Rosa's TurbineBenchmark

Percentage Accurate: 84.2% → 99.7%

Time: 9.4s

Alternatives: 14

Speedup: 1.6×

• 52.7% 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: 84.2% 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

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

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(Float64(2.0 / Float64(r * r)) + 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[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - 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]

Derivation

1. Initial program 83.0%

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

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;t\_1 \leq -500000:\\ \;\;\;\;\left(\left(\left(r \cdot w\right) \cdot -0.375\right) \cdot w\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

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 (- INFINITY))
     (- t_0 (fma (* (* 0.25 (* r r)) w) w 1.5))
     (if (<= t_1 -500000.0) (* (* (* (* r w) -0.375) w) 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 <= -((double) INFINITY)) {
		tmp = t_0 - fma(((0.25 * (r * r)) * w), w, 1.5);
	} else if (t_1 <= -500000.0) {
		tmp = (((r * w) * -0.375) * w) * 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 <= Float64(-Inf))
		tmp = Float64(t_0 - fma(Float64(Float64(0.25 * Float64(r * r)) * w), w, 1.5));
	elseif (t_1 <= -500000.0)
		tmp = Float64(Float64(Float64(Float64(r * w) * -0.375) * w) * r);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(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[(t$95$0 - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -500000.0], N[(N[(N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 80.0%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

   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))) < -5e5

   1. Initial program 95.3%

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

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

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

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

      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 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 66.8%

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

   10. Applied rewrites 72.8%

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

   if -5e5 < (-.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 83.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 - \frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (- -1.5 (* (/ (pow (* w r) 2.0) (- 1.0 v)) (* 0.125 (fma v -2.0 3.0))))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 - ((pow((w * r), 2.0) / (1.0 - v)) * (0.125 * fma(v, -2.0, 3.0))));
}

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64((Float64(w * r) ^ 2.0) / Float64(1.0 - v)) * Float64(0.125 * fma(v, -2.0, 3.0)))))
end

Wolfram

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

Derivation

1. Initial program 83.0%

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

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right)} \]
7. Add Preprocessing

Alternative 4: 89.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -500000:\\ \;\;\;\;\left(\left(\left(r \cdot w\right) \cdot -0.375\right) \cdot w\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -500000.0)
     (* (* (* (* r w) -0.375) w) r)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -500000.0) {
		tmp = (((r * w) * -0.375) * w) * r;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-500000.0d0)) then
        tmp = (((r * w) * (-0.375d0)) * w) * r
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -500000.0) {
		tmp = (((r * w) * -0.375) * w) * r;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -500000.0:
		tmp = (((r * w) * -0.375) * w) * r
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(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))) <= -500000.0)
		tmp = Float64(Float64(Float64(Float64(r * w) * -0.375) * w) * r);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -500000.0)
		tmp = (((r * w) * -0.375) * w) * r;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(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], -500000.0], N[(N[(N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision] * w), $MachinePrecision] * r), $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))) < -5e5

   1. Initial program 83.1%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      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 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 81.6%

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

   10. Applied rewrites 82.1%

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

   if -5e5 < (-.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 83.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 5: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -500000:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

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)))
        -500000.0)
     (* (* (* -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))) <= -500000.0) {
		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))) <= (-500000.0d0)) 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))) <= -500000.0) {
		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))) <= -500000.0:
		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))) <= -500000.0)
		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))) <= -500000.0)
		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], -500000.0], 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))) < -5e5

   1. Initial program 83.1%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      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 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 81.6%

