Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 99.8%

Time: 12.8s

Alternatives: 15

Speedup: 1.8×

• 51.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. 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. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

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

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. unswap-sqr N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 91.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ t_0 3.0)
         (/ (* (* (- 3.0 (* 2.0 v)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))
        3.0)
     (- (- 3.0 (* (* (* 0.375 r) w) (* 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 tmp;
	if (((t_0 + 3.0) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= 3.0) {
		tmp = (3.0 - (((0.375 * r) * w) * (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) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) - ((((3.0d0 - (2.0d0 * v)) * 0.125d0) * (((w * w) * r) * r)) / (1.0d0 - v))) <= 3.0d0) then
        tmp = (3.0d0 - (((0.375d0 * r) * w) * (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 tmp;
	if (((t_0 + 3.0) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= 3.0) {
		tmp = (3.0 - (((0.375 * r) * w) * (r * w))) - 4.5;
	} 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) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= 3.0:
		tmp = (3.0 - (((0.375 * r) * w) * (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))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= 3.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.375 * r) * w) * Float64(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);
	tmp = 0.0;
	if (((t_0 + 3.0) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= 3.0)
		tmp = (3.0 - (((0.375 * r) * w) * (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]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], N[(N[(3.0 - N[(N[(N[(0.375 * r), $MachinePrecision] * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) - \frac{9}{2} \]
   6. 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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 74.8

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

   7. Applied rewrites 74.8%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 72.7%

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

   12. Applied rewrites 84.3%

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

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

   1. Initial program 86.4%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 90.9%

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

Alternative 5: 91.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ t_0 3.0)
         (/ (* (* (- 3.0 (* 2.0 v)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))
        3.0)
     (fma -0.375 (* (* r w) (* r w)) -1.5)
     (- 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) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= 3.0) {
		tmp = fma(-0.375, ((r * w) * (r * w)), -1.5);
	} 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(3.0 - Float64(2.0 * v)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= 3.0)
		tmp = fma(-0.375, Float64(Float64(r * w) * Float64(r * w)), -1.5);
	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]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 84.3%

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

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

   1. Initial program 86.4%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 90.9%

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

Alternative 6: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) - \frac{\left(\left(3 - 2 \cdot v\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -1000:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ t_0 3.0)
         (/ (* (* (- 3.0 (* 2.0 v)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))
        -1000.0)
     (* (* (* -0.375 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -1000.0) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) - ((((3.0d0 - (2.0d0 * v)) * 0.125d0) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-1000.0d0)) then
        tmp = (((-0.375d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -1000.0) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -1000.0:
		tmp = ((-0.375 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(3.0 - Float64(2.0 * v)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= -1000.0)
		tmp = Float64(Float64(Float64(-0.375 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) - ((((3.0 - (2.0 * v)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -1000.0)
		tmp = ((-0.375 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1000.0], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -1e3

   1. Initial program 82.6%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 77.4%

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

   if -1e3 < (-.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 84.5%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.4

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

   5. Applied rewrites 94.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(3 - 2 \cdot v\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -1000:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\\ \mathbf{if}\;r \leq 1.35 \cdot 10^{+148}:\\ \;\;\;\;\left(3 - \mathsf{fma}\left(\frac{w}{1 - v} \cdot \left(\left(r \cdot w\right) \cdot r\right), t\_0, 4.5\right)\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{r}{1 - v} \cdot \left(\left(w \cdot t\_0\right) \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* 0.125 (fma -2.0 v 3.0))))
   (if (<= r 1.35e+148)
     (+
      (- 3.0 (fma (* (/ w (- 1.0 v)) (* (* r w) r)) t_0 4.5))
      (/ 2.0 (* r r)))
     (- (- 3.0 (* (/ r (- 1.0 v)) (* (* w t_0) (* r w)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 0.125 * fma(-2.0, v, 3.0);
	double tmp;
	if (r <= 1.35e+148) {
		tmp = (3.0 - fma(((w / (1.0 - v)) * ((r * w) * r)), t_0, 4.5)) + (2.0 / (r * r));
	} else {
		tmp = (3.0 - ((r / (1.0 - v)) * ((w * t_0) * (r * w)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(0.125 * fma(-2.0, v, 3.0))
	tmp = 0.0
	if (r <= 1.35e+148)
		tmp = Float64(Float64(3.0 - fma(Float64(Float64(w / Float64(1.0 - v)) * Float64(Float64(r * w) * r)), t_0, 4.5)) + Float64(2.0 / Float64(r * r)));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r / Float64(1.0 - v)) * Float64(Float64(w * t_0) * Float64(r * w)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.35e+148], N[(N[(3.0 - N[(N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(w * t$95$0), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.35000000000000009e148

