Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.8%

Time: 11.6s

Alternatives: 17

Speedup: 1.6×

• 53.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 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 82.6%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. 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 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 93.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (- 3.0 (* v 2.0)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -0.25 (* r r)) w) w)
     (if (<= t_1 3.0)
       (- (fma (* (* (* -0.375 w) r) w) r 3.0) 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) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_1 <= 3.0) {
		tmp = fma((((-0.375 * w) * r) * w), r, 3.0) - 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(3.0 - Float64(v * 2.0)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / 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(fma(Float64(Float64(Float64(-0.375 * w) * r) * w), r, 3.0) - 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[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(N[(N[(N[(-0.375 * w), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] * r + 3.0), $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 83.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 88.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 92.0%

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

   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 91.9%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 80.8%

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

   10. Applied rewrites 84.2%

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

   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 78.2%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 94.2%

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

Alternative 3: 93.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (- 3.0 (* v 2.0)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -0.25 (* r r)) w) w)
     (if (<= t_1 3.0)
       (- (fma (* (* -0.375 w) (* w r)) r 3.0) 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) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_1 <= 3.0) {
		tmp = fma(((-0.375 * w) * (w * r)), r, 3.0) - 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(3.0 - Float64(v * 2.0)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / 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(fma(Float64(Float64(-0.375 * w) * Float64(w * r)), r, 3.0) - 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[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(N[(N[(-0.375 * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * r + 3.0), $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 83.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 88.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 92.0%

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

   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 91.9%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 80.8%

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

   10. Applied rewrites 84.1%

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

   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 78.2%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 94.2%

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

Alternative 4: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_1 \leq -50000000:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot -0.375\right) \cdot r\right) \cdot r - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (- 3.0 (* v 2.0)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -0.25 (* r r)) w) w)
     (if (<= t_1 -50000000.0)
       (- (* (* (* (* w w) -0.375) r) r) 4.5)
       (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_1 <= -50000000.0) {
		tmp = ((((w * w) * -0.375) * r) * r) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_1 <= -50000000.0) {
		tmp = ((((w * w) * -0.375) * r) * r) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((-0.25 * (r * r)) * w) * w
	elif t_1 <= -50000000.0:
		tmp = ((((w * w) * -0.375) * r) * r) - 4.5
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / 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 <= -50000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(w * w) * -0.375) * r) * r) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((-0.25 * (r * r)) * w) * w;
	elseif (t_1 <= -50000000.0)
		tmp = ((((w * w) * -0.375) * r) * r) - 4.5;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], If[LessEqual[t$95$1, -50000000.0], N[(N[(N[(N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 #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 83.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 88.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 92.0%

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

   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))) < -5e7

   1. Initial program 99.6%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 63.9%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 75.2%

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

   if -5e7 < (-.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 79.3%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -50000000:\\ \;\;\;\;\left(\left(\left(w \cdot w\right) \cdot -0.375\right) \cdot r\right) \cdot r - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;t\_1 \leq -50000000:\\ \;\;\;\;\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (+ t_0 3.0)
          (/ (* (* (- 3.0 (* v 2.0)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))))
   (if (<= t_1 (- INFINITY))
     (* (* (* -0.25 (* r r)) w) w)
     (if (<= t_1 -50000000.0) (* (* -0.375 (* r r)) (* w w)) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_1 <= -50000000.0) {
		tmp = (-0.375 * (r * r)) * (w * w);
	} 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 = (t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else if (t_1 <= -50000000.0) {
		tmp = (-0.375 * (r * r)) * (w * w);
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((-0.25 * (r * r)) * w) * w
	elif t_1 <= -50000000.0:
		tmp = (-0.375 * (r * r)) * (w * w)
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / 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 <= -50000000.0)
		tmp = Float64(Float64(-0.375 * Float64(r * r)) * Float64(w * w));
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((-0.25 * (r * r)) * w) * w;
	elseif (t_1 <= -50000000.0)
		tmp = (-0.375 * (r * r)) * (w * w);
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], If[LessEqual[t$95$1, -50000000.0], N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $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 83.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 88.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 92.0%

