Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 97.5%

Time: 15.0s

Alternatives: 18

Speedup: 1.7×

• 52.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% 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: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 + -1.5\\ \mathbf{if}\;v \leq -8 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), t\_1\right)\\ \mathbf{elif}\;v \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ t_0 -1.5)))
   (if (<= v -8e-49)
     (fma -0.25 (* r (* w (* r w))) t_1)
     (if (<= v 1.5e+15)
       (-
        (+
         (+ t_0 3.0)
         (/ (* (* 0.125 (- 3.0 (* v 2.0))) (* (* r w) (* r w))) (+ v -1.0)))
        4.5)
       (fma (* -0.25 w) (* r (* r w)) t_1)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 + -1.5;
	double tmp;
	if (v <= -8e-49) {
		tmp = fma(-0.25, (r * (w * (r * w))), t_1);
	} else if (v <= 1.5e+15) {
		tmp = ((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * ((r * w) * (r * w))) / (v + -1.0))) - 4.5;
	} else {
		tmp = fma((-0.25 * w), (r * (r * w)), t_1);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 + -1.5)
	tmp = 0.0
	if (v <= -8e-49)
		tmp = fma(-0.25, Float64(r * Float64(w * Float64(r * w))), t_1);
	elseif (v <= 1.5e+15)
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v * 2.0))) * Float64(Float64(r * w) * Float64(r * w))) / Float64(v + -1.0))) - 4.5);
	else
		tmp = fma(Float64(-0.25 * w), Float64(r * Float64(r * w)), t_1);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.5), $MachinePrecision]}, If[LessEqual[v, -8e-49], N[(-0.25 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[v, 1.5e+15], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-0.25 * w), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -7.99999999999999949e-49

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 91.6%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      17. *-lowering-*.f64 98.5

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

   7. Applied egg-rr 98.5%

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

   if -7.99999999999999949e-49 < v < 1.5e15

   1. Initial program 88.7%

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

      1. associate-*l* N/A

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

      2. unswap-sqr N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if 1.5e15 < v

   1. Initial program 74.3%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 87.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 96.5

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

   7. Applied egg-rr 96.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -8 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r} + -1.5\right)\\ \mathbf{elif}\;v \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\\ t_2 := \mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, 3\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq 3:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v))))
        (t_2 (fma (* r w) (* -0.25 (* r w)) -1.5)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -4e+70)
       (- (fma (* r (* w (* r w))) -0.375 3.0) 4.5)
       (if (<= t_1 3.0) t_2 (+ t_0 -1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double t_2 = fma((r * w), (-0.25 * (r * w)), -1.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -4e+70) {
		tmp = fma((r * (w * (r * w))), -0.375, 3.0) - 4.5;
	} else if (t_1 <= 3.0) {
		tmp = t_2;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v)))
	t_2 = fma(Float64(r * w), Float64(-0.25 * Float64(r * w)), -1.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -4e+70)
		tmp = Float64(fma(Float64(r * Float64(w * Float64(r * w))), -0.375, 3.0) - 4.5);
	elseif (t_1 <= 3.0)
		tmp = t_2;
	else
		tmp = 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]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -4e+70], N[(N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, 3.0], t$95$2, N[(t$95$0 + -1.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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 or -4.00000000000000029e70 < (-.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))) < 3

   1. Initial program 80.8%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 84.4%

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

   6. Taylor expanded in r around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 78.5

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

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot r, -0.25 \cdot \left(w \cdot w\right), -1.5\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 94.5

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

   10. Applied egg-rr 94.5%

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

   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))) < -4.00000000000000029e70

   1. Initial program 99.3%

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

      1. associate-*l* N/A

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

      2. unswap-sqr N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in v around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 81.6

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

   7. Simplified 81.6%

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

   8. Taylor expanded in r around inf

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

   10. Simplified 81.6%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 82.1

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

   12. Applied egg-rr 82.1%

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

   if 3 < (-.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 85.4%

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

   3. Taylor expanded in 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

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

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 99.9

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -4 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, 3\right) - 4.5\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq 3:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\\ t_2 := \mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(w, w \cdot -0.375, 0\right), 0\right), -1.5\right)\\ \mathbf{elif}\;t\_1 \leq 3:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v))))
        (t_2 (fma (* r w) (* -0.25 (* r w)) -1.5)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -4e+70)
       (fma r (fma r (fma w (* w -0.375) 0.0) 0.0) -1.5)
       (if (<= t_1 3.0) t_2 (+ t_0 -1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double t_2 = fma((r * w), (-0.25 * (r * w)), -1.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -4e+70) {
		tmp = fma(r, fma(r, fma(w, (w * -0.375), 0.0), 0.0), -1.5);
	} else if (t_1 <= 3.0) {
		tmp = t_2;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v)))
	t_2 = fma(Float64(r * w), Float64(-0.25 * Float64(r * w)), -1.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -4e+70)
		tmp = fma(r, fma(r, fma(w, Float64(w * -0.375), 0.0), 0.0), -1.5);
	elseif (t_1 <= 3.0)
		tmp = t_2;
	else
		tmp = 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]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -4e+70], N[(r * N[(r * N[(w * N[(w * -0.375), $MachinePrecision] + 0.0), $MachinePrecision] + 0.0), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[t$95$1, 3.0], t$95$2, N[(t$95$0 + -1.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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 or -4.00000000000000029e70 < (-.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))) < 3

   1. Initial program 80.8%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 84.4%

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

   6. Taylor expanded in r around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 78.5

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

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot r, -0.25 \cdot \left(w \cdot w\right), -1.5\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 94.5

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

   10. Applied egg-rr 94.5%

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

   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))) < -4.00000000000000029e70

