Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.7%

Time: 11.8s

Alternatives: 13

Speedup: 1.8×

• 52.7% 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: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \mathsf{fma}\left(-0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), t\_0 - 1.5\right)\\ \mathbf{if}\;v \leq -4 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 10^{+54}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{\left(\left(w \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (fma -0.25 (* (* w r) (* w r)) (- t_0 1.5))))
   (if (<= v -4e+22)
     t_1
     (if (<= v 1e+54)
       (-
        (-
         (+ 3.0 t_0)
         (/ (* (* (* w r) (fma -0.25 v 0.375)) (* w r)) (- 1.0 v)))
        4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = fma(-0.25, ((w * r) * (w * r)), (t_0 - 1.5));
	double tmp;
	if (v <= -4e+22) {
		tmp = t_1;
	} else if (v <= 1e+54) {
		tmp = ((3.0 + t_0) - ((((w * r) * fma(-0.25, v, 0.375)) * (w * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = fma(-0.25, Float64(Float64(w * r) * Float64(w * r)), Float64(t_0 - 1.5))
	tmp = 0.0
	if (v <= -4e+22)
		tmp = t_1;
	elseif (v <= 1e+54)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(w * r) * fma(-0.25, v, 0.375)) * Float64(w * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = 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[(-0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -4e+22], t$95$1, If[LessEqual[v, 1e+54], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(w * r), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -4e22 or 1.0000000000000001e54 < v

   1. Initial program 85.0%

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 90.9%

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

   7. Applied rewrites 99.9%

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

   if -4e22 < v < 1.0000000000000001e54

   1. Initial program 87.7%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lower-fma.f64 87.7

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

   5. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(r \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. swap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 99.8

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

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      13. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      14. lower-*.f64 99.8

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

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      16. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      17. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{elif}\;v \leq 10^{+54}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(w \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{2}{r \cdot r} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot r\right) \cdot r\\ t_1 := \left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot t\_0}{1 - v}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+267}:\\ \;\;\;\;\left(\left(w \cdot \left(r \cdot r\right)\right) \cdot -0.25\right) \cdot w - 4.5\\ \mathbf{elif}\;t\_1 \leq -100000000000:\\ \;\;\;\;t\_0 \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* w w) r) r))
        (t_1
         (-
          (+ 3.0 (/ 2.0 (* r r)))
          (/ (* (* (- 3.0 (* v 2.0)) 0.125) t_0) (- 1.0 v)))))
   (if (<= t_1 -2e+267)
     (- (* (* (* w (* r r)) -0.25) w) 4.5)
     (if (<= t_1 -100000000000.0) (* t_0 -0.375) (+ -1.5 (/ (/ 2.0 r) r))))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = (3.0 + (2.0 / (r * r))) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	double tmp;
	if (t_1 <= -2e+267) {
		tmp = (((w * (r * r)) * -0.25) * w) - 4.5;
	} else if (t_1 <= -100000000000.0) {
		tmp = t_0 * -0.375;
	} else {
		tmp = -1.5 + ((2.0 / r) / r);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((w * w) * r) * r
    t_1 = (3.0d0 + (2.0d0 / (r * r))) - ((((3.0d0 - (v * 2.0d0)) * 0.125d0) * t_0) / (1.0d0 - v))
    if (t_1 <= (-2d+267)) then
        tmp = (((w * (r * r)) * (-0.25d0)) * w) - 4.5d0
    else if (t_1 <= (-100000000000.0d0)) then
        tmp = t_0 * (-0.375d0)
    else
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = (3.0 + (2.0 / (r * r))) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	double tmp;
	if (t_1 <= -2e+267) {
		tmp = (((w * (r * r)) * -0.25) * w) - 4.5;
	} else if (t_1 <= -100000000000.0) {
		tmp = t_0 * -0.375;
	} else {
		tmp = -1.5 + ((2.0 / r) / r);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((w * w) * r) * r
	t_1 = (3.0 + (2.0 / (r * r))) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v))
	tmp = 0
	if t_1 <= -2e+267:
		tmp = (((w * (r * r)) * -0.25) * w) - 4.5
	elif t_1 <= -100000000000.0:
		tmp = t_0 * -0.375
	else:
		tmp = -1.5 + ((2.0 / r) / r)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * w) * r) * r)
	t_1 = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125) * t_0) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= -2e+267)
		tmp = Float64(Float64(Float64(Float64(w * Float64(r * r)) * -0.25) * w) - 4.5);
	elseif (t_1 <= -100000000000.0)
		tmp = Float64(t_0 * -0.375);
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((w * w) * r) * r;
	t_1 = (3.0 + (2.0 / (r * r))) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	tmp = 0.0;
	if (t_1 <= -2e+267)
		tmp = (((w * (r * r)) * -0.25) * w) - 4.5;
	elseif (t_1 <= -100000000000.0)
		tmp = t_0 * -0.375;
	else
		tmp = -1.5 + ((2.0 / r) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+267], N[(N[(N[(N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * w), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -100000000000.0], N[(t$95$0 * -0.375), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $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))) < -1.9999999999999999e267

