Rosa's TurbineBenchmark

Percentage Accurate: 85.3% → 97.7%

Time: 15.3s

Alternatives: 16

Speedup: 1.6×

• 52.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 3 + t\_0\\ t_2 := 0.125 \cdot \left(3 - 2 \cdot v\right)\\ \mathbf{if}\;t\_1 + \frac{t\_2 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -\infty:\\ \;\;\;\;\left(t\_0 - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.25\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \frac{t\_2 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (+ 3.0 t_0))
        (t_2 (* 0.125 (- 3.0 (* 2.0 v)))))
   (if (<= (+ t_1 (/ (* t_2 (* r (* r (* w w)))) (+ v -1.0))) (- INFINITY))
     (- (- t_0 (* (* r (* r w)) (* w 0.25))) 4.5)
     (- (+ t_1 (/ (* t_2 (* (* r w) (* r w))) (+ v -1.0))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double t_2 = 0.125 * (3.0 - (2.0 * v));
	double tmp;
	if ((t_1 + ((t_2 * (r * (r * (w * w)))) / (v + -1.0))) <= -((double) INFINITY)) {
		tmp = (t_0 - ((r * (r * w)) * (w * 0.25))) - 4.5;
	} else {
		tmp = (t_1 + ((t_2 * ((r * w) * (r * w))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double t_2 = 0.125 * (3.0 - (2.0 * v));
	double tmp;
	if ((t_1 + ((t_2 * (r * (r * (w * w)))) / (v + -1.0))) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 - ((r * (r * w)) * (w * 0.25))) - 4.5;
	} else {
		tmp = (t_1 + ((t_2 * ((r * w) * (r * w))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 3.0 + t_0
	t_2 = 0.125 * (3.0 - (2.0 * v))
	tmp = 0
	if (t_1 + ((t_2 * (r * (r * (w * w)))) / (v + -1.0))) <= -math.inf:
		tmp = (t_0 - ((r * (r * w)) * (w * 0.25))) - 4.5
	else:
		tmp = (t_1 + ((t_2 * ((r * w) * (r * w))) / (v + -1.0))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(3.0 + t_0)
	t_2 = Float64(0.125 * Float64(3.0 - Float64(2.0 * v)))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(t_2 * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(Float64(r * Float64(r * w)) * Float64(w * 0.25))) - 4.5);
	else
		tmp = Float64(Float64(t_1 + Float64(Float64(t_2 * Float64(Float64(r * w) * Float64(r * w))) / Float64(v + -1.0))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = 3.0 + t_0;
	t_2 = 0.125 * (3.0 - (2.0 * v));
	tmp = 0.0;
	if ((t_1 + ((t_2 * (r * (r * (w * w)))) / (v + -1.0))) <= -Inf)
		tmp = (t_0 - ((r * (r * w)) * (w * 0.25))) - 4.5;
	else
		tmp = (t_1 + ((t_2 * ((r * w) * (r * w))) / (v + -1.0))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(t$95$2 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(t$95$0 - N[(N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 + N[(N[(t$95$2 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -inf.0

   1. Initial program 77.1%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 85.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 85.4%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \frac{1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(r \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot w\right), \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot \left(r \cdot w\right)\right), \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot w\right)\right), \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right), \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{*.f64}\left(w, \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in r around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{2}{{r}^{2}}\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{*.f64}\left(w, \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{*.f64}\left(w, \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{*.f64}\left(w, \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      3. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{*.f64}\left(w, \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   10. Simplified 98.8%

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

   if -inf.0 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 85.6%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(w \cdot r\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(r \cdot w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      7. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -\infty:\\ \;\;\;\;\left(\frac{2}{r \cdot r} - \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot 0.25\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), \frac{r \cdot w}{1 - v}, 4.5\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ 3.0 (/ 2.0 (* r r)))
  (fma (* (* r w) (fma v -0.25 0.375)) (/ (* r w) (- 1.0 v)) 4.5)))

