Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.7%

Time: 11.5s

Alternatives: 21

Speedup: 1.4×

• 53.1% 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: 84.8% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 94.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, t\_0 - 1.5\right)\\ \mathbf{elif}\;t\_1 \leq -2000000:\\ \;\;\;\;\left(\left(\frac{w}{1 - v} \cdot \left(\mathsf{fma}\left(-2, v, 3\right) \cdot w\right)\right) \cdot \left(-0.125 \cdot r\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \end{array} \end{array} \]

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)))))
   (if (<= t_1 (- INFINITY))
     (fma (* (* -0.25 (* r r)) w) w (- t_0 1.5))
     (if (<= t_1 -2000000.0)
       (* (* (* (/ w (- 1.0 v)) (* (fma -2.0 v 3.0) w)) (* -0.125 r)) r)
       (+ (/ (/ 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));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((-0.25 * (r * r)) * w), w, (t_0 - 1.5));
	} else if (t_1 <= -2000000.0) {
		tmp = (((w / (1.0 - v)) * (fma(-2.0, v, 3.0) * w)) * (-0.125 * r)) * r;
	} 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(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)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(-0.25 * Float64(r * r)) * w), w, Float64(t_0 - 1.5));
	elseif (t_1 <= -2000000.0)
		tmp = Float64(Float64(Float64(Float64(w / Float64(1.0 - v)) * Float64(fma(-2.0, v, 3.0) * w)) * Float64(-0.125 * r)) * r);
	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[(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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2000000.0], N[(N[(N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * v + 3.0), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + -1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 85.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 94.9%

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

   if -inf.0 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2e6

   1. Initial program 98.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 79.0

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

   5. Applied rewrites 79.0%

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

   7. Applied rewrites 96.5%

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

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

   1. Initial program 82.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.8

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

   9. Applied rewrites 94.8%

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

Alternative 3: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+247}:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r\\ \mathbf{elif}\;t\_0 \leq -2000000:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.375 \cdot w, w, \frac{-1.5}{r \cdot r}\right) \cdot r\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \end{array} \end{array} \]

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)))))
   (if (<= t_0 -5e+247)
     (* (* (* -0.25 (* w w)) r) r)
     (if (<= t_0 -2000000.0)
       (* (* (fma (* -0.375 w) w (/ -1.5 (* r r))) r) r)
       (+ (/ (/ 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));
	double tmp;
	if (t_0 <= -5e+247) {
		tmp = ((-0.25 * (w * w)) * r) * r;
	} else if (t_0 <= -2000000.0) {
		tmp = (fma((-0.375 * w), w, (-1.5 / (r * r))) * r) * r;
	} else {
		tmp = ((2.0 / r) / r) + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = 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)))
	tmp = 0.0
	if (t_0 <= -5e+247)
		tmp = Float64(Float64(Float64(-0.25 * Float64(w * w)) * r) * r);
	elseif (t_0 <= -2000000.0)
		tmp = Float64(Float64(fma(Float64(-0.375 * w), w, Float64(-1.5 / Float64(r * r))) * r) * r);
	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[(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]}, If[LessEqual[t$95$0, -5e+247], N[(N[(N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$0, -2000000.0], N[(N[(N[(N[(-0.375 * w), $MachinePrecision] * w + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + -1.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -5.00000000000000023e247

   1. Initial program 86.0%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 91.2%

       \[\leadsto \left(\left(-0.25 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r \]

   if -5.00000000000000023e247 < (-.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))) < -2e6

   1. Initial program 98.3%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 73.1%

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

   10. Applied rewrites 83.8%

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

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

   1. Initial program 82.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.8

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

   9. Applied rewrites 94.8%

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

Alternative 4: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+247}:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r\\ \mathbf{elif}\;t\_0 \leq -2000000:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot 3\right) \cdot \left(-0.125 \cdot r\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \end{array} \end{array} \]