   1. Initial program 99.3%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

   5. Simplified 76.8%

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

   6. Taylor expanded in r around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

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

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

   8. Simplified 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(w, w \cdot -0.375, 0\right), 0\right), -1.5\right)} \]

   if 3 < (-.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 85.4%

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

   3. Taylor expanded in 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

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

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 99.9

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -4 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(w, w \cdot -0.375, 0\right), 0\right), -1.5\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq 3:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\\ t_2 := \mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+70}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{elif}\;t\_1 \leq 3:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v))))
        (t_2 (fma (* r w) (* -0.25 (* r w)) -1.5)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -4e+70)
       (* r (* r (* (* w w) -0.375)))
       (if (<= t_1 3.0) t_2 (+ t_0 -1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double t_2 = fma((r * w), (-0.25 * (r * w)), -1.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -4e+70) {
		tmp = r * (r * ((w * w) * -0.375));
	} else if (t_1 <= 3.0) {
		tmp = t_2;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v)))
	t_2 = fma(Float64(r * w), Float64(-0.25 * Float64(r * w)), -1.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -4e+70)
		tmp = Float64(r * Float64(r * Float64(Float64(w * w) * -0.375)));
	elseif (t_1 <= 3.0)
		tmp = t_2;
	else
		tmp = 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]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -4e+70], N[(r * N[(r * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.0], t$95$2, N[(t$95$0 + -1.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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 or -4.00000000000000029e70 < (-.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))) < 3

   1. Initial program 80.8%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 84.4%

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

   6. Taylor expanded in r around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 78.5

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

   8. Simplified 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot r, -0.25 \cdot \left(w \cdot w\right), -1.5\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 94.5

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

   10. Applied egg-rr 94.5%

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

   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))) < -4.00000000000000029e70

   1. Initial program 99.3%

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

      1. associate-*l* N/A

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

      2. unswap-sqr N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in v around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 81.6

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

   7. Simplified 81.6%

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

   8. Taylor expanded in r around inf

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

   10. Simplified 81.6%

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

   11. Taylor expanded in r around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 81.5

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

   13. Simplified 81.5%

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

   if 3 < (-.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 85.4%

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

   3. Taylor expanded in 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

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

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 99.9

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -4 \cdot 10^{+70}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq 3:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\\ t_2 := \mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+70}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{elif}\;t\_1 \leq 3:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v))))
        (t_2 (fma (* -0.25 w) (* r (* r w)) -1.5)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -4e+70)
       (* r (* r (* (* w w) -0.375)))
       (if (<= t_1 3.0) t_2 (+ t_0 -1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double t_2 = fma((-0.25 * w), (r * (r * w)), -1.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -4e+70) {
		tmp = r * (r * ((w * w) * -0.375));
	} else if (t_1 <= 3.0) {
		tmp = t_2;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v)))
	t_2 = fma(Float64(-0.25 * w), Float64(r * Float64(r * w)), -1.5)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -4e+70)
		tmp = Float64(r * Float64(r * Float64(Float64(w * w) * -0.375)));
	elseif (t_1 <= 3.0)
		tmp = t_2;
	else
		tmp = 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]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.25 * w), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -4e+70], N[(r * N[(r * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.0], t$95$2, N[(t$95$0 + -1.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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 or -4.00000000000000029e70 < (-.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))) < 3

   1. Initial program 80.8%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 84.4%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 95.1

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

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in r around inf

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

   10. Simplified 91.6%

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

   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))) < -4.00000000000000029e70

   1. Initial program 99.3%

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

      1. associate-*l* N/A

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

      2. unswap-sqr N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in v around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 81.6

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

   7. Simplified 81.6%

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

   8. Taylor expanded in r around inf

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

   10. Simplified 81.6%

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

   11. Taylor expanded in r around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. *-commutative N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 81.5

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

   13. Simplified 81.5%

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

   if 3 < (-.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 85.4%

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

   3. Taylor expanded in 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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

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

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 99.9

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -4 \cdot 10^{+70}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq 3:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\\ t_2 := r \cdot \left(r \cdot w\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, t\_2, t\_0\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+70}:\\ \;\;\;\;\left(3 + \frac{\left(w \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot \mathsf{fma}\left(0.125, v \cdot -2, 0.375\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, t\_2, t\_0 + -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v))))
        (t_2 (* r (* r w))))
   (if (<= t_1 (- INFINITY))
     (fma (* -0.25 w) t_2 t_0)
     (if (<= t_1 -4e+70)
       (-
        (+
         3.0
         (/ (* (* w (* r w)) (* r (fma 0.125 (* v -2.0) 0.375))) (+ v -1.0)))
        4.5)
       (fma (* -0.25 w) t_2 (+ t_0 -1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double t_2 = r * (r * w);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((-0.25 * w), t_2, t_0);
	} else if (t_1 <= -4e+70) {
		tmp = (3.0 + (((w * (r * w)) * (r * fma(0.125, (v * -2.0), 0.375))) / (v + -1.0))) - 4.5;
	} else {
		tmp = fma((-0.25 * w), t_2, (t_0 + -1.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v)))
	t_2 = Float64(r * Float64(r * w))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(-0.25 * w), t_2, t_0);
	elseif (t_1 <= -4e+70)
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(w * Float64(r * w)) * Float64(r * fma(0.125, Float64(v * -2.0), 0.375))) / Float64(v + -1.0))) - 4.5);
	else
		tmp = fma(Float64(-0.25 * w), t_2, 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]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.25 * w), $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -4e+70], N[(N[(3.0 + N[(N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r * N[(0.125 * N[(v * -2.0), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-0.25 * w), $MachinePrecision] * t$95$2 + N[(t$95$0 + -1.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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 75.6%