C

double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) - fma(((r * w) * fma(v, -0.25, 0.375)), ((r * w) / (1.0 - v)), 4.5);
}

Julia

function code(v, w, r)
	return Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - fma(Float64(Float64(r * w) * fma(v, -0.25, 0.375)), Float64(Float64(r * w) / Float64(1.0 - v)), 4.5))
end

Wolfram

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

Derivation

1. Initial program 83.0%

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

   1. associate--l- N/A

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

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

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

   7. associate-/l* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 80.8%

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

   1. associate-*r/ N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. swap-sqr N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. cancel-sign-sub-inv N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

6. Applied egg-rr 97.1%

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

Alternative 3: 91.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 460000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;r \leq 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(\left(w \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot \left(0 - w\right), \frac{r \cdot r}{1 - v}, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 460000.0)
   (+ -1.5 (fma (* w (* (* r r) -0.25)) w (/ 2.0 (* r r))))
   (if (<= r 1e+147)
     (fma (* (* w (fma -0.25 v 0.375)) (- 0.0 w)) (/ (* r r) (- 1.0 v)) -1.5)
     (- (- 3.0 (* (* r w) (* r (* w 0.25)))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 460000.0) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, (2.0 / (r * r)));
	} else if (r <= 1e+147) {
		tmp = fma(((w * fma(-0.25, v, 0.375)) * (0.0 - w)), ((r * r) / (1.0 - v)), -1.5);
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 460000.0)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, Float64(2.0 / Float64(r * r))));
	elseif (r <= 1e+147)
		tmp = fma(Float64(Float64(w * fma(-0.25, v, 0.375)) * Float64(0.0 - w)), Float64(Float64(r * r) / Float64(1.0 - v)), -1.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.25)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 460000.0], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1e+147], N[(N[(N[(w * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] * N[(0.0 - w), $MachinePrecision]), $MachinePrecision] * N[(N[(r * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.6e5

   1. Initial program 81.6%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2} \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 88.9%

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

   if 4.6e5 < r < 9.9999999999999998e146

   1. Initial program 92.9%

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

      1. associate--l- N/A

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

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

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 85.5%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. swap-sqr N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in r around inf

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

      1. distribute-lft-in N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

         \[\leadsto -1 \cdot \left(\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}\right) + -1 \cdot \frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{1 - v} \]

      5. associate-*l* N/A

         \[\leadsto -1 \cdot \left(\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)\right) + -1 \cdot \frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{1 - v} \]

      6. lft-mult-inverse N/A

         \[\leadsto -1 \cdot \left(\frac{3}{2} \cdot 1\right) + -1 \cdot \frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{1 - v} \]

      7. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{3}{2} + -1 \cdot \frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{1 - v} \]

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

   9. Simplified 96.3%

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

   if 9.9999999999999998e146 < r

   1. Initial program 83.4%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 61.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 61.4%

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

   6. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 61.4%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \frac{1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(r \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(w \cdot \frac{1}{4}\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right), \frac{9}{2}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(w \cdot \frac{1}{4}\right) \cdot r\right) \cdot \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\left(\left(w \cdot \frac{1}{4}\right) \cdot r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(w \cdot \frac{1}{4}\right), r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{4}\right), r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

      9. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{4}\right), r\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \frac{9}{2}\right) \]

   10. Applied egg-rr 85.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 460000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;r \leq 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(\left(w \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot \left(0 - w\right), \frac{r \cdot r}{1 - v}, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 7.6:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, t\_0\right)\\ \mathbf{elif}\;r \leq 1.65 \cdot 10^{+81}:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 7.6)
     (+ -1.5 (fma (* w (* (* r r) -0.25)) w t_0))
     (if (<= r 1.65e+81)
       (+ t_0 (fma (* r (* r (* w w))) -0.375 -1.5))
       (- (- 3.0 (* (* r w) (* r (* w 0.25)))) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 7.6) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, t_0);
	} else if (r <= 1.65e+81) {
		tmp = t_0 + fma((r * (r * (w * w))), -0.375, -1.5);
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 7.6)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, t_0));
	elseif (r <= 1.65e+81)
		tmp = Float64(t_0 + fma(Float64(r * Float64(r * Float64(w * w))), -0.375, -1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.25)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 7.6], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.65e+81], N[(t$95$0 + N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.5999999999999996