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

   if -5e5 < (-.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 83.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 6: 88.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot w\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_0 \cdot r\right)}{1 - v} \leq -500000:\\ \;\;\;\;\left(r \cdot -0.375\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w w) r)) (t_1 (/ 2.0 (* r r))))
   (if (<=
        (- (+ 3.0 t_1) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_0 r)) (- 1.0 v)))
        -500000.0)
     (* (* r -0.375) t_0)
     (- 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 tmp;
	if (((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) <= -500000.0) {
		tmp = (r * -0.375) * t_0;
	} 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) :: tmp
    t_0 = (w * w) * r
    t_1 = 2.0d0 / (r * r)
    if (((3.0d0 + t_1) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (t_0 * r)) / (1.0d0 - v))) <= (-500000.0d0)) then
        tmp = (r * (-0.375d0)) * t_0
    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 tmp;
	if (((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) <= -500000.0) {
		tmp = (r * -0.375) * t_0;
	} 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)
	tmp = 0
	if ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) <= -500000.0:
		tmp = (r * -0.375) * t_0
	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))
	tmp = 0.0
	if (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))) <= -500000.0)
		tmp = Float64(Float64(r * -0.375) * t_0);
	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);
	tmp = 0.0;
	if (((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) <= -500000.0)
		tmp = (r * -0.375) * t_0;
	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]}, If[LessEqual[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], -500000.0], N[(N[(r * -0.375), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$1 - 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))) < -5e5

   1. Initial program 83.1%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      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 82.2

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 81.6%

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

   10. Applied rewrites 79.3%

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

   if -5e5 < (-.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 83.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 7: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;w \leq 2.1 \cdot 10^{+186}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\frac{\left(\left(w \cdot r\right) \cdot w\right) \cdot r}{1 - v}, \mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot \left(w \cdot r\right), w \cdot r, t\_0\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* r r)) 3.0)))
   (if (<= w 2.1e+186)
     (-
      t_0
      (fma (/ (* (* (* w r) w) r) (- 1.0 v)) (* (fma -2.0 v 3.0) 0.125) 4.5))
     (- (fma (* -0.375 (* w r)) (* w r) t_0) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) + 3.0;
	double tmp;
	if (w <= 2.1e+186) {
		tmp = t_0 - fma(((((w * r) * w) * r) / (1.0 - v)), (fma(-2.0, v, 3.0) * 0.125), 4.5);
	} else {
		tmp = fma((-0.375 * (w * r)), (w * r), t_0) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	tmp = 0.0
	if (w <= 2.1e+186)
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(Float64(w * r) * w) * r) / Float64(1.0 - v)), Float64(fma(-2.0, v, 3.0) * 0.125), 4.5));
	else
		tmp = Float64(fma(Float64(-0.375 * Float64(w * r)), Float64(w * r), t_0) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[w, 2.1e+186], N[(t$95$0 - N[(N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision] + t$95$0), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.1e186

   1. Initial program 84.7%

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 97.2

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

   6. Applied rewrites 97.2%

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

   if 2.1e186 < w

   1. Initial program 67.2%

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

   3. Taylor expanded in v around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 100.0%

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

Alternative 8: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 850000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}, \mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 850000.0)
   (- (/ 2.0 (* r r)) (fma (* (* 0.375 (* r r)) w) w 1.5))
   (-
    3.0
    (fma (* (* r w) (/ (* r w) (- 1.0 v))) (* (fma -2.0 v 3.0) 0.125) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 850000.0) {
		tmp = (2.0 / (r * r)) - fma(((0.375 * (r * r)) * w), w, 1.5);
	} else {
		tmp = 3.0 - fma(((r * w) * ((r * w) / (1.0 - v))), (fma(-2.0, v, 3.0) * 0.125), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 850000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(0.375 * Float64(r * r)) * w), w, 1.5));
	else
		tmp = Float64(3.0 - fma(Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))), Float64(fma(-2.0, v, 3.0) * 0.125), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 850000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.5e5

   1. Initial program 80.0%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 8.5e5 < r

   1. Initial program 96.0%

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 99.9%

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

       1. lift--.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lift--.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 9: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 850000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\frac{\left(\left(w \cdot r\right) \cdot w\right) \cdot r}{1 - v}, \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 850000.0)
   (- (/ 2.0 (* r r)) (fma (* (* 0.375 (* r r)) w) w 1.5))
   (- 3.0 (fma (/ (* (* (* w r) w) r) (- 1.0 v)) (fma v -0.25 0.375) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 850000.0) {
		tmp = (2.0 / (r * r)) - fma(((0.375 * (r * r)) * w), w, 1.5);
	} else {
		tmp = 3.0 - fma(((((w * r) * w) * r) / (1.0 - v)), fma(v, -0.25, 0.375), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 850000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(0.375 * Float64(r * r)) * w), w, 1.5));
	else
		tmp = Float64(3.0 - fma(Float64(Float64(Float64(Float64(w * r) * w) * r) / Float64(1.0 - v)), fma(v, -0.25, 0.375), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 850000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(v * -0.25 + 0.375), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.5e5