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)))))
   (if (<= t_0 -5e+247)
     (* (* (* -0.25 (* w w)) r) r)
     (if (<= t_0 -2000000.0)
       (* (* (* (* w w) 3.0) (* -0.125 r)) r)
       (+ (/ (/ 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));
	double tmp;
	if (t_0 <= -5e+247) {
		tmp = ((-0.25 * (w * w)) * r) * r;
	} else if (t_0 <= -2000000.0) {
		tmp = (((w * w) * 3.0) * (-0.125 * r)) * r;
	} else {
		tmp = ((2.0 / r) / r) + -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 = (3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))
    if (t_0 <= (-5d+247)) then
        tmp = (((-0.25d0) * (w * w)) * r) * r
    else if (t_0 <= (-2000000.0d0)) then
        tmp = (((w * w) * 3.0d0) * ((-0.125d0) * r)) * r
    else
        tmp = ((2.0d0 / r) / r) + (-1.5d0)
    end if
    code = tmp
end function

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));
	double tmp;
	if (t_0 <= -5e+247) {
		tmp = ((-0.25 * (w * w)) * r) * r;
	} else if (t_0 <= -2000000.0) {
		tmp = (((w * w) * 3.0) * (-0.125 * r)) * r;
	} 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))
	tmp = 0
	if t_0 <= -5e+247:
		tmp = ((-0.25 * (w * w)) * r) * r
	elif t_0 <= -2000000.0:
		tmp = (((w * w) * 3.0) * (-0.125 * r)) * r
	else:
		tmp = ((2.0 / r) / r) + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = 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)))
	tmp = 0.0
	if (t_0 <= -5e+247)
		tmp = Float64(Float64(Float64(-0.25 * Float64(w * w)) * r) * r);
	elseif (t_0 <= -2000000.0)
		tmp = Float64(Float64(Float64(Float64(w * w) * 3.0) * Float64(-0.125 * r)) * r);
	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));
	tmp = 0.0;
	if (t_0 <= -5e+247)
		tmp = ((-0.25 * (w * w)) * r) * r;
	elseif (t_0 <= -2000000.0)
		tmp = (((w * w) * 3.0) * (-0.125 * r)) * r;
	else
		tmp = ((2.0 / r) / r) + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -5e+247], N[(N[(N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$0, -2000000.0], N[(N[(N[(N[(w * w), $MachinePrecision] * 3.0), $MachinePrecision] * N[(-0.125 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + -1.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -5.00000000000000023e247

   1. Initial program 86.0%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 91.2%

       \[\leadsto \left(\left(-0.25 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r \]

   if -5.00000000000000023e247 < (-.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))) < -2e6

   1. Initial program 98.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 82.6

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 81.2%

       \[\leadsto \left(\left(\left(w \cdot w\right) \cdot 3\right) \cdot \left(-0.125 \cdot r\right)\right) \cdot r \]

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

   1. Initial program 82.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.8

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

   9. Applied rewrites 94.8%

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

Alternative 5: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+247}:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r\\ \mathbf{elif}\;t\_1 \leq -2000000:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot 3\right) \cdot \left(-0.125 \cdot r\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))))
   (if (<= t_1 -5e+247)
     (* (* (* -0.25 (* w w)) r) r)
     (if (<= t_1 -2000000.0)
       (* (* (* (* w w) 3.0) (* -0.125 r)) r)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v));
	double tmp;
	if (t_1 <= -5e+247) {
		tmp = ((-0.25 * (w * w)) * r) * r;
	} else if (t_1 <= -2000000.0) {
		tmp = (((w * w) * 3.0) * (-0.125 * r)) * 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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))
    if (t_1 <= (-5d+247)) then
        tmp = (((-0.25d0) * (w * w)) * r) * r
    else if (t_1 <= (-2000000.0d0)) then
        tmp = (((w * w) * 3.0d0) * ((-0.125d0) * r)) * 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 t_1 = (3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v));
	double tmp;
	if (t_1 <= -5e+247) {
		tmp = ((-0.25 * (w * w)) * r) * r;
	} else if (t_1 <= -2000000.0) {
		tmp = (((w * w) * 3.0) * (-0.125 * r)) * r;
	} 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))
	tmp = 0
	if t_1 <= -5e+247:
		tmp = ((-0.25 * (w * w)) * r) * r
	elif t_1 <= -2000000.0:
		tmp = (((w * w) * 3.0) * (-0.125 * r)) * r
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(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)))
	tmp = 0.0
	if (t_1 <= -5e+247)
		tmp = Float64(Float64(Float64(-0.25 * Float64(w * w)) * r) * r);
	elseif (t_1 <= -2000000.0)
		tmp = Float64(Float64(Float64(Float64(w * w) * 3.0) * Float64(-0.125 * r)) * 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);
	t_1 = (3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v));
	tmp = 0.0;
	if (t_1 <= -5e+247)
		tmp = ((-0.25 * (w * w)) * r) * r;
	elseif (t_1 <= -2000000.0)
		tmp = (((w * w) * 3.0) * (-0.125 * r)) * 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]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5e+247], N[(N[(N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$1, -2000000.0], N[(N[(N[(N[(w * w), $MachinePrecision] * 3.0), $MachinePrecision] * N[(-0.125 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -5.00000000000000023e247