   1. Initial program 81.5%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2} \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 88.9%

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

   if 7.5999999999999996 < r < 1.65e81

   1. Initial program 88.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(w \cdot r\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(r \cdot w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      7. *-lowering-*.f64 88.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 88.9%

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

   5. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 86.0%

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

   7. Simplified 86.0%

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

   if 1.65e81 < r

   1. Initial program 88.1%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 70.9%

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

   6. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 70.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \frac{1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(r \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(w \cdot \frac{1}{4}\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right), \frac{9}{2}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(w \cdot \frac{1}{4}\right) \cdot r\right) \cdot \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\left(\left(w \cdot \frac{1}{4}\right) \cdot r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(w \cdot \frac{1}{4}\right), r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{4}\right), r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

      9. *-lowering-*.f64 88.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{4}\right), r\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \frac{9}{2}\right) \]

   10. Applied egg-rr 88.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.6:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;r \leq 1.65 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1100:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), \frac{r \cdot w}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1100.0)
   (+ -1.5 (fma (* w (* (* r r) -0.25)) w (/ 2.0 (* r r))))
   (- 3.0 (fma (* (* r w) (fma v -0.25 0.375)) (/ (* r w) (- 1.0 v)) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1100.0) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, (2.0 / (r * r)));
	} else {
		tmp = 3.0 - fma(((r * w) * fma(v, -0.25, 0.375)), ((r * w) / (1.0 - v)), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1100.0)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, Float64(2.0 / Float64(r * r))));
	else
		tmp = Float64(3.0 - fma(Float64(Float64(r * w) * fma(v, -0.25, 0.375)), Float64(Float64(r * w) / Float64(1.0 - v)), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1100.0], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(N[(r * w), $MachinePrecision] * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1100

   1. Initial program 81.6%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2} \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 88.9%

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

   if 1100 < r

   1. Initial program 88.1%

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

      1. associate--l- N/A

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

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

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 80.1%

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

      1. associate-*r/ N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. swap-sqr N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{fma.f64}\left(v, \frac{-1}{4}, \frac{3}{8}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \frac{9}{2}\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 99.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1100:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), \frac{r \cdot w}{1 - v}, 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 0.75:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot -0.375\right), r \cdot w, t\_0 + -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 0.75)
     (+ -1.5 (fma (* w (* (* r r) -0.25)) w t_0))
     (fma (* r (* w -0.375)) (* r w) (+ t_0 -1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 0.75) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, t_0);
	} else {
		tmp = fma((r * (w * -0.375)), (r * w), (t_0 + -1.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 0.75)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, t_0));
	else
		tmp = fma(Float64(r * Float64(w * -0.375)), Float64(r * w), Float64(t_0 + -1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 0.75], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision] + N[(t$95$0 + -1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 0.75

   1. Initial program 81.5%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2} \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 88.9%

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

   if 0.75 < r

   1. Initial program 88.3%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \left(\frac{1}{r}\right), \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{\_.f64}\left(3, \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \color{blue}{\left(3 - \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(3 + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right), \frac{9}{2}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\left({w}^{2}\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\left(w \cdot w\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\mathsf{neg}\left({r}^{2} \cdot \frac{3}{8}\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left({r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left({r}^{2} \cdot \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

      19. *-lowering-*.f64 70.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

   7. Simplified 70.7%

      \[\leadsto \mathsf{fma}\left(\frac{2}{r}, \frac{1}{r}, \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, 3\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. frac-times N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. associate--l+ N/A

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

      5. metadata-eval N/A

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

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

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

   9. Applied egg-rr 79.8%

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

       1. associate-+l+ N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       13. *-lowering-*.f64 86.0%