   1. Initial program 79.8%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lower-fma.f64 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right)} - \frac{9}{2} \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 87.4%

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

   if -1.9999999999999999e267 < (-.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))) < -1e11

   1. Initial program 98.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(r \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. swap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 99.4

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

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      13. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      14. lower-*.f64 99.4

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

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      16. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      17. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

   10. Applied rewrites 66.5%

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

   11. Taylor expanded in v around 0

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

   13. Applied rewrites 76.9%

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

   if -1e11 < (-.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 89.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in r around 0

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

   7. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 94.9

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

   9. Applied rewrites 94.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2 \cdot 10^{+267}:\\ \;\;\;\;\left(\left(w \cdot \left(r \cdot r\right)\right) \cdot -0.25\right) \cdot w - 4.5\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -100000000000:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot r\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(3 + t\_1\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot t\_0}{1 - v}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+267}:\\ \;\;\;\;\left(\left(w \cdot \left(r \cdot r\right)\right) \cdot -0.25\right) \cdot w - 4.5\\ \mathbf{elif}\;t\_2 \leq -100000000000:\\ \;\;\;\;t\_0 \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* w w) r) r))
        (t_1 (/ 2.0 (* r r)))
        (t_2
         (- (+ 3.0 t_1) (/ (* (* (- 3.0 (* v 2.0)) 0.125) t_0) (- 1.0 v)))))
   (if (<= t_2 -2e+267)
     (- (* (* (* w (* r r)) -0.25) w) 4.5)
     (if (<= t_2 -100000000000.0) (* t_0 -0.375) (- t_1 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = (3.0 + t_1) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	double tmp;
	if (t_2 <= -2e+267) {
		tmp = (((w * (r * r)) * -0.25) * w) - 4.5;
	} else if (t_2 <= -100000000000.0) {
		tmp = t_0 * -0.375;
	} else {
		tmp = t_1 - 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((w * w) * r) * r
    t_1 = 2.0d0 / (r * r)
    t_2 = (3.0d0 + t_1) - ((((3.0d0 - (v * 2.0d0)) * 0.125d0) * t_0) / (1.0d0 - v))
    if (t_2 <= (-2d+267)) then
        tmp = (((w * (r * r)) * (-0.25d0)) * w) - 4.5d0
    else if (t_2 <= (-100000000000.0d0)) then
        tmp = t_0 * (-0.375d0)
    else
        tmp = t_1 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = (3.0 + t_1) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	double tmp;
	if (t_2 <= -2e+267) {
		tmp = (((w * (r * r)) * -0.25) * w) - 4.5;
	} else if (t_2 <= -100000000000.0) {
		tmp = t_0 * -0.375;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((w * w) * r) * r
	t_1 = 2.0 / (r * r)
	t_2 = (3.0 + t_1) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v))
	tmp = 0
	if t_2 <= -2e+267:
		tmp = (((w * (r * r)) * -0.25) * w) - 4.5
	elif t_2 <= -100000000000.0:
		tmp = t_0 * -0.375
	else:
		tmp = t_1 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * w) * r) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125) * t_0) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_2 <= -2e+267)
		tmp = Float64(Float64(Float64(Float64(w * Float64(r * r)) * -0.25) * w) - 4.5);
	elseif (t_2 <= -100000000000.0)
		tmp = Float64(t_0 * -0.375);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((w * w) * r) * r;
	t_1 = 2.0 / (r * r);
	t_2 = (3.0 + t_1) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	tmp = 0.0;
	if (t_2 <= -2e+267)
		tmp = (((w * (r * r)) * -0.25) * w) - 4.5;
	elseif (t_2 <= -100000000000.0)
		tmp = t_0 * -0.375;
	else
		tmp = t_1 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+267], N[(N[(N[(N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * w), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$2, -100000000000.0], N[(t$95$0 * -0.375), $MachinePrecision], N[(t$95$1 - 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))) < -1.9999999999999999e267

