-*.f64 80.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 80.1%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(\frac{-3}{8} \cdot w\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8}\right) \cdot w\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8}\right), w\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-3}{8} \cdot \left(r \cdot \left(r \cdot w\right)\right)\right), w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \left(r \cdot \left(r \cdot w\right)\right)\right), w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(r, \left(r \cdot w\right)\right)\right), w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 93.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right)\right), w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   9. Applied egg-rr 93.3%

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

   if 1.25000000000000006e222 < r

   1. Initial program 87.9%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 92.7%

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

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right) \]

      15. +-lowering-+.f64 77.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 77.7%

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

      1. associate-/l* N/A

         \[\leadsto \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \color{blue}{\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{r}\right) \]

      4. associate-*r* N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \color{blue}{w}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot w\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1} \cdot \left(r \cdot w\right)\right), \color{blue}{\left(r \cdot w\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right), \left(r \cdot w\right)\right), \left(\color{blue}{r} \cdot w\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(v + -1\right)\right), \left(r \cdot w\right)\right), \left(r \cdot w\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(v + -1\right)\right), \left(r \cdot w\right)\right), \left(r \cdot w\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(v + -1\right)\right), \left(r \cdot w\right)\right), \left(r \cdot w\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \left(r \cdot w\right)\right), \left(r \cdot w\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \left(r \cdot w\right)\right) \]

      15. *-lowering-*.f64 89.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right) \]

   9. Applied egg-rr 89.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.25 \cdot 10^{+222}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + w \cdot \left(-0.375 \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. sub-neg N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

   3. associate--l+ N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

3. Simplified 88.5%

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

5. Taylor expanded in v around 0

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

   1. associate--l+ N/A

      \[\leadsto \frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right), \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \left(2 \cdot \color{blue}{\frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right)\right) \]

   18. *-lowering-*.f64 79.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

7. Simplified 79.3%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(r \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

   10. *-lowering-*.f64 93.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right), \frac{-3}{2}\right)\right) \]

9. Applied egg-rr 93.8%

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

10. Final simplification 93.8%

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

Alternative 9: 66.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7e+81) (+ (/ 2.0 (* r r)) -1.5) (* (* r w) (* r (* -0.25 w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7e+81) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * w) * (r * (-0.25 * w));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 7d+81) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (r * w) * (r * ((-0.25d0) * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7e+81) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * w) * (r * (-0.25 * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7e+81:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (r * w) * (r * (-0.25 * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7e+81)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(r * w) * Float64(r * Float64(-0.25 * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7e+81)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (r * w) * (r * (-0.25 * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7e+81], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(r * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 7.0000000000000001e81

   1. Initial program 85.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 87.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right) \]

      8. *-lowering-*.f64 63.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right) \]

   7. Simplified 63.8%

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

   if 7.0000000000000001e81 < r

   1. Initial program 89.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 95.2%

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

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right) \]

      15. +-lowering-+.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 76.2%

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

   8. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-1}{4}} \]

      2. associate-*l* N/A

         \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right) \]

      4. associate-*l* N/A

         \[\leadsto r \cdot \color{blue}{\left(r \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right)\right) \]

      9. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right)\right) \]

   10. Simplified 78.5%

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

       1. *-commutative N/A

          \[\leadsto r \cdot \left(\left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right) \cdot \color{blue}{r}\right) \]

       2. associate-*l* N/A

          \[\leadsto r \cdot \left(\left(w \cdot \left(w \cdot \frac{-1}{4}\right)\right) \cdot r\right) \]

       3. associate-*l* N/A

          \[\leadsto r \cdot \left(w \cdot \color{blue}{\left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right)}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot w\right), \color{blue}{\left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(w \cdot r\right), \left(\color{blue}{\left(w \cdot \frac{-1}{4}\right)} \cdot r\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, r\right), \left(\color{blue}{\left(w \cdot \frac{-1}{4}\right)} \cdot r\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, r\right), \left(r \cdot \color{blue}{\left(w \cdot \frac{-1}{4}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, r\right), \mathsf{*.f64}\left(r, \color{blue}{\left(w \cdot \frac{-1}{4}\right)}\right)\right) \]

       10. *-lowering-*.f64 81.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, r\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

   12. Applied egg-rr 81.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.4e+81) (+ (/ 2.0 (* r r)) -1.5) (* (* r w) (* r (* w -0.375)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.4e+81) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.4d+81) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (r * w) * (r * (w * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.4e+81) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.4e+81:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (r * w) * (r * (w * -0.375))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.4e+81)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(r * w) * Float64(r * Float64(w * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.4e+81)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (r * w) * (r * (w * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.4e+81], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.3999999999999999e81

   1. Initial program 85.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 87.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right) \]

      8. *-lowering-*.f64 63.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right) \]

   7. Simplified 63.8%

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

   if 5.3999999999999999e81 < r

   1. Initial program 89.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 95.2%

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

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right) \]

      15. +-lowering-+.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 76.2%

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

      1. associate-/l* N/A

         \[\leadsto \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \color{blue}{\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \frac{\color{blue}{\frac{3}{8} + v \cdot \frac{-1}{4}}}{v + -1} \]