   1. Initial program 86.0%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 91.2%

       \[\leadsto \left(\left(-0.25 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r \]

   if -5.00000000000000023e247 < (-.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))) < -2e6

   1. Initial program 98.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 82.6

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 81.2%

       \[\leadsto \left(\left(\left(w \cdot w\right) \cdot 3\right) \cdot \left(-0.125 \cdot r\right)\right) \cdot r \]

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

   1. Initial program 82.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 6: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot w\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_0 \cdot r\right)}{1 - v}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+247}:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r\\ \mathbf{elif}\;t\_2 \leq -2000000:\\ \;\;\;\;\left(t\_0 \cdot -0.375\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w w) r))
        (t_1 (/ 2.0 (* r r)))
        (t_2
         (-
          (+ 3.0 t_1)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_0 r)) (- 1.0 v)))))
   (if (<= t_2 -5e+247)
     (* (* (* -0.25 (* w w)) r) r)
     (if (<= t_2 -2000000.0) (* (* t_0 -0.375) r) (- t_1 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = (w * w) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = (3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v));
	double tmp;
	if (t_2 <= -5e+247) {
		tmp = ((-0.25 * (w * w)) * r) * r;
	} else if (t_2 <= -2000000.0) {
		tmp = (t_0 * -0.375) * r;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (w * w) * r
    t_1 = 2.0d0 / (r * r)
    t_2 = (3.0d0 + t_1) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (t_0 * r)) / (1.0d0 - v))
    if (t_2 <= (-5d+247)) then
        tmp = (((-0.25d0) * (w * w)) * r) * r
    else if (t_2 <= (-2000000.0d0)) then
        tmp = (t_0 * (-0.375d0)) * r
    else
        tmp = t_1 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (w * w) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = (3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v));
	double tmp;
	if (t_2 <= -5e+247) {
		tmp = ((-0.25 * (w * w)) * r) * r;
	} else if (t_2 <= -2000000.0) {
		tmp = (t_0 * -0.375) * r;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (w * w) * r
	t_1 = 2.0 / (r * r)
	t_2 = (3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))
	tmp = 0
	if t_2 <= -5e+247:
		tmp = ((-0.25 * (w * w)) * r) * r
	elif t_2 <= -2000000.0:
		tmp = (t_0 * -0.375) * r
	else:
		tmp = t_1 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * w) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_0 * r)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_2 <= -5e+247)
		tmp = Float64(Float64(Float64(-0.25 * Float64(w * w)) * r) * r);
	elseif (t_2 <= -2000000.0)
		tmp = Float64(Float64(t_0 * -0.375) * r);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (w * w) * r;
	t_1 = 2.0 / (r * r);
	t_2 = (3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v));
	tmp = 0.0;
	if (t_2 <= -5e+247)
		tmp = ((-0.25 * (w * w)) * r) * r;
	elseif (t_2 <= -2000000.0)
		tmp = (t_0 * -0.375) * r;
	else
		tmp = t_1 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+247], N[(N[(N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$2, -2000000.0], N[(N[(t$95$0 * -0.375), $MachinePrecision] * r), $MachinePrecision], N[(t$95$1 - 1.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -5.00000000000000023e247

   1. Initial program 86.0%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in v around inf

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

   10. Applied rewrites 91.2%

       \[\leadsto \left(\left(-0.25 \cdot \left(w \cdot w\right)\right) \cdot r\right) \cdot r \]

   if -5.00000000000000023e247 < (-.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))) < -2e6