   1. Initial program 83.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 96.9

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

   8. Applied rewrites 96.9%

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

   if 1.35000000000000009e148 < r

   1. Initial program 83.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.35 \cdot 10^{+148}:\\ \;\;\;\;\left(3 - \mathsf{fma}\left(\frac{w}{1 - v} \cdot \left(\left(r \cdot w\right) \cdot r\right), 0.125 \cdot \mathsf{fma}\left(-2, v, 3\right), 4.5\right)\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{r}{1 - v} \cdot \left(\left(w \cdot \left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right)\right) \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.7

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.7

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 9: 91.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.8 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(0.375 \cdot w\right) \cdot \left(r \cdot r\right)\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{r}{1 - v} \cdot \left(\left(w \cdot \left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right)\right) \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.8e+20)
   (- (- (+ (/ 2.0 (* r r)) 3.0) (* (* (* 0.375 w) (* r r)) w)) 4.5)
   (-
    (- 3.0 (* (/ r (- 1.0 v)) (* (* w (* 0.125 (fma -2.0 v 3.0))) (* r w))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.8e+20) {
		tmp = (((2.0 / (r * r)) + 3.0) - (((0.375 * w) * (r * r)) * w)) - 4.5;
	} else {
		tmp = (3.0 - ((r / (1.0 - v)) * ((w * (0.125 * fma(-2.0, v, 3.0))) * (r * w)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.8e+20)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(Float64(0.375 * w) * Float64(r * r)) * w)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r / Float64(1.0 - v)) * Float64(Float64(w * Float64(0.125 * fma(-2.0, v, 3.0))) * Float64(r * w)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.8e+20], N[(N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(0.375 * w), $MachinePrecision] * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.8e20

   1. Initial program 82.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) - \frac{9}{2} \]
   6. 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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 87.7

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

   7. Applied rewrites 87.7%

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

   9. Applied rewrites 87.6%

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

   if 1.8e20 < r

   1. Initial program 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 98.2%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.8 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(0.375 \cdot w\right) \cdot \left(r \cdot r\right)\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{r}{1 - v} \cdot \left(\left(w \cdot \left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right)\right) \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, r \cdot w, -1.5\right) + t\_0\\ \mathbf{if}\;v \leq -1.7 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(r \cdot w\right) \cdot \left(r \cdot w\right), t\_0 - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (+ (fma (* (* -0.25 r) w) (* r w) -1.5) t_0)))
   (if (<= v -1.7e+74)
     t_1
     (if (<= v 0.75) (fma -0.375 (* (* r w) (* r w)) (- t_0 1.5)) t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = fma(((-0.25 * r) * w), (r * w), -1.5) + t_0;
	double tmp;
	if (v <= -1.7e+74) {
		tmp = t_1;
	} else if (v <= 0.75) {
		tmp = fma(-0.375, ((r * w) * (r * w)), (t_0 - 1.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(fma(Float64(Float64(-0.25 * r) * w), Float64(r * w), -1.5) + t_0)
	tmp = 0.0
	if (v <= -1.7e+74)
		tmp = t_1;
	elseif (v <= 0.75)
		tmp = fma(-0.375, Float64(Float64(r * w) * Float64(r * w)), Float64(t_0 - 1.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(r * w), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[v, -1.7e+74], t$95$1, If[LessEqual[v, 0.75], N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.7e74 or 0.75 < v