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

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 94.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 95.1

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

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in r around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 95.1

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

   10. Simplified 95.1%

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

   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))) < -4.00000000000000029e70

   1. Initial program 99.3%

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

      1. associate-*l* N/A

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

      2. unswap-sqr N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.1

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

   4. Applied egg-rr 99.1%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

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

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

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

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

      12. metadata-eval N/A

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

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

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

      14. *-lowering-*.f64 99.5

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in r around inf

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

   9. Simplified 99.5%

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

   if -4.00000000000000029e70 < (-.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 87.3%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 90.2%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 98.6

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -4 \cdot 10^{+70}:\\ \;\;\;\;\left(3 + \frac{\left(w \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot \mathsf{fma}\left(0.125, v \cdot -2, 0.375\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\\ t_2 := r \cdot \left(r \cdot w\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, t\_2, t\_0\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \frac{\left(w \cdot w\right) \cdot \left(-0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, r, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, t\_2, t\_0 + -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v))))
        (t_2 (* r (* r w))))
   (if (<= t_1 (- INFINITY))
     (fma (* -0.25 w) t_2 t_0)
     (if (<= t_1 -5e+54)
       (fma (* r (/ (* (* w w) (* -0.125 (fma v -2.0 3.0))) (- 1.0 v))) r -1.5)
       (fma (* -0.25 w) t_2 (+ t_0 -1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double t_2 = r * (r * w);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((-0.25 * w), t_2, t_0);
	} else if (t_1 <= -5e+54) {
		tmp = fma((r * (((w * w) * (-0.125 * fma(v, -2.0, 3.0))) / (1.0 - v))), r, -1.5);
	} else {
		tmp = fma((-0.25 * w), t_2, (t_0 + -1.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v)))
	t_2 = Float64(r * Float64(r * w))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(-0.25 * w), t_2, t_0);
	elseif (t_1 <= -5e+54)
		tmp = fma(Float64(r * Float64(Float64(Float64(w * w) * Float64(-0.125 * fma(v, -2.0, 3.0))) / Float64(1.0 - v))), r, -1.5);
	else
		tmp = fma(Float64(-0.25 * w), t_2, 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]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.25 * w), $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -5e+54], N[(N[(r * N[(N[(N[(w * w), $MachinePrecision] * N[(-0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r + -1.5), $MachinePrecision], N[(N[(-0.25 * w), $MachinePrecision] * t$95$2 + N[(t$95$0 + -1.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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 75.6%

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

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

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

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

   5. Simplified 94.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-lowering-*.f64 95.1

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

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in r around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 95.1

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

   10. Simplified 95.1%

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

   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))) < -5.00000000000000005e54

   1. Initial program 99.3%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

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

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

      3. distribute-neg-in N/A

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

      4. distribute-lft-in N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lft-mult-inverse N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

   5. Simplified 94.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(r \cdot \frac{\color{blue}{\left(\left(w \cdot w\right) \cdot \frac{-1}{8}\right)} \cdot \left(v \cdot -2 + 3\right)}{1 - v}, r, \frac{-3}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(r \cdot \frac{\color{blue}{\left(w \cdot w\right) \cdot \left(\frac{-1}{8} \cdot \left(v \cdot -2 + 3\right)\right)}}{1 - v}, r, \frac{-3}{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(r \cdot \frac{\color{blue}{\left(w \cdot w\right) \cdot \left(\frac{-1}{8} \cdot \left(v \cdot -2 + 3\right)\right)}}{1 - v}, r, \frac{-3}{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(r \cdot \frac{\color{blue}{\left(w \cdot w\right)} \cdot \left(\frac{-1}{8} \cdot \left(v \cdot -2 + 3\right)\right)}{1 - v}, r, \frac{-3}{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \frac{\left(w \cdot w\right) \cdot \color{blue}{\left(\frac{-1}{8} \cdot \left(v \cdot -2 + 3\right)\right)}}{1 - v}, r, \frac{-3}{2}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \frac{\left(w \cdot w\right) \cdot \left(\frac{-1}{8} \cdot \color{blue}{\mathsf{fma}\left(v, -2, 3\right)}\right)}{1 - v}, r, \frac{-3}{2}\right) \]

      12. --lowering--.f64 99.3

          \[\leadsto \mathsf{fma}\left(r \cdot \frac{\left(w \cdot w\right) \cdot \left(-0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{\color{blue}{1 - v}}, r, -1.5\right) \]

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \frac{\left(w \cdot w\right) \cdot \left(-0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, r, -1.5\right)} \]

   if -5.00000000000000005e54 < (-.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 87.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 90.2%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{w \cdot \left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto w \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto w \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(w \cdot \frac{-1}{4}\right) \cdot \left(w \cdot \left(r \cdot r\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \frac{-1}{4}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \frac{-1}{4}}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{\left(r \cdot r\right) \cdot w}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \color{blue}{\left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 98.6

