Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 99.8%

Time: 6.9s

Alternatives: 13

Speedup: 1.5×

• 52.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lower--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 2: 94.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq -1 \cdot 10^{+55}\right):\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 -1e+55)))
     (- t_0 (fma (* (* w (* 0.25 r)) r) w 1.5))
     (- (* (* (* -0.375 (* w w)) r) r) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= -1e+55)) {
		tmp = t_0 - fma(((w * (0.25 * r)) * r), w, 1.5);
	} else {
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= -1e+55))
		tmp = Float64(t_0 - fma(Float64(Float64(w * Float64(0.25 * r)) * r), w, 1.5));
	else
		tmp = Float64(Float64(Float64(Float64(-0.375 * Float64(w * w)) * r) * r) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, -1e+55]], $MachinePrecision]], N[(t$95$0 - N[(N[(N[(w * N[(0.25 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0 or -1.00000000000000001e55 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 82.9%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 96.8%

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

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.00000000000000001e55

   1. Initial program 99.4%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 77.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty \lor \neg \left(\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (- t_0 (fma (* (* 0.25 (* r r)) w) w 1.5))
     (if (<= t_1 -5e+40)
       (- (* (* (* -0.375 (* w w)) r) r) 4.5)
       (- (/ (/ 2.0 r) r) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 - fma(((0.25 * (r * r)) * w), w, 1.5);
	} else if (t_1 <= -5e+40) {
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5;
	} else {
		tmp = ((2.0 / r) / r) - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 - fma(Float64(Float64(0.25 * Float64(r * r)) * w), w, 1.5));
	elseif (t_1 <= -5e+40)
		tmp = Float64(Float64(Float64(Float64(-0.375 * Float64(w * w)) * r) * r) - 4.5);
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+40], N[(N[(N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 81.3%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -5.00000000000000003e40

   1. Initial program 99.3%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 76.2%

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

   if -5.00000000000000003e40 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 83.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.3%

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

Alternative 4: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot -0.25\right) \cdot r\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0
         (-
          (-
           (+ 3.0 (/ 2.0 (* r r)))
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_0 (- INFINITY))
     (* (* (* (* w r) w) -0.25) r)
     (if (<= t_0 -5e+40)
       (- (* (* (* -0.375 (* w w)) r) r) 4.5)
       (- (/ (/ 2.0 r) r) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((w * r) * w) * -0.25) * r;
	} else if (t_0 <= -5e+40) {
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5;
	} else {
		tmp = ((2.0 / r) / r) - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (((w * r) * w) * -0.25) * r;
	} else if (t_0 <= -5e+40) {
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5;
	} else {
		tmp = ((2.0 / r) / r) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (((w * r) * w) * -0.25) * r
	elif t_0 <= -5e+40:
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5
	else:
		tmp = ((2.0 / r) / r) - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = 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)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(w * r) * w) * -0.25) * r);
	elseif (t_0 <= -5e+40)
		tmp = Float64(Float64(Float64(Float64(-0.375 * Float64(w * w)) * r) * r) - 4.5);
	else
		tmp = Float64(Float64(Float64(2.0 / r) / r) - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (((w * r) * w) * -0.25) * r;
	elseif (t_0 <= -5e+40)
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5;
	else
		tmp = ((2.0 / r) / r) - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * -0.25), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$0, -5e+40], N[(N[(N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - 1.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 81.3%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 91.2%

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

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -5.00000000000000003e40

   1. Initial program 99.3%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 76.2%

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

   if -5.00000000000000003e40 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 83.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.3%

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

Alternative 5: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot -0.25\right) \cdot r\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (* (* (* (* w r) w) -0.25) r)
     (if (<= t_1 -5e+40) (- (* (* (* -0.375 (* w w)) r) r) 4.5) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((w * r) * w) * -0.25) * r;
	} else if (t_1 <= -5e+40) {
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (((w * r) * w) * -0.25) * r;
	} else if (t_1 <= -5e+40) {
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (((w * r) * w) * -0.25) * r
	elif t_1 <= -5e+40:
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - 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)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(w * r) * w) * -0.25) * r);
	elseif (t_1 <= -5e+40)
		tmp = Float64(Float64(Float64(Float64(-0.375 * Float64(w * w)) * r) * r) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (((w * r) * w) * -0.25) * r;
	elseif (t_1 <= -5e+40)
		tmp = (((-0.375 * (w * w)) * r) * r) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * -0.25), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$1, -5e+40], N[(N[(N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 81.3%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 91.2%

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

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -5.00000000000000003e40

   1. Initial program 99.3%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 76.2%

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

   if -5.00000000000000003e40 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 83.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 6: 97.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1

   1. Initial program 85.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 98.3%

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

   if -1 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 82.9%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

Alternative 7: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot -0.25\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        -5e+40)
     (* (* (* (* w r) w) -0.25) r)
     (- 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))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+40) {
		tmp = (((w * r) * w) * -0.25) * r;
	} 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))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-5d+40)) then
        tmp = (((w * r) * w) * (-0.25d0)) * r
    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))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+40) {
		tmp = (((w * r) * w) * -0.25) * r;
	} 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))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+40:
		tmp = (((w * r) * w) * -0.25) * r
	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(Float64(3.0 + t_0) - 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) <= -5e+40)
		tmp = Float64(Float64(Float64(Float64(w * r) * w) * -0.25) * r);
	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))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+40)
		tmp = (((w * r) * w) * -0.25) * r;
	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[(N[(3.0 + t$95$0), $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], -5e+40], N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * -0.25), $MachinePrecision] * r), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -5.00000000000000003e40

