Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 99.1%

Time: 10.6s

Alternatives: 13

Speedup: 1.4×

• 52.2% 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: 85.1% 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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{-1}, \frac{\left(r \cdot w\right) \cdot \left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right)}{1 - v}, \mathsf{fma}\left({r}^{-2}, 2, 3\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \frac{r}{1 - v}\right) \cdot \left(r \cdot w\right)\right) \cdot w\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= (* w w) 1e+14)
   (-
    (fma
     (/ r -1.0)
     (/ (* (* r w) (* (* 0.125 (fma -2.0 v 3.0)) w)) (- 1.0 v))
     (fma (pow r -2.0) 2.0 3.0))
    4.5)
   (-
    (-
     (+ (/ 2.0 (* r r)) 3.0)
     (* (* (* (fma -0.25 v 0.375) (/ r (- 1.0 v))) (* r w)) w))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((w * w) <= 1e+14) {
		tmp = fma((r / -1.0), (((r * w) * ((0.125 * fma(-2.0, v, 3.0)) * w)) / (1.0 - v)), fma(pow(r, -2.0), 2.0, 3.0)) - 4.5;
	} else {
		tmp = (((2.0 / (r * r)) + 3.0) - (((fma(-0.25, v, 0.375) * (r / (1.0 - v))) * (r * w)) * w)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(w * w) <= 1e+14)
		tmp = Float64(fma(Float64(r / -1.0), Float64(Float64(Float64(r * w) * Float64(Float64(0.125 * fma(-2.0, v, 3.0)) * w)) / Float64(1.0 - v)), fma((r ^ -2.0), 2.0, 3.0)) - 4.5);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(Float64(fma(-0.25, v, 0.375) * Float64(r / Float64(1.0 - v))) * Float64(r * w)) * w)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(w * w), $MachinePrecision], 1e+14], N[(N[(N[(r / -1.0), $MachinePrecision] * N[(N[(N[(r * w), $MachinePrecision] * N[(N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + N[(N[Power[r, -2.0], $MachinePrecision] * 2.0 + 3.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1e14

   1. Initial program 92.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. times-frac N/A

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

   4. Applied rewrites 99.9%

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

   if 1e14 < (*.f64 w w)

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 80.4

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

   5. Applied rewrites 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.5

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

   7. Applied rewrites 92.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

   9. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{r}{-1}, \frac{\left(r \cdot w\right) \cdot \left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right)}{1 - v}, \mathsf{fma}\left({r}^{-2}, 2, 3\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \frac{r}{1 - v}\right) \cdot \left(r \cdot w\right)\right) \cdot w\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ t_1 := t\_0 - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(-0.125 \cdot r\right) \cdot \left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot w\right) \cdot \frac{w}{1 - v}\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(0.375 \cdot w\right) \cdot r\right) \cdot r\right) \cdot w\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) 3.0))
        (t_1
         (-
          t_0
          (/ (* (* (* r (* w w)) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -0.25 r) w) (* r w))
     (if (<= t_1 -1e+22)
       (* (* (* -0.125 r) (* (* (fma -2.0 v 3.0) w) (/ w (- 1.0 v)))) r)
       (- (- t_0 (* (* (* (* 0.375 w) r) r) w)) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + 3.0;
	double t_1 = t_0 - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-0.25 * r) * w) * (r * w);
	} else if (t_1 <= -1e+22) {
		tmp = ((-0.125 * r) * ((fma(-2.0, v, 3.0) * w) * (w / (1.0 - v)))) * r;
	} else {
		tmp = (t_0 - ((((0.375 * w) * r) * r) * w)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	t_1 = Float64(t_0 - Float64(Float64(Float64(Float64(r * Float64(w * w)) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-0.25 * r) * w) * Float64(r * w));
	elseif (t_1 <= -1e+22)
		tmp = Float64(Float64(Float64(-0.125 * r) * Float64(Float64(fma(-2.0, v, 3.0) * w) * Float64(w / Float64(1.0 - v)))) * r);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(0.375 * w) * r) * r) * w)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e+22], N[(N[(N[(-0.125 * r), $MachinePrecision] * N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * w), $MachinePrecision] * N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(t$95$0 - N[(N[(N[(N[(0.375 * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $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 80.5%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 90.1%

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

   10. Applied rewrites 95.7%

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

   if -inf.0 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -1e22

   1. Initial program 99.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 81.0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 98.5%

