Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.8%

Time: 11.4s

Alternatives: 14

Speedup: 0.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.8%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 95.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-0.25 * Float64(r * r)) * w) * w);
	elseif (t_1 <= 3.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(Float64(w * r) * w) * r) * fma(-0.25, v, 0.375)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(3.0 - N[(N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $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 83.6%

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

   3. Taylor expanded in r around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 92.6%

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

   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))) < 3

   1. Initial program 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 99.4%

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

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

   1. Initial program 90.5%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 97.1%

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

Alternative 3: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(-0.125 \cdot r\right) \cdot \left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right) \cdot \frac{w}{1 - v}\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -0.25 (* r r)) w) w)
     (if (<= t_1 -1e+26)
       (* (* (* -0.125 r) (* (* (fma v -2.0 3.0) w) (/ w (- 1.0 v)))) r)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_1 <= -1e+26) {
		tmp = ((-0.125 * r) * ((fma(v, -2.0, 3.0) * w) * (w / (1.0 - v)))) * 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(t_0 + 3.0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-0.25 * Float64(r * r)) * w) * w);
	elseif (t_1 <= -1e+26)
		tmp = Float64(Float64(Float64(-0.125 * r) * Float64(Float64(fma(v, -2.0, 3.0) * w) * Float64(w / Float64(1.0 - v)))) * 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[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], If[LessEqual[t$95$1, -1e+26], N[(N[(N[(-0.125 * r), $MachinePrecision] * N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * w), $MachinePrecision] * N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 83.6%

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

   3. Taylor expanded in r around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 92.6%

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

   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))) < -1.00000000000000005e26

   1. Initial program 98.8%

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

   3. Taylor expanded in r around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 68.3

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 99.1%

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

   if -1.00000000000000005e26 < (-.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 90.9%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 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. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(-0.125 \cdot r\right) \cdot \left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right) \cdot \frac{w}{1 - v}\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot w\right) \cdot r\\ t_2 := \left(t\_0 + 3\right) - \frac{\left(t\_1 \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(-0.375 \cdot r\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (* (* w w) r))
        (t_2
         (-
          (+ t_0 3.0)
          (/ (* (* t_1 r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))))
   (if (<= t_2 (- INFINITY))
     (* (* (* -0.25 (* r r)) w) w)
     (if (<= t_2 -1e+26) (* (* -0.375 r) t_1) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double t_2 = (t_0 + 3.0) - (((t_1 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_2 <= -1e+26) {
		tmp = (-0.375 * r) * t_1;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double t_2 = (t_0 + 3.0) - (((t_1 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_2 <= -1e+26) {
		tmp = (-0.375 * r) * t_1;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (w * w) * r
	t_2 = (t_0 + 3.0) - (((t_1 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = ((-0.25 * (r * r)) * w) * w
	elif t_2 <= -1e+26:
		tmp = (-0.375 * r) * t_1
	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(w * w) * r)
	t_2 = Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(t_1 * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-0.25 * Float64(r * r)) * w) * w);
	elseif (t_2 <= -1e+26)
		tmp = Float64(Float64(-0.375 * r) * t_1);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (w * w) * r;
	t_2 = (t_0 + 3.0) - (((t_1 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = ((-0.25 * (r * r)) * w) * w;
	elseif (t_2 <= -1e+26)
		tmp = (-0.375 * r) * t_1;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(t$95$1 * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], If[LessEqual[t$95$2, -1e+26], N[(N[(-0.375 * r), $MachinePrecision] * t$95$1), $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 83.6%

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

   3. Taylor expanded in r around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 92.6%

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

   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))) < -1.00000000000000005e26

   1. Initial program 98.8%

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

   3. Taylor expanded in r around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. 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 41.7%

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

   10. Applied rewrites 68.5%

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

   if -1.00000000000000005e26 < (-.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 90.9%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 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. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(-0.375 \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.8%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.6 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot {r}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.6e-124)
   (* 2.0 (pow r -2.0))
   (-
    (-
     (+ (/ 2.0 (* r r)) 3.0)
     (/ (* (* (* (* w r) w) r) (fma -0.25 v 0.375)) (- 1.0 v)))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.6e-124) {
		tmp = 2.0 * pow(r, -2.0);
	} else {
		tmp = (((2.0 / (r * r)) + 3.0) - (((((w * r) * w) * r) * fma(-0.25, v, 0.375)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.6e-124)
		tmp = Float64(2.0 * (r ^ -2.0));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(Float64(Float64(Float64(w * r) * w) * r) * fma(-0.25, v, 0.375)) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.6e-124], N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.59999999999999996e-124

   1. Initial program 88.4%

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

   3. Taylor expanded in 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 59.3

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 59.4%

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

   if 5.59999999999999996e-124 < r

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 94.3

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 94.3

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

   7. Applied rewrites 94.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.6 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot {r}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(-0.25 \cdot r\right) \cdot \left(\left(w \cdot r\right) \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ t_0 3.0)
         (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        -1e+26)
     (* (* -0.25 r) (* (* w r) w))
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+26) {
		tmp = (-0.25 * r) * ((w * r) * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) - (((((w * w) * r) * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-1d+26)) then
        tmp = ((-0.25d0) * r) * ((w * r) * w)
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+26) {
		tmp = (-0.25 * r) * ((w * r) * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+26:
		tmp = (-0.25 * r) * ((w * r) * w)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -1e+26)
		tmp = Float64(Float64(-0.25 * r) * Float64(Float64(w * r) * w));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+26)
		tmp = (-0.25 * r) * ((w * r) * w);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+26], N[(N[(-0.25 * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -1.00000000000000005e26

