Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.7%

Time: 10.6s

Alternatives: 15

Speedup: 1.8×

• 53.5% 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: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, w \cdot r, -1.5\right) + t\_0\\ t_2 := \left(w \cdot w\right) \cdot r\\ t_3 := \left(3 + t\_0\right) - \frac{\left(t\_2 \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(3 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot t\_2\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (+ (fma (* (* -0.25 r) w) (* w r) -1.5) t_0))
        (t_2 (* (* w w) r))
        (t_3
         (-
          (+ 3.0 t_0)
          (/ (* (* t_2 r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -5e+21)
       (- (- 3.0 (/ (* (* (fma -0.25 v 0.375) t_2) r) (- 1.0 v))) 4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = fma(((-0.25 * r) * w), (w * r), -1.5) + t_0;
	double t_2 = (w * w) * r;
	double t_3 = (3.0 + t_0) - (((t_2 * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -5e+21) {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * t_2) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(fma(Float64(Float64(-0.25 * r) * w), Float64(w * r), -1.5) + t_0)
	t_2 = Float64(Float64(w * w) * r)
	t_3 = Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(t_2 * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -5e+21)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * t_2) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = 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[(N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(t$95$2 * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -5e+21], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * t$95$2), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]]]

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))) < -inf.0 or -5e21 < (-.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 84.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 95.7

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

   7. Applied rewrites 95.7%

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

   9. Applied rewrites 97.8%

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

   if -inf.0 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -5e21

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 99.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, w \cdot r, -1.5\right) + \frac{2}{r \cdot r}\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(3 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, w \cdot r, -1.5\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.0], N[(N[(3.0 - N[(N[(0.125 * v + 0.375), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $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 81.5%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 95.8%

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

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

   1. Initial program 93.5%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in v around 0

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

       1. distribute-rgt-out-- N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-out-- N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out-- N/A

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

       7. metadata-eval N/A

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

       8. *-rgt-identity N/A

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

       9. distribute-rgt-out N/A

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

       10. lower-*.f64 N/A

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

   11. Applied rewrites 79.7%

       \[\leadsto \left(3 - \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \mathsf{fma}\left(0.125, v, 0.375\right)}\right) - 4.5 \]

   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 86.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 94.7%

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

Alternative 4: 90.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$0, 3.0], N[(N[(3.0 - N[(N[(0.125 * v + 0.375), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $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 81.5%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 64.6%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 91.9

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

   11. Applied rewrites 91.9%

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

   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 93.5%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in v around 0

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

       1. distribute-rgt-out-- N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-out-- N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out-- N/A

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

       7. metadata-eval N/A

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

       8. *-rgt-identity N/A

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

       9. distribute-rgt-out N/A

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

       10. lower-*.f64 N/A

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

   11. Applied rewrites 79.7%

       \[\leadsto \left(3 - \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \mathsf{fma}\left(0.125, v, 0.375\right)}\right) - 4.5 \]

   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 86.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 93.1%

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

Alternative 5: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 - v \cdot 2\right) \cdot 0.125\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := 3 + t\_1\\ \mathbf{if}\;t\_2 - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot t\_0}{1 - v} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, w \cdot r, -1.5\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 - \frac{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot t\_0}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (- 3.0 (* v 2.0)) 0.125))
        (t_1 (/ 2.0 (* r r)))
        (t_2 (+ 3.0 t_1)))
   (if (<= (- t_2 (/ (* (* (* (* w w) r) r) t_0) (- 1.0 v))) (- INFINITY))
     (+ (fma (* (* -0.25 r) w) (* w r) -1.5) t_1)
     (- (- t_2 (/ (* (* (* w r) (* w r)) t_0) (- 1.0 v))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = (3.0 - (v * 2.0)) * 0.125;
	double t_1 = 2.0 / (r * r);
	double t_2 = 3.0 + t_1;
	double tmp;
	if ((t_2 - (((((w * w) * r) * r) * t_0) / (1.0 - v))) <= -((double) INFINITY)) {
		tmp = fma(((-0.25 * r) * w), (w * r), -1.5) + t_1;
	} else {
		tmp = (t_2 - ((((w * r) * (w * r)) * t_0) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(3.0 + t_1)
	tmp = 0.0
	if (Float64(t_2 - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * t_0) / Float64(1.0 - v))) <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(-0.25 * r) * w), Float64(w * r), -1.5) + t_1);
	else
		tmp = Float64(Float64(t_2 - Float64(Float64(Float64(Float64(w * r) * Float64(w * r)) * t_0) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(t$95$2 - N[(N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $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))) < -inf.0

   1. Initial program 81.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

   9. Applied rewrites 97.5%

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

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

   1. Initial program 88.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 98.8%

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

Alternative 6: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(3 - \left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (+ 3.0 (/ 2.0 (* r r)))
       (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
      -5e+21)
   (- (- 3.0 (* (* (* 0.25 (* r r)) w) w)) 4.5)
   (+ -1.5 (/ (/ 2.0 r) r))))