          \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \frac{\left(w \cdot w\right) \cdot \left(-0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right)}{1 - v}, r, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\\ t_2 := r \cdot \left(r \cdot w\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, t\_2, t\_0\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, t\_2, t\_0 + -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v))))
        (t_2 (* r (* r w))))
   (if (<= t_1 (- INFINITY))
     (fma (* -0.25 w) t_2 t_0)
     (if (<= t_1 -4e+70)
       (- (fma (* r (* w (* r w))) -0.375 3.0) 4.5)
       (fma (* -0.25 w) t_2 (+ t_0 -1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double t_2 = r * (r * w);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((-0.25 * w), t_2, t_0);
	} else if (t_1 <= -4e+70) {
		tmp = fma((r * (w * (r * w))), -0.375, 3.0) - 4.5;
	} else {
		tmp = fma((-0.25 * w), t_2, (t_0 + -1.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v)))
	t_2 = Float64(r * Float64(r * w))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(-0.25 * w), t_2, t_0);
	elseif (t_1 <= -4e+70)
		tmp = Float64(fma(Float64(r * Float64(w * Float64(r * w))), -0.375, 3.0) - 4.5);
	else
		tmp = fma(Float64(-0.25 * w), t_2, 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]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.25 * w), $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -4e+70], N[(N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-0.25 * w), $MachinePrecision] * t$95$2 + N[(t$95$0 + -1.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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 75.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 94.0%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{w \cdot \left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto w \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto w \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(w \cdot \frac{-1}{4}\right) \cdot \left(w \cdot \left(r \cdot r\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \frac{-1}{4}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \frac{-1}{4}}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{\left(r \cdot r\right) \cdot w}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \color{blue}{\left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 95.1

          \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   8. Taylor expanded in r around 0

      \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{{r}^{2}}}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{{r}^{2}}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}}\right) \]

      3. *-lowering-*.f64 95.1

         \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}}\right) \]

   10. Simplified 95.1%

       \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r}}\right) \]

   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))) < -4.00000000000000029e70

   1. Initial program 99.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. unswap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-lowering-*.f64 99.1

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied egg-rr 99.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} - \frac{9}{2} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} - \frac{9}{2} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \color{blue}{\frac{-3}{8}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) - \frac{9}{2} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right)} - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({r}^{2} \cdot {w}^{2}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right)} - \frac{9}{2} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right), \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right), \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, \color{blue}{3 + 2 \cdot \frac{1}{{r}^{2}}}\right) - \frac{9}{2} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) - \frac{9}{2} \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \frac{\color{blue}{2}}{{r}^{2}}\right) - \frac{9}{2} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \color{blue}{\frac{2}{{r}^{2}}}\right) - \frac{9}{2} \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \frac{2}{\color{blue}{r \cdot r}}\right) - \frac{9}{2} \]

      17. *-lowering-*.f64 81.6

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, 3 + \frac{2}{\color{blue}{r \cdot r}}\right) - 4.5 \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, 3 + \frac{2}{r \cdot r}\right)} - 4.5 \]

   8. Taylor expanded in r around inf

      \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, \color{blue}{3}\right) - \frac{9}{2} \]
   9. Step-by-step derivation

   10. Simplified 81.6%

       \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, \color{blue}{3}\right) - 4.5 \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}, \frac{-3}{8}, 3\right) - \frac{9}{2} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot w\right), \frac{-3}{8}, 3\right) - \frac{9}{2} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}, \frac{-3}{8}, 3\right) - \frac{9}{2} \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot w\right), \frac{-3}{8}, 3\right) - \frac{9}{2} \]

       5. *-lowering-*.f64 82.1

          \[\leadsto \mathsf{fma}\left(r \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot w\right), -0.375, 3\right) - 4.5 \]

   12. Applied egg-rr 82.1%

       \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}, -0.375, 3\right) - 4.5 \]

   if -4.00000000000000029e70 < (-.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 87.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 90.2%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{w \cdot \left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto w \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto w \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(w \cdot \frac{-1}{4}\right) \cdot \left(w \cdot \left(r \cdot r\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \frac{-1}{4}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \frac{-1}{4}}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{\left(r \cdot r\right) \cdot w}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \color{blue}{\left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 98.6

          \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -4 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;t\_1 \leq -5:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* r r) (* -0.25 (* w w)))
     (if (<= t_1 -5.0) (* r (* r (* (* w w) -0.375))) (+ t_0 -1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (r * r) * (-0.25 * (w * w));
	} else if (t_1 <= -5.0) {
		tmp = r * (r * ((w * w) * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (r * r) * (-0.25 * (w * w));
	} else if (t_1 <= -5.0) {
		tmp = r * (r * ((w * w) * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (r * r) * (-0.25 * (w * w))
	elif t_1 <= -5.0:
		tmp = r * (r * ((w * w) * -0.375))
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(r * r) * Float64(-0.25 * Float64(w * w)));
	elseif (t_1 <= -5.0)
		tmp = Float64(r * Float64(r * Float64(Float64(w * w) * -0.375)));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (r * r) * (-0.25 * (w * w));
	elseif (t_1 <= -5.0)
		tmp = r * (r * ((w * w) * -0.375));
	else
		tmp = t_0 + -1.5;
	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[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(r * r), $MachinePrecision] * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5.0], N[(r * N[(r * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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 75.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 94.0%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   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. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({r}^{2} \cdot \frac{-1}{4}\right)} \cdot {w}^{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left(\frac{-1}{4} \cdot {w}^{2}\right)} \]

      8. unpow2 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot w\right)}\right) \]

      9. *-lowering-*.f64 86.1

         \[\leadsto \left(r \cdot r\right) \cdot \left(-0.25 \cdot \color{blue}{\left(w \cdot w\right)}\right) \]

   8. Simplified 86.1%

      \[\leadsto \color{blue}{\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)} \]

   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))) < -5

   1. Initial program 99.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. unswap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-lowering-*.f64 99.1

