Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.8%

Time: 14.8s

Alternatives: 11

Speedup: 1.8×

• 53.6% 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: 84.7% 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, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{v + -1}\right)\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+
   (+ 3.0 (/ 2.0 (* r r)))
   (* (* r w) (* (* r w) (/ (fma v -0.25 0.375) (+ v -1.0)))))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) + ((r * w) * ((r * w) * (fma(v, -0.25, 0.375) / (v + -1.0))))) - 4.5;
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(r * w) * Float64(Float64(r * w) * Float64(fma(v, -0.25, 0.375) / Float64(v + -1.0))))) - 4.5)
end

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift--.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-/.f64 97.1

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

4. Applied egg-rr 97.1%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. swap-sqr N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 88.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -4000000:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* r (* w w)))) (+ v -1.0)))
        -4000000.0)
     (* r (* r (* (* w w) -0.375)))
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -4000000.0) {
		tmp = r * (r * ((w * w) * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) + (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (r * (r * (w * w)))) / (v + (-1.0d0)))) <= (-4000000.0d0)) then
        tmp = r * (r * ((w * w) * (-0.375d0)))
    else
        tmp = t_0 + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -4000000.0) {
		tmp = r * (r * ((w * w) * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -4000000.0:
		tmp = r * (r * ((w * w) * -0.375))
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= -4000000.0)
		tmp = Float64(r * Float64(r * Float64(Float64(w * w) * -0.375)));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * (r * (w * w)))) / (v + -1.0))) <= -4000000.0)
		tmp = r * (r * ((w * w) * -0.375));
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4000000.0], N[(r * N[(r * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -4e6

   1. Initial program 91.9%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 84.8

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

   5. Simplified 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 93.0

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

   7. Applied egg-rr 93.0%

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

   8. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 88.5

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

   10. Simplified 88.5%

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

   if -4e6 < (-.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 77.0%

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

   3. Taylor expanded in w around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 95.2

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

   5. Simplified 95.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -4000000:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r \cdot w, 0.375 \cdot \left(r \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.1e-10)
   (- (/ 2.0 (* r r)) (fma (* r w) (* 0.375 (* r w)) 1.5))
   (-
    (+ 3.0 (* (* r (* w (* r w))) (/ (* 0.125 (fma v -2.0 3.0)) (+ v -1.0))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.1e-10) {
		tmp = (2.0 / (r * r)) - fma((r * w), (0.375 * (r * w)), 1.5);
	} else {
		tmp = (3.0 + ((r * (w * (r * w))) * ((0.125 * fma(v, -2.0, 3.0)) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.1e-10)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(r * w), Float64(0.375 * Float64(r * w)), 1.5));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.125 * fma(v, -2.0, 3.0)) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.1e-10], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.1e-10

   1. Initial program 82.2%

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

   3. 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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 97.4

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

   7. Applied egg-rr 97.4%

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

   if 2.1e-10 < r

   1. Initial program 89.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in r around inf

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

   7. Simplified 99.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r \cdot w, 0.375 \cdot \left(r \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 5 \cdot 10^{-6}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(r \cdot w, 0.375 \cdot \left(r \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 5e-6)
     (- t_0 (fma (* r w) (* 0.375 (* r w)) 1.5))
     (- (- (+ 3.0 t_0) (* (* r w) (* r (* w 0.25)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 5e-6) {
		tmp = t_0 - fma((r * w), (0.375 * (r * w)), 1.5);
	} else {
		tmp = ((3.0 + t_0) - ((r * w) * (r * (w * 0.25)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 5e-6)
		tmp = Float64(t_0 - fma(Float64(r * w), Float64(0.375 * Float64(r * w)), 1.5));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(r * w) * Float64(r * Float64(w * 0.25)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 5e-6], N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(0.375 * N[(r * w), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 5.00000000000000041e-6

   1. Initial program 86.9%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.1

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

   5. Simplified 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 98.0

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

   7. Applied egg-rr 98.0%

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

   if 5.00000000000000041e-6 < v

   1. Initial program 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 96.5

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

   4. Applied egg-rr 96.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. swap-sqr N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 99.8

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

   9. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r \cdot w, 0.375 \cdot \left(r \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 38000000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 38000000.0)
   (+ -1.5 (fma (* w (* (* r r) -0.25)) w (/ 2.0 (* r r))))
   (fma (* r (* r (* w w))) -0.375 -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 38000000.0) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, (2.0 / (r * r)));
	} else {
		tmp = fma((r * (r * (w * w))), -0.375, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 38000000.0)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, Float64(2.0 / Float64(r * r))));
	else
		tmp = fma(Float64(r * Float64(r * Float64(w * w))), -0.375, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 38000000.0], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.8e7