C

double code(double v, double w, double r) {
	double tmp;
	if (((3.0 + (2.0 / (r * r))) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21) {
		tmp = (3.0 - (((0.25 * (r * r)) * w) * w)) - 4.5;
	} else {
		tmp = -1.5 + ((2.0 / r) / r);
	}
	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 (((3.0d0 + (2.0d0 / (r * r))) - (((((w * w) * r) * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-5d+21)) then
        tmp = (3.0d0 - (((0.25d0 * (r * r)) * w) * w)) - 4.5d0
    else
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (((3.0 + (2.0 / (r * r))) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21) {
		tmp = (3.0 - (((0.25 * (r * r)) * w) * w)) - 4.5;
	} else {
		tmp = -1.5 + ((2.0 / r) / r);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if ((3.0 + (2.0 / (r * r))) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21:
		tmp = (3.0 - (((0.25 * (r * r)) * w) * w)) - 4.5
	else:
		tmp = -1.5 + ((2.0 / r) / r)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -5e+21)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.25 * Float64(r * r)) * w) * w)) - 4.5);
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (((3.0 + (2.0 / (r * r))) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21)
		tmp = (3.0 - (((0.25 * (r * r)) * w) * w)) - 4.5;
	else
		tmp = -1.5 + ((2.0 / r) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+21], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $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))) < -5e21

   1. Initial program 84.1%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 63.8%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 83.0

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

   11. Applied rewrites 83.0%

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

   if -5e21 < (-.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 87.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.8

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

   9. Applied rewrites 97.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(3 - \left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (+ 3.0 (/ 2.0 (* r r)))
       (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
      -5e+21)
   (* (* (* -0.375 (* r r)) w) w)
   (+ -1.5 (/ (/ 2.0 r) r))))

C

double code(double v, double w, double r) {
	double tmp;
	if (((3.0 + (2.0 / (r * r))) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = -1.5 + ((2.0 / r) / r);
	}
	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 (((3.0d0 + (2.0d0 / (r * r))) - (((((w * w) * r) * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-5d+21)) then
        tmp = (((-0.375d0) * (r * r)) * w) * w
    else
        tmp = (-1.5d0) + ((2.0d0 / r) / r)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (((3.0 + (2.0 / (r * r))) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = -1.5 + ((2.0 / r) / r);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if ((3.0 + (2.0 / (r * r))) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21:
		tmp = ((-0.375 * (r * r)) * w) * w
	else:
		tmp = -1.5 + ((2.0 / r) / r)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -5e+21)
		tmp = Float64(Float64(Float64(-0.375 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(-1.5 + Float64(Float64(2.0 / r) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (((3.0 + (2.0 / (r * r))) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21)
		tmp = ((-0.375 * (r * r)) * w) * w;
	else
		tmp = -1.5 + ((2.0 / r) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+21], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $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))) < -5e21

   1. Initial program 84.1%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 80.2%

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

   if -5e21 < (-.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 87.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.8

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

   9. Applied rewrites 97.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* (* (* w w) r) r) (* (- 3.0 (* v 2.0)) 0.125)) (- 1.0 v)))
        -5e+21)
     (* (* (* -0.375 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((((w * w) * r) * r) * ((3.0d0 - (v * 2.0d0)) * 0.125d0)) / (1.0d0 - v))) <= (-5d+21)) then
        tmp = (((-0.375d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21:
		tmp = ((-0.375 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(Float64(w * w) * r) * r) * Float64(Float64(3.0 - Float64(v * 2.0)) * 0.125)) / Float64(1.0 - v))) <= -5e+21)
		tmp = Float64(Float64(Float64(-0.375 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((((w * w) * r) * r) * ((3.0 - (v * 2.0)) * 0.125)) / (1.0 - v))) <= -5e+21)
		tmp = ((-0.375 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] * N[(N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+21], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -5e21