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

   11. Applied egg-rr 86.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.75:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot -0.375\right), r \cdot w, \frac{2}{r \cdot r} + -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 0.41:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(r, w \cdot \left(r \cdot \left(w \cdot -0.375\right)\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 0.41)
     (+ -1.5 (fma (* w (* (* r r) -0.25)) w t_0))
     (+ -1.5 (fma r (* w (* r (* w -0.375))) t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 0.41) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, t_0);
	} else {
		tmp = -1.5 + fma(r, (w * (r * (w * -0.375))), t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 0.41)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, t_0));
	else
		tmp = Float64(-1.5 + fma(r, Float64(w * Float64(r * Float64(w * -0.375))), t_0));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 0.41], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r * N[(w * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 0.409999999999999976

   1. Initial program 81.5%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2} \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 88.9%

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

   if 0.409999999999999976 < r

   1. Initial program 88.3%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \left(\frac{1}{r}\right), \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{\_.f64}\left(3, \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \color{blue}{\left(3 - \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(3 + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right), \frac{9}{2}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\left({w}^{2}\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\left(w \cdot w\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\mathsf{neg}\left({r}^{2} \cdot \frac{3}{8}\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left({r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left({r}^{2} \cdot \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

      19. *-lowering-*.f64 70.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

   7. Simplified 70.7%

      \[\leadsto \mathsf{fma}\left(\frac{2}{r}, \frac{1}{r}, \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, 3\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. frac-times N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. associate--l+ N/A

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

      5. metadata-eval N/A

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

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

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

   9. Applied egg-rr 79.8%

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

       1. associate-*r* N/A

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

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

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

       3. associate-*l* N/A

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

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

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

       5. *-lowering-*.f64 86.0%

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

   11. Applied egg-rr 86.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.41:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(r, w \cdot \left(r \cdot \left(w \cdot -0.375\right)\right), \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 460000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;r \leq 3.7 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot r, \left(w \cdot w\right) \cdot -0.375, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 460000.0)
   (+ -1.5 (fma (* w (* (* r r) -0.25)) w (/ 2.0 (* r r))))
   (if (<= r 3.7e+81)
     (fma (* r r) (* (* w w) -0.375) -1.5)
     (- (- 3.0 (* (* r w) (* r (* w 0.25)))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 460000.0) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, (2.0 / (r * r)));
	} else if (r <= 3.7e+81) {
		tmp = fma((r * r), ((w * w) * -0.375), -1.5);
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 460000.0)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, Float64(2.0 / Float64(r * r))));
	elseif (r <= 3.7e+81)
		tmp = fma(Float64(r * r), Float64(Float64(w * w) * -0.375), -1.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.25)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 460000.0], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 3.7e+81], N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.6e5

   1. Initial program 81.6%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + \color{blue}{2} \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 88.9%

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

   if 4.6e5 < r < 3.7000000000000001e81

   1. Initial program 88.2%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \left(\frac{1}{r}\right), \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{\_.f64}\left(3, \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \color{blue}{\left(3 - \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(3 + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right), \frac{9}{2}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left(\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left(\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right) + 3\right)\right), \frac{9}{2}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + 3\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\left({w}^{2}\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\left(w \cdot w\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{-3}{8} \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right), 3\right)\right), \frac{9}{2}\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\mathsf{neg}\left({r}^{2} \cdot \frac{3}{8}\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left({r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right), 3\right)\right), \frac{9}{2}\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \left({r}^{2} \cdot \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

      19. *-lowering-*.f64 85.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(2, r\right), \mathsf{/.f64}\left(1, r\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \frac{-3}{8}\right), 3\right)\right), \frac{9}{2}\right) \]

   7. Simplified 85.0%

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

   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. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 85.0%

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

   10. Simplified 85.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot r, \left(w \cdot w\right) \cdot -0.375, -1.5\right)} \]

   if 3.7000000000000001e81 < r

   1. Initial program 88.1%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 70.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 70.9%