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied egg-rr 99.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} - \frac{9}{2} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} - \frac{9}{2} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \color{blue}{\frac{-3}{8}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) - \frac{9}{2} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right)} - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({r}^{2} \cdot {w}^{2}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right)} - \frac{9}{2} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right), \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right), \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, \color{blue}{3 + 2 \cdot \frac{1}{{r}^{2}}}\right) - \frac{9}{2} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) - \frac{9}{2} \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \frac{\color{blue}{2}}{{r}^{2}}\right) - \frac{9}{2} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \color{blue}{\frac{2}{{r}^{2}}}\right) - \frac{9}{2} \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \frac{2}{\color{blue}{r \cdot r}}\right) - \frac{9}{2} \]

      17. *-lowering-*.f64 70.4

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, 3 + \frac{2}{\color{blue}{r \cdot r}}\right) - 4.5 \]

   7. Simplified 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, 3 + \frac{2}{r \cdot r}\right)} - 4.5 \]

   8. Taylor expanded in r around inf

      \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, \color{blue}{3}\right) - \frac{9}{2} \]
   9. Step-by-step derivation

   10. Simplified 70.4%

       \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, \color{blue}{3}\right) - 4.5 \]

   11. Taylor expanded in r around inf

       \[\leadsto \color{blue}{\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto r \cdot \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto r \cdot \left(r \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto r \cdot \left(r \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto r \cdot \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{-3}{8}\right)\right) \]

       11. *-lowering-*.f64 70.3

           \[\leadsto r \cdot \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot -0.375\right)\right) \]

   13. Simplified 70.3%

       \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)} \]

   if -5 < (-.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 87.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 98.0

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -5:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -5:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ t_0 3.0)
         (/ (* (* r (* r (* w w))) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v)))
        -5.0)
     (* r (* r (* (* w w) -0.375)))
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))) <= -5.0) {
		tmp = r * (r * ((w * w) * -0.375));
	} else {
		tmp = t_0 + -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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) + (((r * (r * (w * w))) * (0.125d0 * ((v * 2.0d0) - 3.0d0))) / (1.0d0 - v))) <= (-5.0d0)) then
        tmp = r * (r * ((w * w) * (-0.375d0)))
    else
        tmp = t_0 + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))) <= -5.0) {
		tmp = r * (r * ((w * w) * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))) <= -5.0:
		tmp = r * (r * ((w * w) * -0.375))
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v))) <= -5.0)
		tmp = Float64(r * Float64(r * Float64(Float64(w * w) * -0.375)));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) + (((r * (r * (w * w))) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))) <= -5.0)
		tmp = r * (r * ((w * w) * -0.375));
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5.0], N[(r * N[(r * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $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))) < -5

   1. Initial program 80.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. unswap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-lowering-*.f64 85.3

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied egg-rr 85.3%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} - \frac{9}{2} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} - \frac{9}{2} \]

      2. metadata-eval N/A

         \[\leadsto \left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \color{blue}{\frac{-3}{8}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) - \frac{9}{2} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right)} - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({r}^{2} \cdot {w}^{2}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right)} - \frac{9}{2} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(r \cdot {w}^{2}\right)}, \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right), \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right), \frac{-3}{8}, 3 + 2 \cdot \frac{1}{{r}^{2}}\right) - \frac{9}{2} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, \color{blue}{3 + 2 \cdot \frac{1}{{r}^{2}}}\right) - \frac{9}{2} \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) - \frac{9}{2} \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \frac{\color{blue}{2}}{{r}^{2}}\right) - \frac{9}{2} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \color{blue}{\frac{2}{{r}^{2}}}\right) - \frac{9}{2} \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, 3 + \frac{2}{\color{blue}{r \cdot r}}\right) - \frac{9}{2} \]

      17. *-lowering-*.f64 76.1

          \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, 3 + \frac{2}{\color{blue}{r \cdot r}}\right) - 4.5 \]

   7. Simplified 76.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, 3 + \frac{2}{r \cdot r}\right)} - 4.5 \]

   8. Taylor expanded in r around inf

      \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, \color{blue}{3}\right) - \frac{9}{2} \]
   9. Step-by-step derivation

   10. Simplified 76.1%

       \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, \color{blue}{3}\right) - 4.5 \]

   11. Taylor expanded in r around inf

       \[\leadsto \color{blue}{\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right)} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto r \cdot \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto r \cdot \left(r \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto r \cdot \left(r \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)}\right) \]

       10. unpow2 N/A

           \[\leadsto r \cdot \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{-3}{8}\right)\right) \]

       11. *-lowering-*.f64 76.1

           \[\leadsto r \cdot \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot -0.375\right)\right) \]

   13. Simplified 76.1%

       \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)} \]

   if -5 < (-.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 87.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. *-lowering-*.f64 98.0

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 98.0%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v} \leq -5:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 + -1.5\\ \mathbf{if}\;v \leq -8 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), t\_1\right)\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \frac{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.375}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ t_0 -1.5)))
   (if (<= v -8e-49)
     (fma -0.25 (* r (* w (* r w))) t_1)
     (if (<= v 1.9e-60)
       (- (+ (+ t_0 3.0) (/ (* (* (* r w) (* r w)) 0.375) (+ v -1.0))) 4.5)
       (fma (* -0.25 w) (* r (* r w)) t_1)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 + -1.5;
	double tmp;
	if (v <= -8e-49) {
		tmp = fma(-0.25, (r * (w * (r * w))), t_1);
	} else if (v <= 1.9e-60) {
		tmp = ((t_0 + 3.0) + ((((r * w) * (r * w)) * 0.375) / (v + -1.0))) - 4.5;
	} else {
		tmp = fma((-0.25 * w), (r * (r * w)), t_1);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 + -1.5)
	tmp = 0.0
	if (v <= -8e-49)
		tmp = fma(-0.25, Float64(r * Float64(w * Float64(r * w))), t_1);
	elseif (v <= 1.9e-60)
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.375) / Float64(v + -1.0))) - 4.5);
	else
		tmp = fma(Float64(-0.25 * w), Float64(r * Float64(r * w)), t_1);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.5), $MachinePrecision]}, If[LessEqual[v, -8e-49], N[(-0.25 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[v, 1.9e-60], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(-0.25 * w), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -7.99999999999999949e-49