   1. Initial program 85.3%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 79.3%

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

   if -5.00000000000000003e40 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 83.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.2

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

   5. Applied rewrites 97.2%

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

Alternative 8: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -20 \lor \neg \left(v \leq 4.9 \cdot 10^{-68}\right):\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot r\right) \cdot w, w \cdot r, 3 + t\_0\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -20.0) (not (<= v 4.9e-68)))
     (- t_0 (fma (* (* w (* 0.25 r)) r) w 1.5))
     (- (fma (* (* -0.375 r) w) (* w r) (+ 3.0 t_0)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -20.0) || !(v <= 4.9e-68)) {
		tmp = t_0 - fma(((w * (0.25 * r)) * r), w, 1.5);
	} else {
		tmp = fma(((-0.375 * r) * w), (w * r), (3.0 + t_0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -20.0) || !(v <= 4.9e-68))
		tmp = Float64(t_0 - fma(Float64(Float64(w * Float64(0.25 * r)) * r), w, 1.5));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * r) * w), Float64(w * r), Float64(3.0 + t_0)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -20.0], N[Not[LessEqual[v, 4.9e-68]], $MachinePrecision]], N[(t$95$0 - N[(N[(N[(w * N[(0.25 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * r), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision] + N[(3.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -20 or 4.89999999999999977e-68 < v

   1. Initial program 84.4%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.1

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 97.1%

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

   if -20 < v < 4.89999999999999977e-68

   1. Initial program 84.1%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -20 \lor \neg \left(v \leq 4.9 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot r\right) \cdot w, w \cdot r, 3 + \frac{2}{r \cdot r}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -20 \lor \neg \left(v \leq 4.9 \cdot 10^{-68}\right):\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot \left(w \cdot r\right), w \cdot r, 3 + t\_0\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -20.0) (not (<= v 4.9e-68)))
     (- t_0 (fma (* (* w (* 0.25 r)) r) w 1.5))
     (- (fma (* -0.375 (* w r)) (* w r) (+ 3.0 t_0)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -20.0) || !(v <= 4.9e-68)) {
		tmp = t_0 - fma(((w * (0.25 * r)) * r), w, 1.5);
	} else {
		tmp = fma((-0.375 * (w * r)), (w * r), (3.0 + t_0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -20.0) || !(v <= 4.9e-68))
		tmp = Float64(t_0 - fma(Float64(Float64(w * Float64(0.25 * r)) * r), w, 1.5));
	else
		tmp = Float64(fma(Float64(-0.375 * Float64(w * r)), Float64(w * r), Float64(3.0 + t_0)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -20.0], N[Not[LessEqual[v, 4.9e-68]], $MachinePrecision]], N[(t$95$0 - N[(N[(N[(w * N[(0.25 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision] + N[(3.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -20 or 4.89999999999999977e-68 < v

   1. Initial program 84.4%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.1

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 97.1%

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

   if -20 < v < 4.89999999999999977e-68

   1. Initial program 84.1%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -20 \lor \neg \left(v \leq 4.9 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot \left(w \cdot r\right), w \cdot r, 3 + \frac{2}{r \cdot r}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3.1 \cdot 10^{+76}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot r, \left(w \cdot w\right) \cdot r, 3 + t\_0\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 3.1e+76)
     (- t_0 (fma (* (* w (* 0.25 r)) r) w 1.5))
     (- (fma (* -0.375 r) (* (* w w) r) (+ 3.0 t_0)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.1e+76) {
		tmp = t_0 - fma(((w * (0.25 * r)) * r), w, 1.5);
	} else {
		tmp = fma((-0.375 * r), ((w * w) * r), (3.0 + t_0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3.1e+76)
		tmp = Float64(t_0 - fma(Float64(Float64(w * Float64(0.25 * r)) * r), w, 1.5));
	else
		tmp = Float64(fma(Float64(-0.375 * r), Float64(Float64(w * w) * r), Float64(3.0 + t_0)) - 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[r, 3.1e+76], N[(t$95$0 - N[(N[(N[(w * N[(0.25 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] + N[(3.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.10000000000000011e76

   1. Initial program 83.7%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 92.3%

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

   if 3.10000000000000011e76 < r

   1. Initial program 87.6%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 81.9%

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

Alternative 11: 50.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.108:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.108) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.108) {
		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 <= 0.108d0) 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 <= 0.108) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.108:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.108)
		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 <= 0.108)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.108], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 0.107999999999999999

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 0.107999999999999999 < r

   1. Initial program 85.1%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 27.3

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

   5. Applied rewrites 27.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 27.4%

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

Alternative 12: 57.1% 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 84.3%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 57.9

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

5. Applied rewrites 57.9%

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

Alternative 13: 14.2% 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 84.3%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 57.9

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

5. Applied rewrites 57.9%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 11.3%

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

Reproduce

herbie shell --seed 2024332 
(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))