   1. Initial program 84.1%

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

   3. Taylor expanded in v around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 80.2%

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

   if -5e21 < (-.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 87.5%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \left(\left(3 - v \cdot 2\right) \cdot 0.125\right)}{1 - v} \leq -5 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, w \cdot r, -1.5\right) + t\_0\\ \mathbf{if}\;v \leq -20000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), -0.125 \cdot v - 0.375, -1.5\right) + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (+ (fma (* (* -0.25 r) w) (* w r) -1.5) t_0)))
   (if (<= v -20000.0)
     t_1
     (if (<= v 9e-10)
       (+ (fma (* (* w r) (* w r)) (- (* -0.125 v) 0.375) -1.5) t_0)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = fma(((-0.25 * r) * w), (w * r), -1.5) + t_0;
	double tmp;
	if (v <= -20000.0) {
		tmp = t_1;
	} else if (v <= 9e-10) {
		tmp = fma(((w * r) * (w * r)), ((-0.125 * v) - 0.375), -1.5) + t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(fma(Float64(Float64(-0.25 * r) * w), Float64(w * r), -1.5) + t_0)
	tmp = 0.0
	if (v <= -20000.0)
		tmp = t_1;
	elseif (v <= 9e-10)
		tmp = Float64(fma(Float64(Float64(w * r) * Float64(w * r)), Float64(Float64(-0.125 * v) - 0.375), -1.5) + t_0);
	else
		tmp = 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[(N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[v, -20000.0], t$95$1, If[LessEqual[v, 9e-10], N[(N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.125 * v), $MachinePrecision] - 0.375), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e4 or 8.9999999999999999e-10 < v

   1. Initial program 82.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 95.7

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

   7. Applied rewrites 95.7%

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

   9. Applied rewrites 99.7%

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

   if -2e4 < v < 8.9999999999999999e-10

   1. Initial program 90.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 86.1

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

   7. Applied rewrites 86.1%

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

   8. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

   10. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -20000:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, w \cdot r, -1.5\right) + \frac{2}{r \cdot r}\\ \mathbf{elif}\;v \leq 9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), -0.125 \cdot v - 0.375, -1.5\right) + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, w \cdot r, -1.5\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, t\_0 - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot r, \left(w \cdot w\right) \cdot r, -1.5\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 4e+30)
     (fma (* (* -0.375 (* r r)) w) w (- t_0 1.5))
     (+ (fma (* -0.25 r) (* (* w w) r) -1.5) t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 4e+30) {
		tmp = fma(((-0.375 * (r * r)) * w), w, (t_0 - 1.5));
	} else {
		tmp = fma((-0.25 * r), ((w * w) * r), -1.5) + t_0;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 4e+30)
		tmp = fma(Float64(Float64(-0.375 * Float64(r * r)) * w), w, Float64(t_0 - 1.5));
	else
		tmp = Float64(fma(Float64(-0.25 * r), Float64(Float64(w * w) * r), -1.5) + t_0);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4e+30], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] + -1.5), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.0000000000000001e30

   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 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 91.6%

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

   if 4.0000000000000001e30 < r

   1. Initial program 94.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 85.9

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

   7. Applied rewrites 85.9%

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

   9. Applied rewrites 92.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot r, \left(w \cdot w\right) \cdot r, -1.5\right) + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r} - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4.5e+30)
   (fma (* (* -0.375 (* r r)) w) w (- (/ 2.0 (* r r)) 1.5))
   (- (- 3.0 (* (* (* 0.25 (* r r)) w) w)) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.5e+30) {
		tmp = fma(((-0.375 * (r * r)) * w), w, ((2.0 / (r * r)) - 1.5));
	} else {
		tmp = (3.0 - (((0.25 * (r * r)) * w) * w)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4.5e+30)
		tmp = fma(Float64(Float64(-0.375 * Float64(r * r)) * w), w, Float64(Float64(2.0 / Float64(r * r)) - 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(0.25 * Float64(r * r)) * w) * w)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4.5e+30], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.49999999999999995e30

   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 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 91.6%

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

   if 4.49999999999999995e30 < r

   1. Initial program 94.8%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 85.9

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

   11. Applied rewrites 85.9%

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

Alternative 12: 93.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(-0.25 \cdot r\right) \cdot w, w \cdot r, -1.5\right) + \frac{2}{r \cdot r} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+ (fma (* (* -0.25 r) w) (* w r) -1.5) (/ 2.0 (* r r))))

C

double code(double v, double w, double r) {
	return fma(((-0.25 * r) * w), (w * r), -1.5) + (2.0 / (r * r));
}

Julia

function code(v, w, r)
	return Float64(fma(Float64(Float64(-0.25 * r) * w), Float64(w * r), -1.5) + Float64(2.0 / Float64(r * r)))
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(N[(-0.25 * r), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision] + -1.5), $MachinePrecision] + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around inf

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-in N/A

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

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

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 91.2

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

7. Applied rewrites 91.2%

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

9. Applied rewrites 94.5%

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

10. Final simplification 94.5%

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

Alternative 13: 50.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = Float64(Float64(-1.5 * r) / r);
	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 * r) / r;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 82.9%

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

   3. Taylor expanded in 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 56.8

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

   5. Applied rewrites 56.8%

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

   if 1.1499999999999999 < r

   1. Initial program 95.2%

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

   3. Taylor expanded in r around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 20.7%

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

Alternative 14: 57.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 54.9

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

5. Applied rewrites 54.9%

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

Alternative 15: 44.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 43.4

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

5. Applied rewrites 43.4%

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

Reproduce

herbie shell --seed 2024327 
(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))