Rosa's TurbineBenchmark

Percentage Accurate: 85.3% → 99.3%

Time: 10.8s

Alternatives: 13

Speedup: 1.6×

• 53.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\left(3 - \mathsf{fma}\left(\frac{\left(\left(w \cdot r\_m\right) \cdot r\_m\right) \cdot w}{1 - v}, \mathsf{fma}\left(-0.25, v, 0.375\right), 4.5\right)\right) + \frac{2}{r\_m \cdot r\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \frac{r\_m}{1 - v}\right) \cdot \left(w \cdot r\_m\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 5e+215)
   (+
    (- 3.0 (fma (/ (* (* (* w r_m) r_m) w) (- 1.0 v)) (fma -0.25 v 0.375) 4.5))
    (/ 2.0 (* r_m r_m)))
   (-
    (-
     3.0
     (* (* (* (* 0.125 (fma -2.0 v 3.0)) w) (/ r_m (- 1.0 v))) (* w r_m)))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 5e+215) {
		tmp = (3.0 - fma(((((w * r_m) * r_m) * w) / (1.0 - v)), fma(-0.25, v, 0.375), 4.5)) + (2.0 / (r_m * r_m));
	} else {
		tmp = (3.0 - ((((0.125 * fma(-2.0, v, 3.0)) * w) * (r_m / (1.0 - v))) * (w * r_m))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 5e+215)
		tmp = Float64(Float64(3.0 - fma(Float64(Float64(Float64(Float64(w * r_m) * r_m) * w) / Float64(1.0 - v)), fma(-0.25, v, 0.375), 4.5)) + Float64(2.0 / Float64(r_m * r_m)));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(0.125 * fma(-2.0, v, 3.0)) * w) * Float64(r_m / Float64(1.0 - v))) * Float64(w * r_m))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 5e+215], N[(N[(3.0 - N[(N[(N[(N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.0000000000000001e215

   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. 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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 97.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 97.9

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.9

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

   9. Applied rewrites 97.9%

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

   if 5.0000000000000001e215 < r

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+215}:\\ \;\;\;\;\left(3 - \mathsf{fma}\left(\frac{\left(\left(w \cdot r\right) \cdot r\right) \cdot w}{1 - v}, \mathsf{fma}\left(-0.25, v, 0.375\right), 4.5\right)\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(w \cdot r\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.2% accurate, 0.4× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1
         (-
          (+ 3.0 t_0)
          (/
           (* (* (* (* w w) r_m) r_m) (* (- 3.0 (* v 2.0)) 0.125))
           (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (- (- 3.0 (* (* (* 0.25 (* r_m r_m)) w) w)) 4.5)
     (if (<= t_1 3.0)
       (- (- 3.0 (* (* (* (* 0.375 r_m) w) w) r_m)) 4.5)
       (- t_0 1.5)))))

C

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = (3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	} else if (t_1 <= 3.0) {
		tmp = (3.0 - ((((0.375 * r_m) * w) * w) * r_m)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	t_1 = (3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5
	elif t_1 <= 3.0:
		tmp = (3.0 - ((((0.375 * r_m) * w) * w) * r_m)) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(3.0 - N[(N[(N[(N[(0.375 * r$95$m), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.5%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 86.8

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

   11. Applied rewrites 86.8%

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

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

   1. Initial program 87.8%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 82.0%

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

   12. Applied rewrites 82.0%

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

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

   1. Initial program 89.7%

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

   3. Taylor expanded in 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 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 91.1%

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

Alternative 3: 93.2% accurate, 0.4× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1
         (-
          (+ 3.0 t_0)
          (/
           (* (* (* (* w w) r_m) r_m) (* (- 3.0 (* v 2.0)) 0.125))
           (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (- (- 3.0 (* (* (* 0.25 (* r_m r_m)) w) w)) 4.5)
     (if (<= t_1 3.0)
       (- (- 3.0 (* (* 0.375 (* w r_m)) (* w r_m))) 4.5)
       (- t_0 1.5)))))

C

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = (3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	} else if (t_1 <= 3.0) {
		tmp = (3.0 - ((0.375 * (w * r_m)) * (w * r_m))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	t_1 = (3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5
	elif t_1 <= 3.0:
		tmp = (3.0 - ((0.375 * (w * r_m)) * (w * r_m))) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(3.0 - N[(N[(0.375 * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.5%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 86.8