   1. Initial program 76.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.0

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

   7. Applied rewrites 88.0%

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

   9. Applied rewrites 99.7%

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

   if -1.7e74 < v < 0.75

   1. Initial program 89.7%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.7 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, r \cdot w, -1.5\right) + \frac{2}{r \cdot r}\\ \mathbf{elif}\;v \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(r \cdot w\right) \cdot \left(r \cdot w\right), \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, r \cdot w, -1.5\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} - 1.5\\ \mathbf{if}\;v \leq 1.4 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(r \cdot w\right) \cdot \left(r \cdot w\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (* r r)) 1.5)))
   (if (<= v 1.4e+128)
     (fma -0.375 (* (* r w) (* r w)) t_0)
     (fma -0.25 (* (* (* w w) r) r) t_0))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) - 1.5;
	double tmp;
	if (v <= 1.4e+128) {
		tmp = fma(-0.375, ((r * w) * (r * w)), t_0);
	} else {
		tmp = fma(-0.25, (((w * w) * r) * r), t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) - 1.5)
	tmp = 0.0
	if (v <= 1.4e+128)
		tmp = fma(-0.375, Float64(Float64(r * w) * Float64(r * w)), t_0);
	else
		tmp = fma(-0.25, Float64(Float64(Float64(w * w) * r) * r), t_0);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]}, If[LessEqual[v, 1.4e+128], N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(-0.25 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 1.39999999999999991e128

   1. Initial program 85.4%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 95.0%

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

   if 1.39999999999999991e128 < v

   1. Initial program 72.8%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 94.2%

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

Alternative 12: 86.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;r \leq 0.9:\\ \;\;\;\;\mathsf{fma}\left(-0.375, t\_0, \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;r \leq 1.25 \cdot 10^{+154}:\\ \;\;\;\;\left(3 - \left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, t\_0, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))))
   (if (<= r 0.9)
     (fma -0.375 t_0 (/ 2.0 (* r r)))
     (if (<= r 1.25e+154)
       (- (- 3.0 (* (* (* 0.25 (* r r)) w) w)) 4.5)
       (fma -0.375 t_0 -1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if (r <= 0.9) {
		tmp = fma(-0.375, t_0, (2.0 / (r * r)));
	} else if (r <= 1.25e+154) {
		tmp = (3.0 - (((0.25 * (r * r)) * w) * w)) - 4.5;
	} else {
		tmp = fma(-0.375, t_0, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if (r <= 0.9)
		tmp = fma(-0.375, t_0, Float64(2.0 / Float64(r * r)));
	elseif (r <= 1.25e+154)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.25 * Float64(r * r)) * w) * w)) - 4.5);
	else
		tmp = fma(-0.375, t_0, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 0.9], N[(-0.375 * t$95$0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.25e+154], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(-0.375 * t$95$0 + -1.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 0.900000000000000022

   1. Initial program 82.3%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in r around 0

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

   10. Applied rewrites 88.7%

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

   if 0.900000000000000022 < r < 1.25000000000000001e154

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) - \frac{9}{2} \]
   6. 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. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.0

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

   7. Applied rewrites 88.0%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 87.2%

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

   11. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 97.1

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

   13. Applied rewrites 97.1%

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

   if 1.25000000000000001e154 < r

   1. Initial program 82.8%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 85.4%

       \[\leadsto \mathsf{fma}\left(-0.375, \left(r \cdot w\right) \cdot \left(r \cdot w\right), -1.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 93.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in v around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. associate-+l+ N/A

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

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

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 81.8%

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

7. Applied rewrites 92.1%

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

Alternative 14: 57.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 56.7

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

5. Applied rewrites 56.7%

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

Alternative 15: 44.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in 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 45.5

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

5. Applied rewrites 45.5%

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

Reproduce

herbie shell --seed 2024270 
(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))