   1. Initial program 80.0%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 8.5e5 < r

   1. Initial program 96.0%

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 99.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. +-commutative N/A

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

       6. +-commutative N/A

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

       7. distribute-lft-in N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. metadata-eval N/A

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

       14. metadata-eval 99.9

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

   11. Applied rewrites 99.9%

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

Alternative 10: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot \left(w \cdot r\right), w \cdot r, t\_0 + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 3.3e-15)
     (- (fma (* -0.375 (* w r)) (* w r) (+ t_0 3.0)) 4.5)
     (- t_0 (fma (* (* w (* 0.25 r)) r) w 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 3.3e-15) {
		tmp = fma((-0.375 * (w * r)), (w * r), (t_0 + 3.0)) - 4.5;
	} else {
		tmp = t_0 - fma(((w * (0.25 * r)) * r), w, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 3.3e-15)
		tmp = Float64(fma(Float64(-0.375 * Float64(w * r)), Float64(w * r), Float64(t_0 + 3.0)) - 4.5);
	else
		tmp = Float64(t_0 - fma(Float64(Float64(w * Float64(0.25 * r)) * r), w, 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[v, 3.3e-15], N[(N[(N[(-0.375 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision] + N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - N[(N[(N[(w * N[(0.25 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 3.3e-15

   1. Initial program 83.9%

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

   3. Taylor expanded in v around 0

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 97.0%

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

   if 3.3e-15 < v

   1. Initial program 81.4%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.2

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

   5. Applied rewrites 94.2%

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

   7. Applied rewrites 97.6%

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

Alternative 11: 89.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 1.4 \cdot 10^{-17}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot \left(0.25 \cdot r\right)\right) \cdot r, w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 1.4e-17)
     (- t_0 (fma (* (* 0.375 (* r r)) w) w 1.5))
     (- t_0 (fma (* (* w (* 0.25 r)) r) w 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 1.4e-17) {
		tmp = t_0 - fma(((0.375 * (r * r)) * w), w, 1.5);
	} else {
		tmp = t_0 - fma(((w * (0.25 * r)) * r), w, 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 1.4e-17)
		tmp = Float64(t_0 - fma(Float64(Float64(0.375 * Float64(r * r)) * w), w, 1.5));
	else
		tmp = Float64(t_0 - fma(Float64(Float64(w * Float64(0.25 * r)) * r), w, 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[v, 1.4e-17], N[(t$95$0 - N[(N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(N[(w * N[(0.25 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 1.3999999999999999e-17

   1. Initial program 83.9%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if 1.3999999999999999e-17 < v

   1. Initial program 81.4%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.2

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

   5. Applied rewrites 94.2%

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

   7. Applied rewrites 97.6%

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

Alternative 12: 88.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 4.5 \cdot 10^{+147}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(0.25 \cdot r, \left(w \cdot w\right) \cdot r, 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 4.5e+147)
     (- t_0 (fma (* (* 0.375 (* r r)) w) w 1.5))
     (- t_0 (fma (* 0.25 r) (* (* w w) r) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 4.5e+147) {
		tmp = t_0 - fma(((0.375 * (r * r)) * w), w, 1.5);
	} else {
		tmp = t_0 - fma((0.25 * r), ((w * w) * r), 1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 4.5e+147)
		tmp = Float64(t_0 - fma(Float64(Float64(0.375 * Float64(r * r)) * w), w, 1.5));
	else
		tmp = Float64(t_0 - fma(Float64(0.25 * r), Float64(Float64(w * w) * 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[r, 4.5e+147], N[(t$95$0 - N[(N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(0.25 * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.50000000000000008e147

   1. Initial program 82.1%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if 4.50000000000000008e147 < r

   1. Initial program 95.2%

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

   3. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

   7. Applied rewrites 99.0%

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

Alternative 13: 56.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 62.3

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

5. Applied rewrites 62.3%

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

Alternative 14: 43.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 50.0

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

5. Applied rewrites 50.0%

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

Reproduce

herbie shell --seed 2024331 
(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))