   1. Initial program 98.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 82.6

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 81.1%

       \[\leadsto \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot -0.375\right) \cdot r \]

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

   1. Initial program 82.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 7: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2000000:\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.375\right) \cdot w, w, 4.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -2000000.0)
     (+ t_0 (- 3.0 (fma (* (* (* r r) 0.375) w) w 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 tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = t_0 + (3.0 - fma((((r * r) * 0.375) * w), w, 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))
	tmp = 0.0
	if (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))) <= -2000000.0)
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(Float64(Float64(r * r) * 0.375) * w), w, 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]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(t$95$0 + N[(3.0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.375), $MachinePrecision] * w), $MachinePrecision] * w + 4.5), $MachinePrecision]), $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 #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))) < -2e6

   1. Initial program 88.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.9

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

   7. Applied rewrites 88.9%

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

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

   1. Initial program 82.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.8

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

   9. Applied rewrites 94.8%

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

Alternative 8: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2000000:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(\left(\left(-0.375 \cdot r\right) \cdot r\right) \cdot w, w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -2000000.0)
     (+ t_0 (fma (* (* (* -0.375 r) r) w) w -1.5))
     (+ (/ (/ 2.0 r) r) -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))) <= -2000000.0) {
		tmp = t_0 + fma((((-0.375 * r) * r) * w), w, -1.5);
	} else {
		tmp = ((2.0 / r) / r) + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (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))) <= -2000000.0)
		tmp = Float64(t_0 + fma(Float64(Float64(Float64(-0.375 * r) * r) * w), w, -1.5));
	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]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(t$95$0 + N[(N[(N[(N[(-0.375 * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * w + -1.5), $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 #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))) < -2e6

   1. Initial program 88.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 93.8%

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

   7. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 88.9

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

   9. Applied rewrites 88.9%

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

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

   1. Initial program 82.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.8

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

   9. Applied rewrites 94.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(\left(\left(-0.375 \cdot r\right) \cdot r\right) \cdot w, w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2000000:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, t\_0 - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -2000000.0)
     (fma (* (* -0.375 (* r r)) w) w (- t_0 1.5))
     (+ (/ (/ 2.0 r) r) -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))) <= -2000000.0) {
		tmp = fma(((-0.375 * (r * r)) * w), w, (t_0 - 1.5));
	} else {
		tmp = ((2.0 / r) / r) + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (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))) <= -2000000.0)
		tmp = fma(Float64(Float64(-0.375 * Float64(r * r)) * w), w, Float64(t_0 - 1.5));
	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]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000.0], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + N[(t$95$0 - 1.5), $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 #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))) < -2e6

   1. Initial program 88.3%

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

   3. Taylor expanded in v around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 88.9%

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

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

   1. Initial program 82.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.8

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

   9. Applied rewrites 94.8%

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

Alternative 10: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2000000:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -2000000.0)
     (* (* (* -0.375 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-2000000.0d0)) then
        tmp = (((-0.375d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0:
		tmp = ((-0.375 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(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))) <= -2000000.0)
		tmp = Float64(Float64(Float64(-0.375 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0)
		tmp = ((-0.375 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(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], -2000000.0], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 84.9%

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

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

   1. Initial program 82.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 11: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2000000:\\ \;\;\;\;\left(\left(\left(-0.375 \cdot r\right) \cdot r\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -2000000.0)
     (* (* (* (* -0.375 r) r) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = (((-0.375 * r) * r) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-2000000.0d0)) then
        tmp = ((((-0.375d0) * r) * r) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = (((-0.375 * r) * r) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0:
		tmp = (((-0.375 * r) * r) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(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))) <= -2000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.375 * r) * r) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0)
		tmp = (((-0.375 * r) * r) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(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], -2000000.0], N[(N[(N[(N[(-0.375 * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 79.6%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 84.9%