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

   if -1e22 < (-.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 88.6%

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

   3. Taylor expanded in v around 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 93.4

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

   5. Applied rewrites 93.4%

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

   7. Applied rewrites 98.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(-0.125 \cdot r\right) \cdot \left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot w\right) \cdot \frac{w}{1 - v}\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\left(0.375 \cdot w\right) \cdot r\right) \cdot r\right) \cdot w\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{elif}\;t\_1 \leq -100000:\\ \;\;\;\;\left(\left(-0.125 \cdot r\right) \cdot \left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot w\right) \cdot \frac{w}{1 - v}\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (* r (* w w)) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -0.25 r) w) (* r w))
     (if (<= t_1 -100000.0)
       (* (* (* -0.125 r) (* (* (fma -2.0 v 3.0) w) (/ w (- 1.0 v)))) r)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-0.25 * r) * w) * (r * w);
	} else if (t_1 <= -100000.0) {
		tmp = ((-0.125 * r) * ((fma(-2.0, v, 3.0) * w) * (w / (1.0 - v)))) * r;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(r * Float64(w * w)) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-0.25 * r) * w) * Float64(r * w));
	elseif (t_1 <= -100000.0)
		tmp = Float64(Float64(Float64(-0.125 * r) * Float64(Float64(fma(-2.0, v, 3.0) * w) * Float64(w / Float64(1.0 - v)))) * r);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -100000.0], N[(N[(N[(-0.125 * r), $MachinePrecision] * N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * w), $MachinePrecision] * N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $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 80.5%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 90.1%

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

   10. Applied rewrites 95.7%

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

   if -inf.0 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -1e5

   1. Initial program 99.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 96.3%

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

   if -1e5 < (-.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 88.5%

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

   3. Taylor expanded in 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 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -100000:\\ \;\;\;\;\left(\left(-0.125 \cdot r\right) \cdot \left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot w\right) \cdot \frac{w}{1 - v}\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{elif}\;t\_1 \leq -100000:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-2, v, 3\right) \cdot w\right) \cdot w}{1 - v} \cdot \left(-0.125 \cdot \left(r \cdot r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (* r (* w w)) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))))
   (if (<= t_1 -2e+287)
     (* (* (* -0.25 r) w) (* r w))
     (if (<= t_1 -100000.0)
       (* (/ (* (* (fma -2.0 v 3.0) w) w) (- 1.0 v)) (* -0.125 (* r r)))
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -2e+287) {
		tmp = ((-0.25 * r) * w) * (r * w);
	} else if (t_1 <= -100000.0) {
		tmp = (((fma(-2.0, v, 3.0) * w) * w) / (1.0 - v)) * (-0.125 * (r * r));
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(r * Float64(w * w)) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= -2e+287)
		tmp = Float64(Float64(Float64(-0.25 * r) * w) * Float64(r * w));
	elseif (t_1 <= -100000.0)
		tmp = Float64(Float64(Float64(Float64(fma(-2.0, v, 3.0) * w) * w) / Float64(1.0 - v)) * Float64(-0.125 * Float64(r * r)));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+287], N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -100000.0], N[(N[(N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2.0000000000000002e287

   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 w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 87.3%

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

   10. Applied rewrites 93.1%

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

   if -2.0000000000000002e287 < (-.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))) < -1e5

   1. Initial program 99.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -1e5 < (-.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 88.5%

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

   3. Taylor expanded in 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 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -100000:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-2, v, 3\right) \cdot w\right) \cdot w}{1 - v} \cdot \left(-0.125 \cdot \left(r \cdot r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{elif}\;t\_1 \leq 3:\\ \;\;\;\;\left(3 - \left(\left(\left(0.375 \cdot w\right) \cdot r\right) \cdot r\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (* r (* w w)) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))))
   (if (<= t_1 -2e+107)
     (* (* (* -0.25 r) w) (* r w))
     (if (<= t_1 3.0)
       (- (- 3.0 (* (* (* (* 0.375 w) r) r) w)) 4.5)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -2e+107) {
		tmp = ((-0.25 * r) * w) * (r * w);
	} else if (t_1 <= 3.0) {
		tmp = (3.0 - ((((0.375 * w) * r) * r) * w)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (t_0 + 3.0d0) - ((((r * (w * w)) * r) * ((3.0d0 - (2.0d0 * v)) * 0.125d0)) / (1.0d0 - v))
    if (t_1 <= (-2d+107)) then
        tmp = (((-0.25d0) * r) * w) * (r * w)
    else if (t_1 <= 3.0d0) then
        tmp = (3.0d0 - ((((0.375d0 * w) * r) * r) * w)) - 4.5d0
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -2e+107) {
		tmp = ((-0.25 * r) * w) * (r * w);
	} else if (t_1 <= 3.0) {
		tmp = (3.0 - ((((0.375 * w) * r) * r) * w)) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))
	tmp = 0
	if t_1 <= -2e+107:
		tmp = ((-0.25 * r) * w) * (r * w)
	elif t_1 <= 3.0:
		tmp = (3.0 - ((((0.375 * w) * r) * r) * w)) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(r * Float64(w * w)) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= -2e+107)
		tmp = Float64(Float64(Float64(-0.25 * r) * w) * Float64(r * w));
	elseif (t_1 <= 3.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(0.375 * w) * r) * r) * w)) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v));
	tmp = 0.0;
	if (t_1 <= -2e+107)
		tmp = ((-0.25 * r) * w) * (r * w);
	elseif (t_1 <= 3.0)
		tmp = (3.0 - ((((0.375 * w) * r) * r) * w)) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+107], N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(3.0 - N[(N[(N[(N[(0.375 * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * w), $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))) < -1.9999999999999999e107