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

   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))) < -5e7

   1. Initial program 99.6%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 57.9%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 46.6%

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

   if -5e7 < (-.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 79.3%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -50000000:\\ \;\;\;\;\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 82.6%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 7: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -50000000:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ t_0 3.0)
         (/ (* (* (- 3.0 (* v 2.0)) 0.125) (* (* (* w w) r) r)) (- 1.0 v)))
        -50000000.0)
     (* (* (* -0.25 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -50000000.0) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) - ((((3.0d0 - (v * 2.0d0)) * 0.125d0) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-50000000.0d0)) then
        tmp = (((-0.25d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -50000000.0) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -50000000.0:
		tmp = ((-0.25 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= -50000000.0)
		tmp = Float64(Float64(Float64(-0.25 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) - ((((3.0 - (v * 2.0)) * 0.125) * (((w * w) * r) * r)) / (1.0 - v))) <= -50000000.0)
		tmp = ((-0.25 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -50000000.0], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -5e7

   1. Initial program 87.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 75.7%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 74.7%

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

   if -5e7 < (-.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 79.3%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(3 - v \cdot 2\right) \cdot 0.125\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -50000000:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{1 - v}\\ t_1 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;r \leq 4.6 \cdot 10^{+58}:\\ \;\;\;\;\left(t\_1 - \left(\left(\left(w \cdot r\right) \cdot t\_0\right) \cdot \left(0.125 \cdot w\right)\right) \cdot \mathsf{fma}\left(v, -2, 3\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot t\_0\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ r (- 1.0 v))) (t_1 (+ (/ 2.0 (* r r)) 3.0)))
   (if (<= r 4.6e+58)
     (- (- t_1 (* (* (* (* w r) t_0) (* 0.125 w)) (fma v -2.0 3.0))) 4.5)
     (- (- t_1 (* (* (* (* 0.125 (fma -2.0 v 3.0)) w) (* w r)) t_0)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = r / (1.0 - v);
	double t_1 = (2.0 / (r * r)) + 3.0;
	double tmp;
	if (r <= 4.6e+58) {
		tmp = (t_1 - ((((w * r) * t_0) * (0.125 * w)) * fma(v, -2.0, 3.0))) - 4.5;
	} else {
		tmp = (t_1 - ((((0.125 * fma(-2.0, v, 3.0)) * w) * (w * r)) * t_0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(r / Float64(1.0 - v))
	t_1 = Float64(Float64(2.0 / Float64(r * r)) + 3.0)
	tmp = 0.0
	if (r <= 4.6e+58)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(Float64(w * r) * t_0) * Float64(0.125 * w)) * fma(v, -2.0, 3.0))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(Float64(0.125 * fma(-2.0, v, 3.0)) * w) * Float64(w * r)) * t_0)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[r, 4.6e+58], N[(N[(t$95$1 - N[(N[(N[(N[(w * r), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(0.125 * w), $MachinePrecision]), $MachinePrecision] * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.60000000000000005e58

   1. Initial program 80.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.7

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 97.7

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

   6. Applied rewrites 97.7%

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

   if 4.60000000000000005e58 < r

   1. Initial program 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.6 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\left(w \cdot r\right) \cdot \frac{r}{1 - v}\right) \cdot \left(0.125 \cdot w\right)\right) \cdot \mathsf{fma}\left(v, -2, 3\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3 \cdot 10^{-67}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 3e-67)
     (- t_0 (fma (* (* 0.25 (* r r)) w) w 1.5))
     (-
      (-
       (+ t_0 3.0)
       (* (* (* (* 0.125 (fma -2.0 v 3.0)) w) (* w r)) (/ r (- 1.0 v))))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3e-67) {
		tmp = t_0 - fma(((0.25 * (r * r)) * w), w, 1.5);
	} else {
		tmp = ((t_0 + 3.0) - ((((0.125 * fma(-2.0, v, 3.0)) * w) * (w * r)) * (r / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3e-67)
		tmp = Float64(t_0 - fma(Float64(Float64(0.25 * Float64(r * r)) * w), w, 1.5));
	else
		tmp = Float64(Float64(Float64(t_0 + 3.0) - Float64(Float64(Float64(Float64(0.125 * fma(-2.0, v, 3.0)) * w) * Float64(w * r)) * Float64(r / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 3e-67], N[(t$95$0 - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(N[(N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.00000000000000032e-67