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

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

   1. Initial program 82.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 12: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2000000:\\ \;\;\;\;\left(\left(w \cdot \left(-0.25 \cdot r\right)\right) \cdot r\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -2000000.0)
     (* (* (* w (* -0.25 r)) r) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = ((w * (-0.25 * r)) * r) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-2000000.0d0)) then
        tmp = ((w * ((-0.25d0) * r)) * r) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = ((w * (-0.25 * r)) * r) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0:
		tmp = ((w * (-0.25 * r)) * r) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(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))) <= -2000000.0)
		tmp = Float64(Float64(Float64(w * Float64(-0.25 * r)) * r) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0)
		tmp = ((w * (-0.25 * r)) * r) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(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], -2000000.0], N[(N[(N[(w * N[(-0.25 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 79.6%

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

   10. Applied rewrites 80.8%

       \[\leadsto \left(\left(w \cdot \left(-0.25 \cdot r\right)\right) \cdot r\right) \cdot w \]

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

   1. Initial program 82.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 13: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2000000:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -2000000.0)
     (* (* (* -0.25 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-2000000.0d0)) then
        tmp = (((-0.25d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0:
		tmp = ((-0.25 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(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))) <= -2000000.0)
		tmp = Float64(Float64(Float64(-0.25 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2000000.0)
		tmp = ((-0.25 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(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], -2000000.0], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 79.6%

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

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

   1. Initial program 82.6%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

Alternative 14: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ \mathbf{if}\;v \leq -1720:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(t\_1, \frac{0.125}{v} - 0.25, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{t\_1 \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* (* w r) (* w r))))
   (if (<= v -1720.0)
     (+ t_0 (fma t_1 (- (/ 0.125 v) 0.25) -1.5))
     (- (- (+ 3.0 t_0) (/ (* t_1 (fma -0.25 v 0.375)) (- 1.0 v))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * r) * (w * r);
	double tmp;
	if (v <= -1720.0) {
		tmp = t_0 + fma(t_1, ((0.125 / v) - 0.25), -1.5);
	} else {
		tmp = ((3.0 + t_0) - ((t_1 * fma(-0.25, v, 0.375)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(w * r) * Float64(w * r))
	tmp = 0.0
	if (v <= -1720.0)
		tmp = Float64(t_0 + fma(t_1, Float64(Float64(0.125 / v) - 0.25), -1.5));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(t_1 * fma(-0.25, v, 0.375)) / Float64(1.0 - v))) - 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[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1720.0], N[(t$95$0 + N[(t$95$1 * N[(N[(0.125 / v), $MachinePrecision] - 0.25), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(t$95$1 * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1720

   1. Initial program 80.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around inf

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{0.125}{v} - 0.25, -1.5\right)} \]

   if -1720 < v

   1. Initial program 87.5%

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

   3. Taylor expanded in v around inf

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

      1. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. unswap-sqr N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 98.6

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

   8. Applied rewrites 98.6%

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

Alternative 15: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -5.8 \cdot 10^{+184}:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{0.125}{v} - 0.25, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(r \cdot \left(w \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right), w \cdot \frac{r}{1 - v}, 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -5.8e+184)
     (+ t_0 (fma (* (* w r) (* w r)) (- (/ 0.125 v) 0.25) -1.5))
     (+
      t_0
      (-
       3.0
       (fma (* r (* w (fma -0.25 v 0.375))) (* w (/ r (- 1.0 v))) 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -5.8e+184) {
		tmp = t_0 + fma(((w * r) * (w * r)), ((0.125 / v) - 0.25), -1.5);
	} else {
		tmp = t_0 + (3.0 - fma((r * (w * fma(-0.25, v, 0.375))), (w * (r / (1.0 - v))), 4.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -5.8e+184)
		tmp = Float64(t_0 + fma(Float64(Float64(w * r) * Float64(w * r)), Float64(Float64(0.125 / v) - 0.25), -1.5));
	else
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(r * Float64(w * fma(-0.25, v, 0.375))), Float64(w * Float64(r / Float64(1.0 - v))), 4.5)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -5.8e+184], N[(t$95$0 + N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 / v), $MachinePrecision] - 0.25), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(3.0 - N[(N[(r * N[(w * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -5.7999999999999998e184

   1. Initial program 73.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around inf

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

   7. Applied rewrites 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{0.125}{v} - 0.25, -1.5\right)} \]

   if -5.7999999999999998e184 < v

   1. Initial program 86.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 96.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 98.0%