   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

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\left(w \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, w \cdot \left(w \cdot \left(r \cdot r\right)\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, w \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot r\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, w \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\left(w \cdot \left(r \cdot w\right)\right) \cdot r}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \color{blue}{\left(r \cdot w\right)}\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      17. *-lowering-*.f64 98.5

          \[\leadsto \mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   if -7.99999999999999949e-49 < v < 1.89999999999999997e-60

   1. Initial program 88.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. unswap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      7. *-lowering-*.f64 99.8

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)}{1 - v}\right) - 4.5 \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]

   5. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\frac{3}{8}} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - \frac{9}{2} \]
   6. Step-by-step derivation

   7. Simplified 99.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5 \]

   if 1.89999999999999997e-60 < v

   1. Initial program 77.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 87.6%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{w \cdot \left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto w \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto w \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(w \cdot \frac{-1}{4}\right) \cdot \left(w \cdot \left(r \cdot r\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \frac{-1}{4}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \frac{-1}{4}}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{\left(r \cdot r\right) \cdot w}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \color{blue}{\left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 97.0

          \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -8 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r} + -1.5\right)\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.375}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;v \leq -2.45 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), t\_0\right)\\ \mathbf{elif}\;v \leq 1.85 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right), -0.375, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) -1.5)))
   (if (<= v -2.45e+70)
     (fma -0.25 (* r (* w (* r w))) t_0)
     (if (<= v 1.85e-60)
       (fma (* w (* w (* r r))) -0.375 t_0)
       (fma (* -0.25 w) (* r (* r w)) t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (v <= -2.45e+70) {
		tmp = fma(-0.25, (r * (w * (r * w))), t_0);
	} else if (v <= 1.85e-60) {
		tmp = fma((w * (w * (r * r))), -0.375, t_0);
	} else {
		tmp = fma((-0.25 * w), (r * (r * w)), t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if (v <= -2.45e+70)
		tmp = fma(-0.25, Float64(r * Float64(w * Float64(r * w))), t_0);
	elseif (v <= 1.85e-60)
		tmp = fma(Float64(w * Float64(w * Float64(r * r))), -0.375, t_0);
	else
		tmp = fma(Float64(-0.25 * w), Float64(r * Float64(r * w)), t_0);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[v, -2.45e+70], N[(-0.25 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[v, 1.85e-60], N[(N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + t$95$0), $MachinePrecision], N[(N[(-0.25 * w), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.45000000000000014e70

   1. Initial program 80.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 88.2%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\left(w \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, w \cdot \left(w \cdot \left(r \cdot r\right)\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, w \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot r\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, w \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\left(w \cdot \left(r \cdot w\right)\right) \cdot r}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \color{blue}{\left(r \cdot w\right)}\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      17. *-lowering-*.f64 98.0

          \[\leadsto \mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   if -2.45000000000000014e70 < v < 1.85000000000000012e-60

   1. Initial program 89.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) + \frac{2}{r \cdot r}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \left(\frac{-3}{2} + \frac{2}{r \cdot r}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot \left(w \cdot w\right)} + \left(\frac{-3}{2} + \frac{2}{r \cdot r}\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \left(r \cdot r\right)\right)} \cdot \left(w \cdot w\right) + \left(\frac{-3}{2} + \frac{2}{r \cdot r}\right) \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\frac{-3}{8} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)} + \left(\frac{-3}{2} + \frac{2}{r \cdot r}\right) \]

      6. associate-*r* N/A

         \[\leadsto \frac{-3}{8} \cdot \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} + \left(\frac{-3}{2} + \frac{2}{r \cdot r}\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{-3}{8}} + \left(\frac{-3}{2} + \frac{2}{r \cdot r}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{-3}{8} + \color{blue}{\left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(w \cdot w\right) \cdot \left(r \cdot r\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \left(w \cdot \left(r \cdot r\right)\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot w\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \color{blue}{\left(r \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \left(r \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot w\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}, \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \left(w \cdot \color{blue}{\left(r \cdot r\right)}\right), \frac{-3}{8}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right), \frac{-3}{8}, \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right), \frac{-3}{8}, \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      22. *-lowering-*.f64 97.0

          \[\leadsto \mathsf{fma}\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right), -0.375, \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right), -0.375, \frac{2}{r \cdot r} + -1.5\right)} \]

   if 1.85000000000000012e-60 < v

   1. Initial program 77.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 87.6%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{w \cdot \left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto w \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto w \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(w \cdot \frac{-1}{4}\right) \cdot \left(w \cdot \left(r \cdot r\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \frac{-1}{4}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \frac{-1}{4}}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{\left(r \cdot r\right) \cdot w}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \color{blue}{\left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 97.0

          \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.45 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r} + -1.5\right)\\ \mathbf{elif}\;v \leq 1.85 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right), -0.375, \frac{2}{r \cdot r} + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2 \cdot 10^{+126}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), t\_0 + -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2e+126)
     (+ -1.5 (fma (* w (* -0.25 (* r r))) w t_0))
     (fma -0.25 (* r (* w (* 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 <= 2e+126) {
		tmp = -1.5 + fma((w * (-0.25 * (r * r))), w, t_0);
	} else {
		tmp = fma(-0.25, (r * (w * (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 <= 2e+126)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r * r))), w, t_0));
	else
		tmp = fma(-0.25, Float64(r * Float64(w * 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, 2e+126], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + -1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.99999999999999985e126