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

   6. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 70.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \frac{1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(r \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(w \cdot \frac{1}{4}\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right), \frac{9}{2}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(w \cdot \frac{1}{4}\right) \cdot r\right) \cdot \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\left(\left(w \cdot \frac{1}{4}\right) \cdot r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(w \cdot \frac{1}{4}\right), r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{4}\right), r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

      9. *-lowering-*.f64 88.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{4}\right), r\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \frac{9}{2}\right) \]

   10. Applied egg-rr 88.2%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 460000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;r \leq 3.7 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot r, \left(w \cdot w\right) \cdot -0.375, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 8.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 8.8e-18)
   (/ (/ 2.0 r) r)
   (- (- 3.0 (* (* r w) (* r (* w 0.25)))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 8.8e-18) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 8.8d-18) then
        tmp = (2.0d0 / r) / r
    else
        tmp = (3.0d0 - ((r * w) * (r * (w * 0.25d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 8.8e-18) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 8.8e-18:
		tmp = (2.0 / r) / r
	else:
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 8.8e-18)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(r * Float64(w * 0.25)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 8.8e-18)
		tmp = (2.0 / r) / r;
	else
		tmp = (3.0 - ((r * w) * (r * (w * 0.25)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 8.8e-18], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.7999999999999994e-18

   1. Initial program 81.8%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. 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 61.5%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 61.5%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 61.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   7. Applied egg-rr 61.5%

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

   if 8.7999999999999994e-18 < r

   1. Initial program 87.1%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 74.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 74.9%

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

   6. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 72.0%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \frac{1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(r \cdot r\right) \cdot w\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(w \cdot \frac{1}{4}\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right), \frac{9}{2}\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(w \cdot \frac{1}{4}\right) \cdot r\right) \cdot \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\left(\left(w \cdot \frac{1}{4}\right) \cdot r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(w \cdot \frac{1}{4}\right), r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{4}\right), r\right), \left(r \cdot w\right)\right)\right), \frac{9}{2}\right) \]

      9. *-lowering-*.f64 83.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \frac{1}{4}\right), r\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \frac{9}{2}\right) \]

   10. Applied egg-rr 83.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.95 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \left(r \cdot \left(r \cdot w\right)\right), -0.25, 3\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.95e-23)
   (/ (/ 2.0 r) r)
   (- (fma (* w (* r (* r w))) -0.25 3.0) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.95e-23) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = fma((w * (r * (r * w))), -0.25, 3.0) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.95e-23)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(fma(Float64(w * Float64(r * Float64(r * w))), -0.25, 3.0) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.95e-23], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.95e-23

   1. Initial program 82.1%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. 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 61.6%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 61.6%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 61.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   7. Applied egg-rr 61.6%

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

   if 1.95e-23 < r

   1. Initial program 85.9%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 75.7%

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

   6. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 71.3%

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(3 - \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \frac{1}{4}\right)\right), \color{blue}{\frac{9}{2}}\right) \]

   10. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \left(r \cdot \left(r \cdot w\right)\right), -0.25, 3\right) - 4.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.7 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(r \cdot \left(w \cdot w\right)\right), -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.7e-23) (/ (/ 2.0 r) r) (fma -0.25 (* r (* r (* w w))) -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.7e-23) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = fma(-0.25, (r * (r * (w * w))), -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.7e-23)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = fma(-0.25, Float64(r * Float64(r * Float64(w * w))), -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.7e-23], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(-0.25 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.7e-23

   1. Initial program 82.1%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. 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 61.6%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 61.6%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 61.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   7. Applied egg-rr 61.6%

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

   if 1.7e-23 < r

   1. Initial program 85.9%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 75.7%

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

   6. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({r}^{2} \cdot \left(\frac{1}{4} \cdot {w}^{2} + \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\frac{1}{4} \cdot {w}^{2}\right) \cdot {r}^{2} + \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}\right)\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot {w}^{2}\right) \cdot {r}^{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({w}^{2} \cdot {r}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \cdot {r}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\left(\frac{3}{2} \cdot \color{blue}{\frac{1}{{r}^{2}}}\right) \cdot {r}^{2}\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \cdot {r}^{2}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) - \color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) - \frac{3}{2} \cdot \color{blue}{\left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)} \]