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

Alternative 16: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -0.72 \lor \neg \left(v \leq 0.98\right):\\ \;\;\;\;t\_0 + \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{0.125}{v} - 0.25, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot w\right) \cdot r\right) \cdot w, -0.125 \cdot v - 0.375, -1.5\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -0.72) (not (<= v 0.98)))
     (+ t_0 (fma (* (* w r) (* w r)) (- (/ 0.125 v) 0.25) -1.5))
     (+ (fma (* (* (* r w) r) w) (- (* -0.125 v) 0.375) -1.5) t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -0.72) || !(v <= 0.98)) {
		tmp = t_0 + fma(((w * r) * (w * r)), ((0.125 / v) - 0.25), -1.5);
	} else {
		tmp = fma((((r * w) * r) * w), ((-0.125 * v) - 0.375), -1.5) + t_0;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -0.72) || !(v <= 0.98))
		tmp = Float64(t_0 + fma(Float64(Float64(w * r) * Float64(w * r)), Float64(Float64(0.125 / v) - 0.25), -1.5));
	else
		tmp = Float64(fma(Float64(Float64(Float64(r * w) * r) * w), Float64(Float64(-0.125 * v) - 0.375), -1.5) + t_0);
	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, -0.72], N[Not[LessEqual[v, 0.98]], $MachinePrecision]], N[(t$95$0 + N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 / v), $MachinePrecision] - 0.25), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(r * w), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * N[(N[(-0.125 * v), $MachinePrecision] - 0.375), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -0.71999999999999997 or 0.97999999999999998 < v

   1. Initial program 82.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around inf

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

   7. Applied rewrites 99.5%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{0.125}{v} - 0.25, -1.5\right)} \]

   if -0.71999999999999997 < v < 0.97999999999999998

   1. Initial program 88.1%

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

   3. Taylor expanded in v around 0

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

      1. associate--l+ N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. sub-neg N/A

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

      6. lower-+.f64 N/A

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(r \cdot w\right) \cdot r\right) \cdot w, -0.125 \cdot v - 0.375, -1.5\right) + \frac{2}{r \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -0.72 \lor \neg \left(v \leq 0.98\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), \frac{0.125}{v} - 0.25, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot w\right) \cdot r\right) \cdot w, -0.125 \cdot v - 0.375, -1.5\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 89.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(3 - \mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot 0.375\right) \cdot w, w, 4.5\right)\right)\\ \mathbf{elif}\;r \leq 2.55 \cdot 10^{+167}:\\ \;\;\;\;\left(3 - \frac{\left(-0.25 \cdot v\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{w \cdot \left(\left(\left(w \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot -0.125\right) \cdot r\right)}{1 - v} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.8e+151)
   (+ (/ 2.0 (* r r)) (- 3.0 (fma (* (* (* r r) 0.375) w) w 4.5)))
   (if (<= r 2.55e+167)
     (- (- 3.0 (/ (* (* -0.25 v) (* (* (* w w) r) r)) (- 1.0 v))) 4.5)
     (* (/ (* w (* (* (* w (fma v -2.0 3.0)) -0.125) r)) (- 1.0 v)) r))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.8e+151) {
		tmp = (2.0 / (r * r)) + (3.0 - fma((((r * r) * 0.375) * w), w, 4.5));
	} else if (r <= 2.55e+167) {
		tmp = (3.0 - (((-0.25 * v) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((w * (((w * fma(v, -2.0, 3.0)) * -0.125) * r)) / (1.0 - v)) * r;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.8e+151)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(3.0 - fma(Float64(Float64(Float64(r * r) * 0.375) * w), w, 4.5)));
	elseif (r <= 2.55e+167)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(-0.25 * v) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(Float64(w * Float64(Float64(Float64(w * fma(v, -2.0, 3.0)) * -0.125) * r)) / Float64(1.0 - v)) * r);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.8e+151], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(3.0 - N[(N[(N[(N[(r * r), $MachinePrecision] * 0.375), $MachinePrecision] * w), $MachinePrecision] * w + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 2.55e+167], N[(N[(3.0 - N[(N[(N[(-0.25 * v), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(w * N[(N[(N[(w * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.79999999999999987e151