   1. Initial program 83.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 90.1%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   if 1.99999999999999985e126 < r

   1. Initial program 87.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 63.1%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(\left(w \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, w \cdot \left(w \cdot \left(r \cdot r\right)\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, w \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot r\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, w \cdot \left(\color{blue}{\left(r \cdot w\right)} \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\left(w \cdot \left(r \cdot w\right)\right) \cdot r}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \color{blue}{\left(r \cdot w\right)}\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      17. *-lowering-*.f64 93.0

          \[\leadsto \mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r} + -1.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{+126}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, r \cdot \left(w \cdot \left(r \cdot w\right)\right), \frac{2}{r \cdot r} + -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+117}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e+117)
   (+ -1.5 (fma (* w (* -0.25 (* r r))) w (/ 2.0 (* r r))))
   (fma (* r w) (* -0.25 (* r w)) -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e+117) {
		tmp = -1.5 + fma((w * (-0.25 * (r * r))), w, (2.0 / (r * r)));
	} else {
		tmp = fma((r * w), (-0.25 * (r * w)), -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e+117)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r * r))), w, Float64(2.0 / Float64(r * r))));
	else
		tmp = fma(Float64(r * w), Float64(-0.25 * Float64(r * w)), -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e+117], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.99999999999999983e117

   1. Initial program 83.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 90.0%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   if 4.99999999999999983e117 < r

   1. Initial program 89.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-1}{4} \cdot {w}^{2} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \color{blue}{\frac{-3}{2}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({r}^{2}, \frac{-1}{4} \cdot {w}^{2}, \frac{-3}{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot r}, \frac{-1}{4} \cdot {w}^{2}, \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot r}, \frac{-1}{4} \cdot {w}^{2}, \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot r, \color{blue}{\frac{-1}{4} \cdot {w}^{2}}, \frac{-3}{2}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot r, \frac{-1}{4} \cdot \color{blue}{\left(w \cdot w\right)}, \frac{-3}{2}\right) \]

      14. *-lowering-*.f64 66.7

          \[\leadsto \mathsf{fma}\left(r \cdot r, -0.25 \cdot \color{blue}{\left(w \cdot w\right)}, -1.5\right) \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot r, -0.25 \cdot \left(w \cdot w\right), -1.5\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right)} + \frac{-3}{2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-1}{4}} + \frac{-3}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right)} \cdot \frac{-1}{4} + \frac{-3}{2} \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(r \cdot \left(r \cdot w\right)\right)} \cdot w\right) \cdot \frac{-1}{4} + \frac{-3}{2} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{-1}{4}\right)} + \frac{-3}{2} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(w \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)} + \frac{-3}{2} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot w\right)} + \frac{-3}{2} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(r \cdot w\right) \cdot \left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right)} + \frac{-3}{2} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot w, \left(w \cdot \frac{-1}{4}\right) \cdot r, \frac{-3}{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot r}, \left(w \cdot \frac{-1}{4}\right) \cdot r, \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot r}, \left(w \cdot \frac{-1}{4}\right) \cdot r, \frac{-3}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \color{blue}{\left(\frac{-1}{4} \cdot w\right)} \cdot r, \frac{-3}{2}\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \color{blue}{\frac{-1}{4} \cdot \left(w \cdot r\right)}, \frac{-3}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \frac{-1}{4} \cdot \color{blue}{\left(r \cdot w\right)}, \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \color{blue}{\frac{-1}{4} \cdot \left(r \cdot w\right)}, \frac{-3}{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \frac{-1}{4} \cdot \color{blue}{\left(w \cdot r\right)}, \frac{-3}{2}\right) \]

      17. *-lowering-*.f64 93.7

          \[\leadsto \mathsf{fma}\left(w \cdot r, -0.25 \cdot \color{blue}{\left(w \cdot r\right)}, -1.5\right) \]

   10. Applied egg-rr 93.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot r, -0.25 \cdot \left(w \cdot r\right), -1.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+117}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.75)
   (fma (* -0.25 w) (* r (* r w)) (/ 2.0 (* r r)))
   (fma (* r w) (* -0.25 (* r w)) -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.75) {
		tmp = fma((-0.25 * w), (r * (r * w)), (2.0 / (r * r)));
	} else {
		tmp = fma((r * w), (-0.25 * (r * w)), -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.75)
		tmp = fma(Float64(-0.25 * w), Float64(r * Float64(r * w)), Float64(2.0 / Float64(r * r)));
	else
		tmp = fma(Float64(r * w), Float64(-0.25 * Float64(r * w)), -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.75], N[(N[(-0.25 * w), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(-0.25 * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 0.75

   1. Initial program 82.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 89.1%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\right) + \frac{-3}{2}} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{w \cdot \left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      4. associate-*l* N/A

         \[\leadsto w \cdot \color{blue}{\left(\frac{-1}{4} \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto w \cdot \left(\frac{-1}{4} \cdot \color{blue}{\left(w \cdot \left(r \cdot r\right)\right)}\right) + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(w \cdot \frac{-1}{4}\right) \cdot \left(w \cdot \left(r \cdot r\right)\right)} + \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \frac{-1}{4}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \frac{-1}{4}}, w \cdot \left(r \cdot r\right), \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{\left(r \cdot r\right) \cdot w}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, \color{blue}{r \cdot \left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \color{blue}{\left(r \cdot w\right)}, \frac{2}{r \cdot r} + \frac{-3}{2}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r} + \frac{-3}{2}}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r}} + \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 93.0