   1. Initial program 83.2%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 83.8%

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

   10. Applied rewrites 89.2%

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

   if -1.9999999999999999e107 < (-.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 92.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 87.8%

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

   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 88.1%

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

   3. Taylor expanded in 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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -2 \cdot 10^{+107}:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq 3:\\ \;\;\;\;\left(3 - \left(\left(\left(0.375 \cdot w\right) \cdot r\right) \cdot r\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -100000:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ t_0 3.0)
         (/ (* (* (* r (* w w)) r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))
        -100000.0)
     (* (* (* -0.25 r) w) (* r w))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -100000.0) {
		tmp = ((-0.25 * r) * w) * (r * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) - ((((r * (w * w)) * r) * ((3.0d0 - (2.0d0 * v)) * 0.125d0)) / (1.0d0 - v))) <= (-100000.0d0)) then
        tmp = (((-0.25d0) * r) * w) * (r * w)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -100000.0) {
		tmp = ((-0.25 * r) * w) * (r * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -100000.0:
		tmp = ((-0.25 * r) * w) * (r * w)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(r * Float64(w * w)) * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v))) <= -100000.0)
		tmp = Float64(Float64(Float64(-0.25 * r) * w) * Float64(r * w));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) - ((((r * (w * w)) * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -100000.0)
		tmp = ((-0.25 * r) * w) * (r * w);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -100000.0], N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -1e5

   1. Initial program 85.0%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 84.0%

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

   if -1e5 < (-.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 88.5%

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

   3. Taylor expanded in 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 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -100000:\\ \;\;\;\;\left(\left(-0.25 \cdot r\right) \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := r \cdot \left(w \cdot w\right)\\ \mathbf{if}\;\left(t\_0 + 3\right) - \frac{\left(t\_1 \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -100000:\\ \;\;\;\;\left(-0.25 \cdot r\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* r (* w w))))
   (if (<=
        (- (+ t_0 3.0) (/ (* (* t_1 r) (* (- 3.0 (* 2.0 v)) 0.125)) (- 1.0 v)))
        -100000.0)
     (* (* -0.25 r) t_1)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r * (w * w);
	double tmp;
	if (((t_0 + 3.0) - (((t_1 * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -100000.0) {
		tmp = (-0.25 * r) * t_1;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = r * (w * w)
    if (((t_0 + 3.0d0) - (((t_1 * r) * ((3.0d0 - (2.0d0 * v)) * 0.125d0)) / (1.0d0 - v))) <= (-100000.0d0)) then
        tmp = ((-0.25d0) * r) * t_1
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r * (w * w);
	double tmp;
	if (((t_0 + 3.0) - (((t_1 * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -100000.0) {
		tmp = (-0.25 * r) * t_1;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = r * (w * w)
	tmp = 0
	if ((t_0 + 3.0) - (((t_1 * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -100000.0:
		tmp = (-0.25 * r) * t_1
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(r * Float64(w * w))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(t_1 * r) * Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125)) / Float64(1.0 - v))) <= -100000.0)
		tmp = Float64(Float64(-0.25 * r) * t_1);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = r * (w * w);
	tmp = 0.0;
	if (((t_0 + 3.0) - (((t_1 * r) * ((3.0 - (2.0 * v)) * 0.125)) / (1.0 - v))) <= -100000.0)
		tmp = (-0.25 * r) * t_1;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(t$95$1 * r), $MachinePrecision] * N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -100000.0], N[(N[(-0.25 * r), $MachinePrecision] * t$95$1), $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))) < -1e5