   1. Initial program 85.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

   if 2.79999999999999987e151 < r < 2.55000000000000002e167

   1. Initial program 100.0%

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

   3. Taylor expanded in v around inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 100.0%

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

   if 2.55000000000000002e167 < r

   1. Initial program 74.9%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 63.8

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

   5. Applied rewrites 63.8%

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

   7. Applied rewrites 74.0%

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

   9. Applied rewrites 97.7%

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

Alternative 18: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(\left(\left(-0.375 \cdot r\right) \cdot r\right) \cdot w, w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.3e+31)
   (+ (/ 2.0 (* r r)) (fma (* (* (* -0.375 r) r) w) w -1.5))
   (- (- 3.0 (/ (* (* (* w r) (* w r)) (fma -0.25 v 0.375)) (- 1.0 v))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.3e+31) {
		tmp = (2.0 / (r * r)) + fma((((-0.375 * r) * r) * w), w, -1.5);
	} else {
		tmp = (3.0 - ((((w * r) * (w * r)) * fma(-0.25, v, 0.375)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.3e+31)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + fma(Float64(Float64(Float64(-0.375 * r) * r) * w), w, -1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r) * Float64(w * r)) * fma(-0.25, v, 0.375)) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.3e+31], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.375 * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * w + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.3e31

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 90.9%

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

   7. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 91.7

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

   9. Applied rewrites 91.7%

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

   if 1.3e31 < r

   1. Initial program 88.4%

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

   3. Taylor expanded in v around inf

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

      1. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. unswap-sqr N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f64 98.1

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 98.1%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(\left(\left(-0.375 \cdot r\right) \cdot r\right) \cdot w, w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(\left(\left(-0.375 \cdot r\right) \cdot r\right) \cdot w, w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{w \cdot \left(\left(\left(w \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot -0.125\right) \cdot r\right)}{1 - v} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 9e+146)
   (+ (/ 2.0 (* r r)) (fma (* (* (* -0.375 r) r) w) w -1.5))
   (* (/ (* w (* (* (* w (fma v -2.0 3.0)) -0.125) r)) (- 1.0 v)) r)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 9e+146) {
		tmp = (2.0 / (r * r)) + fma((((-0.375 * r) * r) * w), w, -1.5);
	} else {
		tmp = ((w * (((w * fma(v, -2.0, 3.0)) * -0.125) * r)) / (1.0 - v)) * r;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 9e+146)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + fma(Float64(Float64(Float64(-0.375 * r) * r) * w), w, -1.5));
	else
		tmp = Float64(Float64(Float64(w * Float64(Float64(Float64(w * fma(v, -2.0, 3.0)) * -0.125) * r)) / Float64(1.0 - v)) * r);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 9e+146], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.375 * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * w + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(w * N[(N[(N[(w * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.00000000000000051e146

   1. Initial program 85.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 92.6

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

   9. Applied rewrites 92.6%

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

   if 9.00000000000000051e146 < r

   1. Initial program 80.6%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 66.3

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

   5. Applied rewrites 66.3%

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

   7. Applied rewrites 74.3%

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

   9. Applied rewrites 92.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(\left(\left(-0.375 \cdot r\right) \cdot r\right) \cdot w, w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{w \cdot \left(\left(\left(w \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot -0.125\right) \cdot r\right)}{1 - v} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 58.0% 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 85.0%

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

3. Taylor expanded in w around 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 57.1

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} - 1.5 \]

5. Applied rewrites 57.1%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} - 1.5} \]
6. Add Preprocessing

Alternative 21: 44.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (/ 2.0 (* r r)))

C

double code(double v, double w, double r) {
	return 2.0 / (r * r);
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = 2.0d0 / (r * r)
end function

Java

public static double code(double v, double w, double r) {
	return 2.0 / (r * r);
}

Python

def code(v, w, r):
	return 2.0 / (r * r)

Julia

function code(v, w, r)
	return Float64(2.0 / Float64(r * r))
end

MATLAB

function tmp = code(v, w, r)
	tmp = 2.0 / (r * r);
end

Wolfram

code[v_, w_, r_] := N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in 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 43.1

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

5. Applied rewrites 43.1%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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))