          \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}} + -1.5\right) \]

   7. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   8. Taylor expanded in r around 0

      \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{{r}^{2}}}\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{{r}^{2}}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(w \cdot \frac{-1}{4}, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}}\right) \]

      3. *-lowering-*.f64 84.6

         \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \frac{2}{\color{blue}{r \cdot r}}\right) \]

   10. Simplified 84.6%

       \[\leadsto \mathsf{fma}\left(w \cdot -0.25, r \cdot \left(r \cdot w\right), \color{blue}{\frac{2}{r \cdot r}}\right) \]

   if 0.75 < r

   1. Initial program 89.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Simplified 83.4%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-1}{4} \cdot {w}^{2} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right) + \color{blue}{\frac{-3}{2}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({r}^{2}, \frac{-1}{4} \cdot {w}^{2}, \frac{-3}{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot r}, \frac{-1}{4} \cdot {w}^{2}, \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot r}, \frac{-1}{4} \cdot {w}^{2}, \frac{-3}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot r, \color{blue}{\frac{-1}{4} \cdot {w}^{2}}, \frac{-3}{2}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(r \cdot r, \frac{-1}{4} \cdot \color{blue}{\left(w \cdot w\right)}, \frac{-3}{2}\right) \]

      14. *-lowering-*.f64 82.4

          \[\leadsto \mathsf{fma}\left(r \cdot r, -0.25 \cdot \color{blue}{\left(w \cdot w\right)}, -1.5\right) \]

   8. Simplified 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot r, -0.25 \cdot \left(w \cdot w\right), -1.5\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right)} + \frac{-3}{2} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-1}{4}} + \frac{-3}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right)} \cdot \frac{-1}{4} + \frac{-3}{2} \]

      4. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(r \cdot \left(r \cdot w\right)\right)} \cdot w\right) \cdot \frac{-1}{4} + \frac{-3}{2} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot \frac{-1}{4}\right)} + \frac{-3}{2} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(w \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)} + \frac{-3}{2} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot w\right)} + \frac{-3}{2} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(r \cdot w\right) \cdot \left(\left(w \cdot \frac{-1}{4}\right) \cdot r\right)} + \frac{-3}{2} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot w, \left(w \cdot \frac{-1}{4}\right) \cdot r, \frac{-3}{2}\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot r}, \left(w \cdot \frac{-1}{4}\right) \cdot r, \frac{-3}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot r}, \left(w \cdot \frac{-1}{4}\right) \cdot r, \frac{-3}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \color{blue}{\left(\frac{-1}{4} \cdot w\right)} \cdot r, \frac{-3}{2}\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \color{blue}{\frac{-1}{4} \cdot \left(w \cdot r\right)}, \frac{-3}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \frac{-1}{4} \cdot \color{blue}{\left(r \cdot w\right)}, \frac{-3}{2}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \color{blue}{\frac{-1}{4} \cdot \left(r \cdot w\right)}, \frac{-3}{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(w \cdot r, \frac{-1}{4} \cdot \color{blue}{\left(w \cdot r\right)}, \frac{-3}{2}\right) \]

      17. *-lowering-*.f64 93.8

          \[\leadsto \mathsf{fma}\left(w \cdot r, -0.25 \cdot \color{blue}{\left(w \cdot r\right)}, -1.5\right) \]

   10. Applied egg-rr 93.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot r, -0.25 \cdot \left(w \cdot r\right), -1.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot w, r \cdot \left(r \cdot w\right), \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, -0.25 \cdot \left(r \cdot w\right), -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 7e-7) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7e-7) {
		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 <= 7d-7) 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 <= 7e-7) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7e-7:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7e-7)
		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 <= 7e-7)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7e-7], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 6.99999999999999968e-7

   1. Initial program 82.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

      3. *-lowering-*.f64 58.5

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

   5. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]

   if 6.99999999999999968e-7 < r

   1. Initial program 89.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \color{blue}{-1 \cdot \left({r}^{2} \cdot \left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v} + \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({r}^{2} \cdot \left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v} + \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{{r}^{2} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v} + \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto {r}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \color{blue}{{r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

      8. lft-mult-inverse N/A

         \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \color{blue}{\frac{-3}{2}} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({r}^{2}, \mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right), \frac{-3}{2}\right)} \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot r, \frac{\left(-0.125 \cdot \left(w \cdot w\right)\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -1.5\right)} \]

   6. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 30.9%

      \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.7% 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 84.1%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in 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

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

   9. *-lowering-*.f64 58.7

      \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

5. Simplified 58.7%

   \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]

6. Final simplification 58.7%

   \[\leadsto \frac{2}{r \cdot r} + -1.5 \]
7. Add Preprocessing

Alternative 18: 14.6% 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 84.1%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in r around inf

   \[\leadsto \color{blue}{-1 \cdot \left({r}^{2} \cdot \left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v} + \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({r}^{2} \cdot \left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v} + \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v} + \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

   3. distribute-neg-in N/A

      \[\leadsto {r}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

   8. lft-mult-inverse N/A

      \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right) + \color{blue}{\frac{-3}{2}} \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left({r}^{2}, \mathsf{neg}\left(\frac{1}{8} \cdot \frac{{w}^{2} \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right), \frac{-3}{2}\right)} \]

5. Simplified 44.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot r, \frac{\left(-0.125 \cdot \left(w \cdot w\right)\right) \cdot \mathsf{fma}\left(v, -2, 3\right)}{1 - v}, -1.5\right)} \]

6. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{-3}{2}} \]
7. Step-by-step derivation

8. Simplified 15.3%

   \[\leadsto \color{blue}{-1.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(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))