   1. Initial program 79.8%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lower-fma.f64 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right)} - \frac{9}{2} \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 93.1

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

   8. Applied rewrites 93.1%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 87.4%

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

   if -1.9999999999999999e267 < (-.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))) < -1e11

   1. Initial program 98.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(r \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. swap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 99.4

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

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      13. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      14. lower-*.f64 99.4

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

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      16. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      17. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

   10. Applied rewrites 66.5%

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

   11. Taylor expanded in v around 0

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

   13. Applied rewrites 76.9%

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

   if -1e11 < (-.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 89.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2 \cdot 10^{+267}:\\ \;\;\;\;\left(\left(w \cdot \left(r \cdot r\right)\right) \cdot -0.25\right) \cdot w - 4.5\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -100000000000:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot r\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(3 + t\_1\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot t\_0}{1 - v}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+267}:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot -0.25\right) \cdot r\right) \cdot r\\ \mathbf{elif}\;t\_2 \leq -100000000000:\\ \;\;\;\;t\_0 \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* w w) r) r))
        (t_1 (/ 2.0 (* r r)))
        (t_2
         (- (+ 3.0 t_1) (/ (* (* (- 3.0 (* v 2.0)) 0.125) t_0) (- 1.0 v)))))
   (if (<= t_2 -2e+267)
     (* (* (* (* w w) -0.25) r) r)
     (if (<= t_2 -100000000000.0) (* t_0 -0.375) (- t_1 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = (3.0 + t_1) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	double tmp;
	if (t_2 <= -2e+267) {
		tmp = (((w * w) * -0.25) * r) * r;
	} else if (t_2 <= -100000000000.0) {
		tmp = t_0 * -0.375;
	} else {
		tmp = t_1 - 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((w * w) * r) * r
    t_1 = 2.0d0 / (r * r)
    t_2 = (3.0d0 + t_1) - ((((3.0d0 - (v * 2.0d0)) * 0.125d0) * t_0) / (1.0d0 - v))
    if (t_2 <= (-2d+267)) then
        tmp = (((w * w) * (-0.25d0)) * r) * r
    else if (t_2 <= (-100000000000.0d0)) then
        tmp = t_0 * (-0.375d0)
    else
        tmp = t_1 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = (3.0 + t_1) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	double tmp;
	if (t_2 <= -2e+267) {
		tmp = (((w * w) * -0.25) * r) * r;
	} else if (t_2 <= -100000000000.0) {
		tmp = t_0 * -0.375;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((w * w) * r) * r
	t_1 = 2.0 / (r * r)
	t_2 = (3.0 + t_1) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v))
	tmp = 0
	if t_2 <= -2e+267:
		tmp = (((w * w) * -0.25) * r) * r
	elif t_2 <= -100000000000.0:
		tmp = t_0 * -0.375
	else:
		tmp = t_1 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * w) * r) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125) * t_0) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_2 <= -2e+267)
		tmp = Float64(Float64(Float64(Float64(w * w) * -0.25) * r) * r);
	elseif (t_2 <= -100000000000.0)
		tmp = Float64(t_0 * -0.375);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((w * w) * r) * r;
	t_1 = 2.0 / (r * r);
	t_2 = (3.0 + t_1) - ((((3.0 - (v * 2.0)) * 0.125) * t_0) / (1.0 - v));
	tmp = 0.0;
	if (t_2 <= -2e+267)
		tmp = (((w * w) * -0.25) * r) * r;
	elseif (t_2 <= -100000000000.0)
		tmp = t_0 * -0.375;
	else
		tmp = t_1 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+267], N[(N[(N[(N[(w * w), $MachinePrecision] * -0.25), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$2, -100000000000.0], N[(t$95$0 * -0.375), $MachinePrecision], N[(t$95$1 - 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))) < -1.9999999999999999e267