   1. Initial program 82.0%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. associate-*r/ N/A

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

   5. Simplified 92.5%

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

   if 3.8e7 < r

   1. Initial program 91.0%

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

   3. Taylor expanded in v around 0

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

      1. distribute-rgt-out-- N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-out-- N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

   5. Simplified 82.9%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 75.9%

      \[\leadsto \color{blue}{v \cdot \left(\frac{2}{v \cdot \left(r \cdot r\right)} + \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.125 + \frac{-0.375}{v}, \frac{-1.5}{v}\right)\right)} \]

   9. Taylor expanded in r around inf

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

   11. Simplified 82.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \mathsf{fma}\left(-v, 0.125, -0.375\right), -1.5\right)} \]

   12. Taylor expanded in v around 0

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

   14. Simplified 86.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 38000000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in v around 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. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 78.9

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

5. Simplified 78.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. swap-sqr N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 95.7

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

7. Applied egg-rr 95.7%

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

8. Final simplification 95.7%

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

Alternative 7: 65.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.9e-10) (/ (/ 2.0 r) r) (fma (* r (* r (* w w))) -0.375 -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.9e-10) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = fma((r * (r * (w * w))), -0.375, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.9e-10)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = fma(Float64(r * Float64(r * Float64(w * w))), -0.375, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.9e-10], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.8999999999999999e-10

   1. Initial program 82.2%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   5. Simplified 57.1%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 57.1

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

   7. Applied egg-rr 57.1%

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

   if 1.8999999999999999e-10 < r

   1. Initial program 89.9%

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

   3. Taylor expanded in v around 0

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

      1. distribute-rgt-out-- N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-out-- N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

   5. Simplified 80.7%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 74.0%

      \[\leadsto \color{blue}{v \cdot \left(\frac{2}{v \cdot \left(r \cdot r\right)} + \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.125 + \frac{-0.375}{v}, \frac{-1.5}{v}\right)\right)} \]

   9. Taylor expanded in r around inf

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

   11. Simplified 80.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \mathsf{fma}\left(-v, 0.125, -0.375\right), -1.5\right)} \]

   12. Taylor expanded in v around 0

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

   14. Simplified 85.9%

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

Alternative 8: 65.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.375, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.9e-10) (/ 2.0 (* r r)) (fma (* r (* r (* w w))) -0.375 -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.9e-10) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = fma((r * (r * (w * w))), -0.375, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.9e-10)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = fma(Float64(r * Float64(r * Float64(w * w))), -0.375, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.9e-10], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.8999999999999999e-10

   1. Initial program 82.2%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   5. Simplified 57.1%

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

   if 1.8999999999999999e-10 < r

   1. Initial program 89.9%

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

   3. Taylor expanded in v around 0

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

      1. distribute-rgt-out-- N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-out-- N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out-- N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

   5. Simplified 80.7%

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

   6. Taylor expanded in v around inf

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

   8. Simplified 74.0%

      \[\leadsto \color{blue}{v \cdot \left(\frac{2}{v \cdot \left(r \cdot r\right)} + \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), -0.125 + \frac{-0.375}{v}, \frac{-1.5}{v}\right)\right)} \]

   9. Taylor expanded in r around inf

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

   11. Simplified 80.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \mathsf{fma}\left(-v, 0.125, -0.375\right), -1.5\right)} \]

   12. Taylor expanded in v around 0

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

   14. Simplified 85.9%

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

Alternative 9: 50.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 2.1e-10) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.1e-10) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2.1d-10) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.1e-10) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.1e-10:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.1e-10)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.1e-10)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.1e-10], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 2.1e-10

   1. Initial program 82.2%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   5. Simplified 57.1%

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

   if 2.1e-10 < r

   1. Initial program 89.9%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.1

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

   5. Simplified 78.1%

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

   6. Taylor expanded in w around 0

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

   8. Simplified 18.4%

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

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   10. Step-by-step derivation

   11. Simplified 18.2%

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

Alternative 10: 57.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in w around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 52.5

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

5. Simplified 52.5%

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

6. Final simplification 52.5%

   \[\leadsto \frac{2}{r \cdot r} + -1.5 \]
7. Add Preprocessing

Alternative 11: 14.3% accurate, 73.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

3. Taylor expanded in v around 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. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 78.9

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

5. Simplified 78.9%

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

6. Taylor expanded in w around 0

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

8. Simplified 52.5%

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

9. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{-3}{2}} \]
10. Step-by-step derivation

11. Simplified 11.1%

    \[\leadsto \color{blue}{-1.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))