   1. Initial program 78.9%

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

   3. Taylor expanded in 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 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 87.1

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

   8. Applied rewrites 87.1%

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

   if 3.00000000000000032e-67 < r

   1. Initial program 91.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 97.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2000000000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2000000000.0)
   (- (/ 2.0 (* r r)) (fma (* (* 0.375 (* r r)) w) w 1.5))
   (-
    (- 3.0 (* (* (* (* 0.125 (fma -2.0 v 3.0)) w) (* w r)) (/ r (- 1.0 v))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2000000000.0) {
		tmp = (2.0 / (r * r)) - fma(((0.375 * (r * r)) * w), w, 1.5);
	} else {
		tmp = (3.0 - ((((0.125 * fma(-2.0, v, 3.0)) * w) * (w * r)) * (r / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2000000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(0.375 * Float64(r * r)) * w), w, 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(0.125 * fma(-2.0, v, 3.0)) * w) * Float64(w * r)) * Float64(r / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2000000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(0.125 * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2e9

   1. Initial program 79.9%

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

   3. Taylor expanded in 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 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   if 2e9 < 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2000000000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - 1.5\\ \mathbf{if}\;r \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;r \leq 1.7 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\left(w \cdot r\right) \cdot w\right) \cdot r, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (- t_0 1.5)))
   (if (<= r 2.4e-32)
     (- t_0 (fma (* (* 0.25 (* r r)) w) w 1.5))
     (if (<= r 1.7e+203)
       (fma -0.375 (* (* (* w w) r) r) t_1)
       (fma -0.25 (* (* (* w r) w) r) t_1)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - 1.5;
	double tmp;
	if (r <= 2.4e-32) {
		tmp = t_0 - fma(((0.25 * (r * r)) * w), w, 1.5);
	} else if (r <= 1.7e+203) {
		tmp = fma(-0.375, (((w * w) * r) * r), t_1);
	} else {
		tmp = fma(-0.25, (((w * r) * w) * r), t_1);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - 1.5)
	tmp = 0.0
	if (r <= 2.4e-32)
		tmp = Float64(t_0 - fma(Float64(Float64(0.25 * Float64(r * r)) * w), w, 1.5));
	elseif (r <= 1.7e+203)
		tmp = fma(-0.375, Float64(Float64(Float64(w * w) * r) * r), t_1);
	else
		tmp = fma(-0.25, Float64(Float64(Float64(w * r) * w) * r), t_1);
	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[(t$95$0 - 1.5), $MachinePrecision]}, If[LessEqual[r, 2.4e-32], N[(t$95$0 - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.7e+203], N[(-0.375 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] + t$95$1), $MachinePrecision], N[(-0.25 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.4000000000000001e-32

   1. Initial program 78.8%

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

   3. Taylor expanded in r around 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 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 87.2

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

   8. Applied rewrites 87.2%

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

   if 2.4000000000000001e-32 < r < 1.7000000000000001e203

   1. Initial program 94.0%

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

   3. Taylor expanded in v around 0

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

      1. 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 93.1%

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

   if 1.7000000000000001e203 < r

   1. Initial program 90.0%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-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 93.9%