   1. Initial program 79.8%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lower-fma.f64 79.8

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

   5. Applied rewrites 79.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(r \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. swap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 89.4

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

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      13. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      14. lower-*.f64 89.4

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

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      16. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      17. lower-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

   10. Applied rewrites 82.9%

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

   11. Taylor expanded in v around inf

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

   13. Applied rewrites 87.3%

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

   if -1.9999999999999999e267 < (-.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))) < -1e11

   1. Initial program 98.0%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(\left(w \cdot w\right) \cdot r\right)} \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      4. associate-*l* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(r \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      6. swap-sqr N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}{1 - v}\right) - \frac{9}{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - \frac{9}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(\left(w \cdot r\right) \cdot \color{blue}{\left(w \cdot r\right)}\right)}{1 - v}\right) - \frac{9}{2} \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      11. lower-*.f64 99.4

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

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      13. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      14. lower-*.f64 99.4

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

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(w \cdot r\right)}}{1 - v}\right) - \frac{9}{2} \]

      16. *-commutative N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(\frac{-1}{4}, v, \frac{3}{8}\right) \cdot \left(r \cdot w\right)\right) \cdot \color{blue}{\left(r \cdot w\right)}}{1 - v}\right) - \frac{9}{2} \]

      17. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

   10. Applied rewrites 66.5%

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

   11. Taylor expanded in v around 0

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

   13. Applied rewrites 76.9%

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

   if -1e11 < (-.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 89.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2 \cdot 10^{+267}:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot -0.25\right) \cdot r\right) \cdot r\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -100000000000:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -100000000000:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* (- 3.0 (* v 2.0)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))
        -100000000000.0)
     (* (* (* -0.375 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -100000000000.0) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} 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 (((3.0d0 + t_0) - ((((3.0d0 - (v * 2.0d0)) * 0.125d0) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-100000000000.0d0)) then
        tmp = (((-0.375d0) * (r * r)) * w) * w
    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 (((3.0 + t_0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -100000000000.0) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -100000000000.0:
		tmp = ((-0.375 * (r * r)) * w) * w
	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(3.0 + t_0) - Float64(Float64(Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= -100000000000.0)
		tmp = Float64(Float64(Float64(-0.375 * Float64(r * r)) * w) * w);
	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 (((3.0 + t_0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -100000000000.0)
		tmp = ((-0.375 * (r * r)) * w) * w;
	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[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -100000000000.0], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $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))) < -1e11

   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 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. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 79.5%

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

   if -1e11 < (-.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 89.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -100000000000:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := t\_1 - 1.5\\ \mathbf{if}\;v \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, t\_0, t\_2\right)\\ \mathbf{elif}\;v \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot r\right) \cdot w, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{0.125}{v} + -0.25, -1.5\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w r) (* w r))) (t_1 (/ 2.0 (* r r))) (t_2 (- t_1 1.5)))
   (if (<= v -1.1e+15)
     (fma -0.25 t_0 t_2)
     (if (<= v 1.55e-29)
       (fma -0.375 (* (* (* w r) r) w) t_2)
       (+ (fma t_0 (+ (/ 0.125 v) -0.25) -1.5) t_1)))))