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

   11. Applied rewrites 86.8%

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

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

   1. Initial program 87.8%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 82.0%

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

   12. Applied rewrites 82.0%

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

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

   1. Initial program 89.7%

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

   3. Taylor expanded in 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 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 91.1%

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

Alternative 4: 93.2% accurate, 0.4× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1
         (-
          (+ 3.0 t_0)
          (/
           (* (* (* (* w w) r_m) r_m) (* (- 3.0 (* v 2.0)) 0.125))
           (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (- (- 3.0 (* (* (* 0.25 (* r_m r_m)) w) w)) 4.5)
     (if (<= t_1 3.0)
       (- (- 3.0 (* (* (* 0.375 r_m) w) (* w r_m))) 4.5)
       (- t_0 1.5)))))

C

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = (3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	} else if (t_1 <= 3.0) {
		tmp = (3.0 - (((0.375 * r_m) * w) * (w * r_m))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	t_1 = (3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5
	elif t_1 <= 3.0:
		tmp = (3.0 - (((0.375 * r_m) * w) * (w * r_m))) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(3.0 - N[(N[(N[(0.375 * r$95$m), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.5%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 86.8

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

   11. Applied rewrites 86.8%

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

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

   1. Initial program 87.8%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 82.0%

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

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

   1. Initial program 89.7%

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

   3. Taylor expanded in 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 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 91.1%

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

Alternative 5: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(3.0 - N[(N[(N[Power[N[(w * r$95$m), $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$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. 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: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\_m\right) \cdot r\_m\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq 2.9999999343293795:\\ \;\;\;\;\left(3 - \left(\left(0.25 \cdot \left(r\_m \cdot r\_m\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* (* (* w w) r_m) r_m) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        2.9999999343293795)
     (- (- 3.0 (* (* (* 0.25 (* r_m r_m)) w) w)) 4.5)
     (- t_0 1.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= 2.9999999343293795) {
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    if (((3.0d0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= 2.9999999343293795d0) then
        tmp = (3.0d0 - (((0.25d0 * (r_m * r_m)) * w) * w)) - 4.5d0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= 2.9999999343293795) {
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	tmp = 0
	if ((3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= 2.9999999343293795:
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r_m) * r_m) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= 2.9999999343293795)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.25 * Float64(r_m * r_m)) * w) * w)) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 2.0 / (r_m * r_m);
	tmp = 0.0;
	if (((3.0 + t_0) - (((((w * w) * r_m) * r_m) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= 2.9999999343293795)
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.9999999343293795], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $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))) < 2.9999999343293795

   1. Initial program 81.1%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 73.0%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 75.8

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

   11. Applied rewrites 75.8%

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

   if 2.9999999343293795 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 87.6%

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

   3. Taylor expanded in 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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq 2.9999999343293795:\\ \;\;\;\;\left(3 - \left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 180000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(\left(0.375 \cdot \left(r\_m \cdot r\_m\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \frac{r\_m}{1 - v}\right) \cdot \left(w \cdot r\_m\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 180000.0)
   (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* (* 0.375 (* r_m r_m)) w) w)) 4.5)
   (-
    (-
     3.0
     (* (* (* (* 0.125 (fma -2.0 v 3.0)) w) (/ r_m (- 1.0 v))) (* w r_m)))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 180000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - (((0.375 * (r_m * r_m)) * w) * w)) - 4.5;
	} else {
		tmp = (3.0 - ((((0.125 * fma(-2.0, v, 3.0)) * w) * (r_m / (1.0 - v))) * (w * r_m))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 180000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(Float64(0.375 * Float64(r_m * r_m)) * w) * w)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(0.125 * fma(-2.0, v, 3.0)) * w) * Float64(r_m / Float64(1.0 - v))) * Float64(w * r_m))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 180000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.375 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.8e5

   1. Initial program 85.1%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 1.8e5 < r

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 180000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(w \cdot r\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 230000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(\left(0.375 \cdot \left(r\_m \cdot r\_m\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\_m\right)\right) \cdot \frac{r\_m}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 230000.0)
   (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* (* 0.375 (* r_m r_m)) w) w)) 4.5)
   (-
    (- 3.0 (* (* (* (fma -0.25 v 0.375) w) (* w r_m)) (/ r_m (- 1.0 v))))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 230000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - (((0.375 * (r_m * r_m)) * w) * w)) - 4.5;
	} else {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * w) * (w * r_m)) * (r_m / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 230000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(Float64(0.375 * Float64(r_m * r_m)) * w) * w)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * w) * Float64(w * r_m)) * Float64(r_m / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 230000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.375 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.3e5