   1. Initial program 85.0%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 81.1%

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

   if -1e5 < (-.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 88.5%

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

   3. Taylor expanded in 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 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot \left(\left(3 - 2 \cdot v\right) \cdot 0.125\right)}{1 - v} \leq -100000:\\ \;\;\;\;\left(-0.25 \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1 - v}\\ t_1 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-102}:\\ \;\;\;\;\left(t\_1 - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot t\_0\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot t\_0\right) \cdot \left(r \cdot w\right)\right) \cdot w\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ r (- 1.0 v))) (t_1 (+ (/ 2.0 (* r r)) 3.0)))
   (if (<= (* w w) 2e-102)
     (- (- t_1 (* (* (* (fma -0.25 v 0.375) w) (* r w)) t_0)) 4.5)
     (- (- t_1 (* (* (* (fma -0.25 v 0.375) t_0) (* r w)) w)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = r / (1.0 - v);
	double t_1 = (2.0 / (r * r)) + 3.0;
	double tmp;
	if ((w * w) <= 2e-102) {
		tmp = (t_1 - (((fma(-0.25, v, 0.375) * w) * (r * w)) * t_0)) - 4.5;
	} else {
		tmp = (t_1 - (((fma(-0.25, v, 0.375) * t_0) * (r * w)) * w)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(r / Float64(1.0 - v))
	t_1 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	tmp = 0.0
	if (Float64(w * w) <= 2e-102)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * w) * Float64(r * w)) * t_0)) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * t_0) * Float64(r * w)) * w)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e-102], N[(N[(t$95$1 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.99999999999999987e-102

   1. Initial program 91.4%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.4

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

   5. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.4

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

   7. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 1.99999999999999987e-102 < (*.f64 w w)

   1. Initial program 83.6%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 83.6

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

   5. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.3

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

   7. Applied rewrites 93.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-102}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \frac{r}{1 - v}\right) \cdot \left(r \cdot w\right)\right) \cdot w\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1 - v}\\ \mathbf{if}\;r \leq 10^{+121}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot t\_0\right) \cdot \left(r \cdot w\right)\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot t\_0\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ r (- 1.0 v))))
   (if (<= r 1e+121)
     (-
      (- (+ (/ 2.0 (* r r)) 3.0) (* (* (* (fma -0.25 v 0.375) t_0) (* r w)) w))
      4.5)
     (- (- 3.0 (* (* (* r (* w w)) (fma -0.25 v 0.375)) t_0)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = r / (1.0 - v);
	double tmp;
	if (r <= 1e+121) {
		tmp = (((2.0 / (r * r)) + 3.0) - (((fma(-0.25, v, 0.375) * t_0) * (r * w)) * w)) - 4.5;
	} else {
		tmp = (3.0 - (((r * (w * w)) * fma(-0.25, v, 0.375)) * t_0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(r / Float64(1.0 - v))
	tmp = 0.0
	if (r <= 1e+121)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(Float64(fma(-0.25, v, 0.375) * t_0) * Float64(r * w)) * w)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(r * Float64(w * w)) * fma(-0.25, v, 0.375)) * t_0)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1e+121], N[(N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.00000000000000004e121

   1. Initial program 88.1%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 88.1

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

   5. Applied rewrites 88.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.6

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

   7. Applied rewrites 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

   9. Applied rewrites 98.1%

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

   if 1.00000000000000004e121 < r

   1. Initial program 81.5%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 81.5

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

   5. Applied rewrites 81.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 93.0

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 93.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10^{+121}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \frac{r}{1 - v}\right) \cdot \left(r \cdot w\right)\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.46 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.46e+20)
   (fma (* (* -0.25 (* r r)) w) w (- (/ 2.0 (* r r)) 1.5))
   (- (- 3.0 (* (* (* r (* w w)) (fma -0.25 v 0.375)) (/ r (- 1.0 v)))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.46e+20) {
		tmp = fma(((-0.25 * (r * r)) * w), w, ((2.0 / (r * r)) - 1.5));
	} else {
		tmp = (3.0 - (((r * (w * w)) * fma(-0.25, v, 0.375)) * (r / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.46e+20)
		tmp = fma(Float64(Float64(-0.25 * Float64(r * r)) * w), w, Float64(Float64(2.0 / Float64(r * r)) - 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(r * Float64(w * w)) * fma(-0.25, v, 0.375)) * Float64(r / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.46e+20], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.46e20

   1. Initial program 87.1%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   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 93.3%

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

   if 1.46e20 < r

   1. Initial program 86.9%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 86.9

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

   5. Applied rewrites 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 94.1

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

   7. Applied rewrites 94.1%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 94.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.46 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 87.1%

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

   3. Taylor expanded in 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 59.9

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

   5. Applied rewrites 59.9%

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

   if 1.1499999999999999 < r

   1. Initial program 86.9%

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

   3. Taylor expanded in r around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 28.4%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} - 1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in w around 0

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

   1. 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 58.5

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

5. Applied rewrites 58.5%

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

Alternative 13: 14.1% accurate, 73.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 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 r around 0

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 71.8

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

5. Applied rewrites 71.8%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 15.6%

   \[\leadsto -1.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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))