C

double code(double v, double w, double r) {
	double t_0 = (w * r) * (w * r);
	double t_1 = 2.0 / (r * r);
	double t_2 = t_1 - 1.5;
	double tmp;
	if (v <= -1.1e+15) {
		tmp = fma(-0.25, t_0, t_2);
	} else if (v <= 1.55e-29) {
		tmp = fma(-0.375, (((w * r) * r) * w), t_2);
	} else {
		tmp = fma(t_0, ((0.125 / v) + -0.25), -1.5) + t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * r) * Float64(w * r))
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(t_1 - 1.5)
	tmp = 0.0
	if (v <= -1.1e+15)
		tmp = fma(-0.25, t_0, t_2);
	elseif (v <= 1.55e-29)
		tmp = fma(-0.375, Float64(Float64(Float64(w * r) * r) * w), t_2);
	else
		tmp = Float64(fma(t_0, Float64(Float64(0.125 / v) + -0.25), -1.5) + t_1);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - 1.5), $MachinePrecision]}, If[LessEqual[v, -1.1e+15], N[(-0.25 * t$95$0 + t$95$2), $MachinePrecision], If[LessEqual[v, 1.55e-29], N[(-0.375 * N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(t$95$0 * N[(N[(0.125 / v), $MachinePrecision] + -0.25), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.1e15

   1. Initial program 84.8%

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 99.9%

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

   if -1.1e15 < v < 1.55000000000000013e-29

   1. Initial program 88.0%

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

   3. 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. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 87.4%

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

   7. Applied rewrites 98.4%

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

   if 1.55000000000000013e-29 < v

   1. Initial program 85.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around inf

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

   7. Applied rewrites 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), -0.25 + \frac{0.125}{v}, -1.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{elif}\;v \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot r\right) \cdot w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{0.125}{v} + -0.25, -1.5\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} - 1.5\\ t_1 := \mathsf{fma}\left(-0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), t\_0\right)\\ \mathbf{if}\;v \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot r\right) \cdot w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (* r r)) 1.5))
        (t_1 (fma -0.25 (* (* w r) (* w r)) t_0)))
   (if (<= v -1.1e+15)
     t_1
     (if (<= v 1.5e-29) (fma -0.375 (* (* (* w r) r) w) t_0) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) - 1.5;
	double t_1 = fma(-0.25, ((w * r) * (w * r)), t_0);
	double tmp;
	if (v <= -1.1e+15) {
		tmp = t_1;
	} else if (v <= 1.5e-29) {
		tmp = fma(-0.375, (((w * r) * r) * w), t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) - 1.5)
	t_1 = fma(-0.25, Float64(Float64(w * r) * Float64(w * r)), t_0)
	tmp = 0.0
	if (v <= -1.1e+15)
		tmp = t_1;
	elseif (v <= 1.5e-29)
		tmp = fma(-0.375, Float64(Float64(Float64(w * r) * r) * w), t_0);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(-0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[v, -1.1e+15], t$95$1, If[LessEqual[v, 1.5e-29], N[(-0.375 * N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] + t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.1e15 or 1.5000000000000001e-29 < v

   1. Initial program 85.0%

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 90.3%

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

   7. Applied rewrites 99.8%

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

   if -1.1e15 < v < 1.5000000000000001e-29

   1. Initial program 88.0%

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

   3. 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. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 87.4%