   1. Initial program 85.1%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 2.3e5 < r

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.2

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

   10. Applied rewrites 98.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 230000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 2.6 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot \left(r\_m \cdot r\_m\right)\right) \cdot w, w, \frac{2}{r\_m \cdot r\_m} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.375 \cdot r\_m\right) \cdot w\right) \cdot w\right) \cdot r\_m\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 2.6e+124)
   (fma (* (* -0.25 (* r_m r_m)) w) w (- (/ 2.0 (* r_m r_m)) 1.5))
   (- (- 3.0 (* (* (* (* 0.375 r_m) w) w) r_m)) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2.6e+124) {
		tmp = fma(((-0.25 * (r_m * r_m)) * w), w, ((2.0 / (r_m * r_m)) - 1.5));
	} else {
		tmp = (3.0 - ((((0.375 * r_m) * w) * w) * r_m)) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 2.6e+124)
		tmp = fma(Float64(Float64(-0.25 * Float64(r_m * r_m)) * w), w, Float64(Float64(2.0 / Float64(r_m * r_m)) - 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(0.375 * r_m) * w) * w) * r_m)) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 2.6e+124], N[(N[(N[(-0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(0.375 * r$95$m), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.6e124

   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 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. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 89.7%

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

   if 2.6e124 < r

   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

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 65.2%

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

   10. Applied rewrites 91.2%

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

   12. Applied rewrites 91.2%

       \[\leadsto \left(3 - \left(\left(\left(0.375 \cdot r\right) \cdot w\right) \cdot w\right) \cdot \color{blue}{r}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.6 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.375 \cdot r\right) \cdot w\right) \cdot w\right) \cdot r\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 0.6:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r\_m \cdot r\_m\right)\right) \cdot w\right) \cdot w + \frac{2}{r\_m \cdot r\_m}\\ \mathbf{elif}\;r\_m \leq 2.6 \cdot 10^{+124}:\\ \;\;\;\;\left(3 - \left(\left(0.25 \cdot \left(r\_m \cdot r\_m\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.375 \cdot r\_m\right) \cdot w\right) \cdot w\right) \cdot r\_m\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 0.6)
   (+ (* (* (* -0.375 (* r_m r_m)) w) w) (/ 2.0 (* r_m r_m)))
   (if (<= r_m 2.6e+124)
     (- (- 3.0 (* (* (* 0.25 (* r_m r_m)) w) w)) 4.5)
     (- (- 3.0 (* (* (* (* 0.375 r_m) w) w) r_m)) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 0.6) {
		tmp = (((-0.375 * (r_m * r_m)) * w) * w) + (2.0 / (r_m * r_m));
	} else if (r_m <= 2.6e+124) {
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	} else {
		tmp = (3.0 - ((((0.375 * r_m) * w) * w) * r_m)) - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 0.6d0) then
        tmp = ((((-0.375d0) * (r_m * r_m)) * w) * w) + (2.0d0 / (r_m * r_m))
    else if (r_m <= 2.6d+124) then
        tmp = (3.0d0 - (((0.25d0 * (r_m * r_m)) * w) * w)) - 4.5d0
    else
        tmp = (3.0d0 - ((((0.375d0 * r_m) * w) * w) * r_m)) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 0.6) {
		tmp = (((-0.375 * (r_m * r_m)) * w) * w) + (2.0 / (r_m * r_m));
	} else if (r_m <= 2.6e+124) {
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	} else {
		tmp = (3.0 - ((((0.375 * r_m) * w) * w) * r_m)) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 0.6:
		tmp = (((-0.375 * (r_m * r_m)) * w) * w) + (2.0 / (r_m * r_m))
	elif r_m <= 2.6e+124:
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5
	else:
		tmp = (3.0 - ((((0.375 * r_m) * w) * w) * r_m)) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 0.6)
		tmp = Float64(Float64(Float64(Float64(-0.375 * Float64(r_m * r_m)) * w) * w) + Float64(2.0 / Float64(r_m * r_m)));
	elseif (r_m <= 2.6e+124)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.25 * Float64(r_m * r_m)) * w) * w)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(0.375 * r_m) * w) * w) * r_m)) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 0.6)
		tmp = (((-0.375 * (r_m * r_m)) * w) * w) + (2.0 / (r_m * r_m));
	elseif (r_m <= 2.6e+124)
		tmp = (3.0 - (((0.25 * (r_m * r_m)) * w) * w)) - 4.5;
	else
		tmp = (3.0 - ((((0.375 * r_m) * w) * w) * r_m)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 0.6], N[(N[(N[(N[(-0.375 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r$95$m, 2.6e+124], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(0.375 * r$95$m), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 0.599999999999999978