   1. Initial program 86.1%

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

   3. Taylor expanded in r around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 84.4%

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

   10. Applied rewrites 83.5%

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

   if -1.00000000000000005e26 < (-.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 90.9%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 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. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(-0.25 \cdot r\right) \cdot \left(\left(w \cdot r\right) \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot w\right) \cdot r\\ \mathbf{if}\;\left(t\_0 + 3\right) - \frac{\left(t\_1 \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(-0.375 \cdot r\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* (* w w) r)))
   (if (<=
        (- (+ t_0 3.0) (/ (* (* t_1 r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        -1e+26)
     (* (* -0.375 r) t_1)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double tmp;
	if (((t_0 + 3.0) - (((t_1 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+26) {
		tmp = (-0.375 * r) * t_1;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (w * w) * r
    if (((t_0 + 3.0d0) - (((t_1 * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-1d+26)) then
        tmp = ((-0.375d0) * r) * t_1
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double tmp;
	if (((t_0 + 3.0) - (((t_1 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+26) {
		tmp = (-0.375 * r) * t_1;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (w * w) * r
	tmp = 0
	if ((t_0 + 3.0) - (((t_1 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+26:
		tmp = (-0.375 * r) * t_1
	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(w * w) * r)
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(t_1 * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -1e+26)
		tmp = Float64(Float64(-0.375 * r) * t_1);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (w * w) * r;
	tmp = 0.0;
	if (((t_0 + 3.0) - (((t_1 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -1e+26)
		tmp = (-0.375 * r) * t_1;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(t$95$1 * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+26], N[(N[(-0.375 * r), $MachinePrecision] * t$95$1), $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))) < -1.00000000000000005e26

   1. Initial program 86.1%

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

   3. Taylor expanded in r around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   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 78.0%

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

   10. Applied rewrites 82.0%

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

   if -1.00000000000000005e26 < (-.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 90.9%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 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. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(-0.375 \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.15e-130)
   (/ (/ 2.0 r) r)
   (-
    (-
     (+ (/ 2.0 (* r r)) 3.0)
     (/ (* (* (* (* w r) w) r) (fma -0.25 v 0.375)) (- 1.0 v)))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15e-130) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (((2.0 / (r * r)) + 3.0) - (((((w * r) * w) * r) * fma(-0.25, v, 0.375)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15e-130)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - Float64(Float64(Float64(Float64(Float64(w * r) * w) * r) * fma(-0.25, v, 0.375)) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15e-130], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.1500000000000001e-130

   1. Initial program 88.3%

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

   3. Taylor expanded in 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 59.0

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in r around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 59.1

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

   8. Applied rewrites 59.1%

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

   if 1.1500000000000001e-130 < r

   1. Initial program 89.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 94.3

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 94.3

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

   7. Applied rewrites 94.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.15 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, t\_0 - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.15e-94) t_0 (fma -0.25 (* (* (* w w) r) r) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.15e-94) {
		tmp = t_0;
	} else {
		tmp = fma(-0.25, (((w * w) * r) * r), (t_0 - 1.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.15e-94)
		tmp = t_0;
	else
		tmp = fma(-0.25, Float64(Float64(Float64(w * w) * r) * r), Float64(t_0 - 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.15e-94], t$95$0, N[(-0.25 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.1499999999999999e-94

   1. Initial program 87.7%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   if 2.1499999999999999e-94 < r

   1. Initial program 91.4%

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

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 93.2%

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

Alternative 11: 66.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, t\_0 - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.15e-94) t_0 (fma -0.375 (* (* (* w w) r) r) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.15e-94) {
		tmp = t_0;
	} else {
		tmp = fma(-0.375, (((w * w) * r) * r), (t_0 - 1.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.15e-94)
		tmp = t_0;
	else
		tmp = fma(-0.375, Float64(Float64(Float64(w * w) * r) * r), Float64(t_0 - 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.15e-94], t$95$0, N[(-0.375 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.1499999999999999e-94

   1. Initial program 87.7%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   if 2.1499999999999999e-94 < r

   1. Initial program 91.4%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 89.1%

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

Alternative 12: 48.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.0007:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.0007) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.0007) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.0007d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.0007) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.0007:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.0007)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.0007)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.0007], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 6.99999999999999993e-4

   1. Initial program 88.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   if 6.99999999999999993e-4 < r

   1. Initial program 91.5%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 22.7

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

   5. Applied rewrites 22.7%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 22.7%

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

Alternative 13: 56.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.8%

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

3. Taylor expanded in 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 58.7

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

5. Applied rewrites 58.7%

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

Alternative 14: 14.1% accurate, 73.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.8%

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

3. Taylor expanded in 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 58.7

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

5. Applied rewrites 58.7%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 11.5%

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

Reproduce

herbie shell --seed 2024242 
(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))