      3. associate-*l* N/A

         \[\leadsto \left(r \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\color{blue}{r} \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(w \cdot \left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(r \cdot w\right), \color{blue}{\left(w \cdot \left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\color{blue}{w} \cdot \left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \color{blue}{\left(r \cdot \frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \left(\frac{r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v + -1}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right), \color{blue}{\left(v + -1\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right), \left(\color{blue}{v} + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right)\right)\right) \]

      14. +-lowering-+.f64 83.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right)\right) \]

   9. Applied egg-rr 83.5%

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

   10. Taylor expanded in v around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \color{blue}{\left(\frac{-3}{8} \cdot \left(r \cdot w\right)\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\left(\frac{-3}{8} \cdot r\right) \cdot \color{blue}{w}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\left(r \cdot \frac{-3}{8}\right) \cdot w\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(r \cdot \color{blue}{\left(\frac{-3}{8} \cdot w\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{-3}{8} \cdot w\right)}\right)\right) \]

       5. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\frac{-3}{8}, \color{blue}{w}\right)\right)\right) \]

   12. Simplified 73.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 8.2e+81) (+ (/ 2.0 (* r r)) -1.5) (* r (* r (* -0.25 (* w w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 8.2e+81) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = r * (r * (-0.25 * (w * w)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 8.2d+81) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = r * (r * ((-0.25d0) * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 8.2e+81) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = r * (r * (-0.25 * (w * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 8.2e+81:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = r * (r * (-0.25 * (w * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 8.2e+81)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(r * Float64(r * Float64(-0.25 * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 8.2e+81)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = r * (r * (-0.25 * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 8.2e+81], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(r * N[(r * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.20000000000000024e81

   1. Initial program 85.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 87.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right) \]

      8. *-lowering-*.f64 63.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right) \]

   7. Simplified 63.8%

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

   if 8.20000000000000024e81 < r

   1. Initial program 89.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 95.2%

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

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right), \left(v - 1\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v - 1\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(v + -1\right)\right) \]

      15. +-lowering-+.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 76.2%

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

   8. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-1}{4}} \]

      2. associate-*l* N/A

         \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)} \]

      3. unpow2 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right) \]

      4. associate-*l* N/A

         \[\leadsto r \cdot \color{blue}{\left(r \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right)\right) \]

      9. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right)\right) \]

   10. Simplified 78.5%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(r \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.5e+81) (+ (/ 2.0 (* r r)) -1.5) (* r (* r (* -0.375 (* w w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.5e+81) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = r * (r * (-0.375 * (w * w)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.5d+81) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = r * (r * ((-0.375d0) * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.5e+81) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = r * (r * (-0.375 * (w * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.5e+81:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = r * (r * (-0.375 * (w * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.5e+81)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(r * Float64(r * Float64(-0.375 * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.5e+81)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = r * (r * (-0.375 * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.5e+81], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(r * N[(r * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.5000000000000003e81

   1. Initial program 85.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 87.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right) \]

      8. *-lowering-*.f64 63.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right) \]

   7. Simplified 63.8%

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

   if 5.5000000000000003e81 < r

   1. Initial program 89.2%

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

      1. sub-neg N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

      3. associate--l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

   3. Simplified 95.2%

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

   5. Taylor expanded in v around 0

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

      1. associate--l+ N/A

         \[\leadsto \frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right), \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2 \cdot \frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \left(\color{blue}{2} \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \left(2 \cdot \color{blue}{\frac{1}{{r}^{2}}} - \frac{3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{\color{blue}{1}}{{r}^{2}} - \frac{3}{2}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right)\right) \]

      18. *-lowering-*.f64 73.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 73.3%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \frac{-3}{8}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(r \cdot \frac{-3}{8}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)}, \frac{-3}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \left(r \cdot \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(r, r\right)}\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 87.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \frac{-3}{8}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right), \frac{-3}{2}\right)\right) \]

   9. Applied egg-rr 87.7%

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

   10. Taylor expanded in r around inf

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

       1. *-commutative N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-3}{8}} \]

       2. associate-*l* N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \left(\frac{-3}{8} \cdot \color{blue}{{w}^{2}}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right) \]

       5. associate-*l* N/A

          \[\leadsto r \cdot \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\frac{-3}{8}, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\frac{-3}{8}, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 73.0%

           \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   12. Simplified 73.0%

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

Alternative 13: 57.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. sub-neg N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

   3. associate--l+ N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

3. Simplified 88.5%

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

5. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\frac{-3}{2}}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \frac{-3}{2}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \frac{-3}{2}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \frac{-3}{2}\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \frac{-3}{2}\right) \]

   8. *-lowering-*.f64 55.2%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \frac{-3}{2}\right) \]

7. Simplified 55.2%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + -1.5} \]
8. Add Preprocessing

Alternative 14: 44.1% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. sub-neg N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(3 + \frac{2}{r \cdot r}\right)\right) - \frac{9}{2} \]

   3. associate--l+ N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{9}{2}\right)}\right) \]

3. Simplified 88.5%

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

5. Taylor expanded in r around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

   3. *-lowering-*.f64 42.7%

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

7. Simplified 42.7%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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))