   1. Initial program 85.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 99.8%

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

   5. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 79.2

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in w around inf

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

   10. Applied rewrites 80.4%

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

   if 0.599999999999999978 < r < 2.6e124

   1. Initial program 80.4%

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

   3. Taylor expanded in v around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 97.0

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

   11. Applied rewrites 97.0%

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

   if 2.6e124 < r

   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

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 65.2%

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

   10. Applied rewrites 91.2%

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

   12. Applied rewrites 91.2%

       \[\leadsto \left(3 - \left(\left(\left(0.375 \cdot r\right) \cdot w\right) \cdot w\right) \cdot \color{blue}{r}\right) - 4.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.6:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w + \frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 2.6 \cdot 10^{+124}:\\ \;\;\;\;\left(3 - \left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.375 \cdot r\right) \cdot w\right) \cdot w\right) \cdot r\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.15:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m}\\ \mathbf{else}:\\ \;\;\;\;3 - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.15) (/ 2.0 (* r_m r_m)) (- 3.0 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.15) {
		tmp = 2.0 / (r_m * r_m);
	} else {
		tmp = 3.0 - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.15d0) then
        tmp = 2.0d0 / (r_m * r_m)
    else
        tmp = 3.0d0 - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.15) {
		tmp = 2.0 / (r_m * r_m);
	} else {
		tmp = 3.0 - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.15:
		tmp = 2.0 / (r_m * r_m)
	else:
		tmp = 3.0 - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.15)
		tmp = Float64(2.0 / Float64(r_m * r_m));
	else
		tmp = Float64(3.0 - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.15)
		tmp = 2.0 / (r_m * r_m);
	else
		tmp = 3.0 - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.15], N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision], N[(3.0 - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 85.1%

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

   3. Taylor expanded in r around 0

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

      1. 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 57.6

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

   5. Applied rewrites 57.6%

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

   if 1.1499999999999999 < r

   1. Initial program 83.9%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + 3 \cdot {r}^{2}}{{r}^{2}}} - \frac{9}{2} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{2 + 3 \cdot {r}^{2}}{\color{blue}{r \cdot r}} - \frac{9}{2} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}} - \frac{9}{2} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}} - \frac{9}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2 + 3 \cdot {r}^{2}}{r}}}{r} - \frac{9}{2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{3 \cdot {r}^{2} + 2}}{r}}{r} - \frac{9}{2} \]

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 14.9

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

   5. Applied rewrites 14.9%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 24.3%

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

Alternative 12: 58.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{2}{r\_m \cdot r\_m} - 1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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 58.7

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

5. Applied rewrites 58.7%

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

Alternative 13: 13.6% accurate, 18.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ 3 - 4.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (- 3.0 4.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return 3.0 - 4.5;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = 3.0d0 - 4.5d0
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return 3.0 - 4.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return 3.0 - 4.5

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(3.0 - 4.5), $MachinePrecision]

Derivation

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 r around 0

   \[\leadsto \color{blue}{\frac{2 + 3 \cdot {r}^{2}}{{r}^{2}}} - \frac{9}{2} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{2 + 3 \cdot {r}^{2}}{\color{blue}{r \cdot r}} - \frac{9}{2} \]

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}} - \frac{9}{2} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}} - \frac{9}{2} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{2 + 3 \cdot {r}^{2}}{r}}}{r} - \frac{9}{2} \]

   5. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{3 \cdot {r}^{2} + 2}}{r}}{r} - \frac{9}{2} \]

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 54.2

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

5. Applied rewrites 54.2%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 15.0%

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

Reproduce

herbie shell --seed 2024283 
(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))