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

   7. Applied rewrites 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{elif}\;v \leq 1.5 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot r\right) \cdot w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{2}{r \cdot r} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(w \cdot \left(r \cdot r\right)\right) \cdot w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r, -0.25, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e+130)
   (fma -0.375 (* (* w (* r r)) w) (- (/ 2.0 (* r r)) 1.5))
   (fma (* (* (* w w) r) r) -0.25 -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e+130) {
		tmp = fma(-0.375, ((w * (r * r)) * w), ((2.0 / (r * r)) - 1.5));
	} else {
		tmp = fma((((w * w) * r) * r), -0.25, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e+130)
		tmp = fma(-0.375, Float64(Float64(w * Float64(r * r)) * w), Float64(Float64(2.0 / Float64(r * r)) - 1.5));
	else
		tmp = fma(Float64(Float64(Float64(w * w) * r) * r), -0.25, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e+130], N[(-0.375 * N[(N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * -0.25 + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.9999999999999996e130

   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 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. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 91.5%

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

   if 4.9999999999999996e130 < r

   1. Initial program 82.6%

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

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 87.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(w \cdot \left(r \cdot r\right)\right) \cdot w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r, -0.25, -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

3. Taylor expanded in v around 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. +-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. associate-+l+ N/A

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

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

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 85.1%

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

7. Applied rewrites 92.3%

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

8. Final simplification 92.3%

   \[\leadsto \mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot r\right) \cdot w, \frac{2}{r \cdot r} - 1.5\right) \]
9. Add Preprocessing

Alternative 11: 50.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.00015:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;3 - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.00015) (/ 2.0 (* r r)) (- 3.0 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00015) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = 3.0 - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.00015d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = 3.0d0 - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00015) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = 3.0 - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.00015:
		tmp = 2.0 / (r * r)
	else:
		tmp = 3.0 - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.00015)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = Float64(3.0 - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.00015)
		tmp = 2.0 / (r * r);
	else
		tmp = 3.0 - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.00015], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], N[(3.0 - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.49999999999999987e-4

   1. Initial program 84.5%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if 1.49999999999999987e-4 < r

   1. Initial program 90.8%

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

   3. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

      2. lower-fma.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in v around inf

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

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} - \frac{9}{2} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right)} - \frac{9}{2} \]

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

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      7. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 73.9

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

   8. Applied rewrites 73.9%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 73.4%

       \[\leadsto \mathsf{fma}\left(-0.25 \cdot \left(w \cdot w\right), \color{blue}{r \cdot r}, 3\right) - 4.5 \]

   12. Taylor expanded in r around 0

       \[\leadsto 3 - \frac{9}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 33.7%

       \[\leadsto 3 - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.2% 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 86.4%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 59.5

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

5. Applied rewrites 59.5%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
6. Add Preprocessing

Alternative 13: 13.5% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ 3 - 4.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (- 3.0 4.5))

C

double code(double v, double w, double r) {
	return 3.0 - 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 - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return 3.0 - 4.5;
}

Python

def code(v, w, r):
	return 3.0 - 4.5

Julia

function code(v, w, r)
	return Float64(3.0 - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = 3.0 - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(3.0 - 4.5), $MachinePrecision]

Derivation

1. Initial program 86.4%

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

3. Taylor expanded in v around 0

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{-1}{4} \cdot v + \frac{3}{8}\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2} \]

   2. lower-fma.f64 86.4

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

5. Applied rewrites 86.4%

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

6. Taylor expanded in v around inf

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

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(\left(3 + 2 \cdot \frac{1}{{r}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} - \frac{9}{2} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right)} - \frac{9}{2} \]

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

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

   4. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

   5. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

   7. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + \left(3 + 2 \cdot \frac{1}{{r}^{2}}\right)\right) - \frac{9}{2} \]

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f64 86.4

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

8. Applied rewrites 86.4%

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

9. Taylor expanded in r around inf

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

11. Applied rewrites 42.4%

    \[\leadsto \mathsf{fma}\left(-0.25 \cdot \left(w \cdot w\right), \color{blue}{r \cdot r}, 3\right) - 4.5 \]

12. Taylor expanded in r around 0

    \[\leadsto 3 - \frac{9}{2} \]
13. Step-by-step derivation

14. Applied rewrites 17.1%

    \[\leadsto 3 - 4.5 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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))