      10. lft-mult-inverse N/A

          \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) - \frac{3}{2} \cdot 1 \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) - \frac{3}{2} \]

      12. sub-neg N/A

          \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{-3}{2} \]

      14. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, r \cdot \left(r \cdot \left(w \cdot w\right)\right), -1.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(r \cdot \left(w \cdot w\right)\right), -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.3e-23) (/ 2.0 (* r r)) (fma -0.25 (* r (* r (* w w))) -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.3e-23) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = fma(-0.25, (r * (r * (w * w))), -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.3e-23)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = fma(-0.25, Float64(r * Float64(r * Float64(w * w))), -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.3e-23], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.3000000000000001e-23

   1. Initial program 82.1%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. 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 61.6%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 61.6%

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

   if 2.3000000000000001e-23 < r

   1. Initial program 85.9%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 75.7%

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

   6. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({r}^{2} \cdot \left(\frac{1}{4} \cdot {w}^{2} + \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{neg}\left(\left(\left(\frac{1}{4} \cdot {w}^{2}\right) \cdot {r}^{2} + \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}\right)\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{1}{4} \cdot {w}^{2}\right) \cdot {r}^{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({w}^{2} \cdot {r}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \cdot {r}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\left(\frac{3}{2} \cdot \color{blue}{\frac{1}{{r}^{2}}}\right) \cdot {r}^{2}\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \cdot {r}^{2}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) - \color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) - \frac{3}{2} \cdot \color{blue}{\left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)} \]

      10. lft-mult-inverse N/A

          \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) - \frac{3}{2} \cdot 1 \]

      11. metadata-eval N/A

          \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) - \frac{3}{2} \]

      12. sub-neg N/A

          \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{-3}{2} \]

      14. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, r \cdot \left(r \cdot \left(w \cdot w\right)\right), -1.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.15 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.15e-23) (/ 2.0 (* r r)) (* -0.25 (* r (* r (* w w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.15e-23) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -0.25 * (r * (r * (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 <= 2.15d-23) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = (-0.25d0) * (r * (r * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.15e-23) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -0.25 * (r * (r * (w * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.15e-23:
		tmp = 2.0 / (r * r)
	else:
		tmp = -0.25 * (r * (r * (w * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.15e-23)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = Float64(-0.25 * Float64(r * Float64(r * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.15e-23)
		tmp = 2.0 / (r * r);
	else
		tmp = -0.25 * (r * (r * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.15e-23], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.15000000000000001e-23

   1. Initial program 82.1%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. 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 61.6%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 61.6%

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

   if 2.15000000000000001e-23 < r

   1. Initial program 85.9%

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

   3. Taylor expanded in v around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{1}{4}\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left({r}^{2} \cdot \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{1}{4}\right)\right)\right), \frac{9}{2}\right) \]

   5. Simplified 75.7%

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

   6. Taylor expanded in r around inf

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 59.6%

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

   8. Simplified 59.6%

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

Alternative 14: 50.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.055:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.055) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.055) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.055d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.055) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.055:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.055)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.055)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.055], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 0.0550000000000000003

   1. Initial program 81.5%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. 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 61.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 61.2%

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

   if 0.0550000000000000003 < r

   1. Initial program 88.3%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. 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. +-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 27.0%

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

   5. Simplified 27.0%

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

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 25.7%

      \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.6% accurate, 3.7× 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 83.0%

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

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
4. 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. +-commutative N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 59.6%

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

5. Simplified 59.6%

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

6. Final simplification 59.6%

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

Alternative 16: 14.5% accurate, 73.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
4. 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. +-commutative N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 59.6%

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

5. Simplified 59.6%

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

6. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{-3}{2}} \]
7. Step-by-step derivation

8. Simplified 12.5%

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

Reproduce

herbie shell --seed 2024193 
(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))