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

   7. Applied rewrites 94.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{elif}\;r \leq 1.7 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\left(w \cdot r\right) \cdot w\right) \cdot r, \frac{2}{r \cdot r} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 5.8 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot r\right) \cdot w, t\_0 + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\left(w \cdot r\right) \cdot w\right) \cdot r, t\_0 - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 5.8e-20)
     (- (fma -0.375 (* (* (* w r) r) w) (+ t_0 3.0)) 4.5)
     (fma -0.25 (* (* (* w r) w) r) (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 5.8e-20) {
		tmp = fma(-0.375, (((w * r) * r) * w), (t_0 + 3.0)) - 4.5;
	} else {
		tmp = fma(-0.25, (((w * r) * w) * 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 (v <= 5.8e-20)
		tmp = Float64(fma(-0.375, Float64(Float64(Float64(w * r) * r) * w), Float64(t_0 + 3.0)) - 4.5);
	else
		tmp = fma(-0.25, Float64(Float64(Float64(w * r) * w) * 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[v, 5.8e-20], N[(N[(-0.375 * N[(N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision] + N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(-0.25 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 5.8e-20

   1. Initial program 85.3%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 94.4%

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

   if 5.8e-20 < v

   1. Initial program 75.9%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-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 77.4%

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

   7. Applied rewrites 94.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 5.8 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \left(\left(w \cdot r\right) \cdot r\right) \cdot w, \frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \left(\left(w \cdot r\right) \cdot w\right) \cdot r, \frac{2}{r \cdot r} - 1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot w\right) \cdot \left(w \cdot r\right), r, 3\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4e+58)
   (- (/ 2.0 (* r r)) (fma (* (* 0.375 (* r r)) w) w 1.5))
   (- (fma (* (* -0.375 w) (* w r)) r 3.0) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4e+58) {
		tmp = (2.0 / (r * r)) - fma(((0.375 * (r * r)) * w), w, 1.5);
	} else {
		tmp = fma(((-0.375 * w) * (w * r)), r, 3.0) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4e+58)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(0.375 * Float64(r * r)) * w), w, 1.5));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * w) * Float64(w * r)), r, 3.0) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4e+58], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * r + 3.0), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.99999999999999978e58

   1. Initial program 80.8%

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

   3. Taylor expanded in r around 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 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 89.6

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

   8. Applied rewrites 89.6%

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

   if 3.99999999999999978e58 < r

   1. Initial program 90.8%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 90.1%

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

   10. Applied rewrites 93.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{+58}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot w\right) \cdot \left(w \cdot r\right), r, 3\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3200000000000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot w\right) \cdot \left(w \cdot r\right), r, 3\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3200000000000.0)
   (- (/ 2.0 (* r r)) (fma (* (* 0.25 (* r r)) w) w 1.5))
   (- (fma (* (* -0.375 w) (* w r)) r 3.0) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3200000000000.0) {
		tmp = (2.0 / (r * r)) - fma(((0.25 * (r * r)) * w), w, 1.5);
	} else {
		tmp = fma(((-0.375 * w) * (w * r)), r, 3.0) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3200000000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(0.25 * Float64(r * r)) * w), w, 1.5));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * w) * Float64(w * r)), r, 3.0) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3200000000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * r + 3.0), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.2e12

   1. Initial program 80.0%

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

   3. Taylor expanded in 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 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in v around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if 3.2e12 < r

   1. Initial program 91.3%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 92.1%

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

   10. Applied rewrites 94.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3200000000000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot w\right) \cdot \left(w \cdot r\right), r, 3\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		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 <= 1.15d0) 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 <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		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 <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 79.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 64.6

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

   5. Applied rewrites 64.6%

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

   if 1.1499999999999999 < r

   1. Initial program 92.0%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 32.5

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

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 31.4%

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

Alternative 16: 57.2% 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 82.6%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 59.6

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

5. Applied rewrites 59.6%

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

Alternative 17: 14.4% 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 82.6%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 59.6

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

5. Applied rewrites 59.6%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 11.8%

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

Reproduce

herbie shell --seed 2024248 
(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))