Rosa's TurbineBenchmark

Percentage Accurate: 85.4% → 97.8%

Time: 13.1s

Alternatives: 15

Speedup: 1.6×

• 53.3% 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: 85.4% 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: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), w \cdot \left(r \cdot \left(w \cdot \frac{r}{v + -1}\right)\right), \frac{2}{r \cdot r} + -1.5\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (fma
  (fma v -0.25 0.375)
  (* w (* r (* w (/ r (+ v -1.0)))))
  (+ (/ 2.0 (* r r)) -1.5)))

C

double code(double v, double w, double r) {
	return fma(fma(v, -0.25, 0.375), (w * (r * (w * (r / (v + -1.0))))), ((2.0 / (r * r)) + -1.5));
}

Julia

function code(v, w, r)
	return fma(fma(v, -0.25, 0.375), Float64(w * Float64(r * Float64(w * Float64(r / Float64(v + -1.0))))), Float64(Float64(2.0 / Float64(r * r)) + -1.5))
end

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 95.2

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

4. Applied rewrites 95.2%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

6. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), \left(-w\right) \cdot \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right), \frac{2}{r \cdot r} + -1.5\right)} \]

7. Final simplification 98.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), w \cdot \left(r \cdot \left(w \cdot \frac{r}{v + -1}\right)\right), \frac{2}{r \cdot r} + -1.5\right) \]
8. Add Preprocessing

Alternative 2: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot 0.25\right), -w, 3\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t\_0 - r \cdot \left(w \cdot \left(r \cdot \left(0.375 \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* 0.125 (- 3.0 (* v 2.0))) (* r (* r (* w w)))) (+ v -1.0)))))
   (if (<= t_1 (- INFINITY))
     (- (fma (* w (* (* r r) 0.25)) (- w) 3.0) 4.5)
     (if (<= t_1 -2e+24) (- t_0 (* r (* w (* r (* 0.375 w))))) (+ t_0 -1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((w * ((r * r) * 0.25)), -w, 3.0) - 4.5;
	} else if (t_1 <= -2e+24) {
		tmp = t_0 - (r * (w * (r * (0.375 * w))));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v * 2.0))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(w * Float64(Float64(r * r) * 0.25)), Float64(-w), 3.0) - 4.5);
	elseif (t_1 <= -2e+24)
		tmp = Float64(t_0 - Float64(r * Float64(w * Float64(r * Float64(0.375 * w)))));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(w * N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * (-w) + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -2e+24], N[(t$95$0 - N[(r * N[(w * N[(r * N[(0.375 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 84.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

   9. Applied rewrites 93.8%

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

   10. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 88.3

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

   12. Applied rewrites 88.3%

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

   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))) < -2e24

   1. Initial program 97.4%

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 65.0%

      \[\leadsto \frac{2}{r \cdot r} - r \cdot \color{blue}{\left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.3%

       \[\leadsto \frac{2}{r \cdot r} - r \cdot \left(\left(r \cdot \left(w \cdot 0.375\right)\right) \cdot w\right) \]

   if -2e24 < (-.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 86.2%

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

   3. Taylor expanded in w around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot 0.25\right), -w, 3\right) - 4.5\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{r \cdot r} - r \cdot \left(w \cdot \left(r \cdot \left(0.375 \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot 0.25\right), -w, 3\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* 0.125 (- 3.0 (* v 2.0))) (* r (* r (* w w)))) (+ v -1.0)))))
   (if (<= t_1 (- INFINITY))
     (- (fma (* w (* (* r r) 0.25)) (- w) 3.0) 4.5)
     (if (<= t_1 -2e+24)
       (- (- 3.0 (* r (* r (* 0.375 (* w w))))) 4.5)
       (+ t_0 -1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((w * ((r * r) * 0.25)), -w, 3.0) - 4.5;
	} else if (t_1 <= -2e+24) {
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v * 2.0))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(w * Float64(Float64(r * r) * 0.25)), Float64(-w), 3.0) - 4.5);
	elseif (t_1 <= -2e+24)
		tmp = Float64(Float64(3.0 - Float64(r * Float64(r * Float64(0.375 * Float64(w * w))))) - 4.5);
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(w * N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] * (-w) + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -2e+24], N[(N[(3.0 - N[(r * N[(r * N[(0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 84.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

   9. Applied rewrites 93.8%

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

   10. Taylor expanded in v around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 88.3

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

   12. Applied rewrites 88.3%

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

   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))) < -2e24

   1. Initial program 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 65.0

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

   10. Applied rewrites 65.0%

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

   if -2e24 < (-.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 86.2%

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

   3. Taylor expanded in w around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot 0.25\right), -w, 3\right) - 4.5\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(3 - \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot 0.25\right)\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (+
          (+ t_0 3.0)
          (/ (* (* 0.125 (- 3.0 (* v 2.0))) (* r (* r (* w w)))) (+ v -1.0)))))
   (if (<= t_1 (- INFINITY))
     (- (- 3.0 (* (* r r) (* (* w w) 0.25))) 4.5)
     (if (<= t_1 -2e+24)
       (- (- 3.0 (* r (* r (* 0.375 (* w w))))) 4.5)
       (+ t_0 -1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (3.0 - ((r * r) * ((w * w) * 0.25))) - 4.5;
	} else if (t_1 <= -2e+24) {
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (3.0 - ((r * r) * ((w * w) * 0.25))) - 4.5;
	} else if (t_1 <= -2e+24) {
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (3.0 - ((r * r) * ((w * w) * 0.25))) - 4.5
	elif t_1 <= -2e+24:
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v * 2.0))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(3.0 - Float64(Float64(r * r) * Float64(Float64(w * w) * 0.25))) - 4.5);
	elseif (t_1 <= -2e+24)
		tmp = Float64(Float64(3.0 - Float64(r * Float64(r * Float64(0.375 * Float64(w * w))))) - 4.5);
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (3.0 - ((r * r) * ((w * w) * 0.25))) - 4.5;
	elseif (t_1 <= -2e+24)
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(3.0 - N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -2e+24], N[(N[(3.0 - N[(r * N[(r * N[(0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in v around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.7

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

   10. Applied rewrites 83.7%

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

   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))) < -2e24

   1. Initial program 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 65.0

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

   10. Applied rewrites 65.0%

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

   if -2e24 < (-.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 86.2%

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

   3. Taylor expanded in w around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -\infty:\\ \;\;\;\;\left(3 - \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot 0.25\right)\right) - 4.5\\ \mathbf{elif}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ t_0 3.0)
         (/ (* (* 0.125 (- 3.0 (* v 2.0))) (* r (* r (* w w)))) (+ v -1.0)))
        -2e+24)
     (- (- 3.0 (* r (* r (* 0.375 (* w w))))) 4.5)
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2e+24) {
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) + (((0.125d0 * (3.0d0 - (v * 2.0d0))) * (r * (r * (w * w)))) / (v + (-1.0d0)))) <= (-2d+24)) then
        tmp = (3.0d0 - (r * (r * (0.375d0 * (w * w))))) - 4.5d0
    else
        tmp = t_0 + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2e+24) {
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2e+24:
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v * 2.0))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= -2e+24)
		tmp = Float64(Float64(3.0 - Float64(r * Float64(r * Float64(0.375 * Float64(w * w))))) - 4.5);
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2e+24)
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+24], N[(N[(3.0 - N[(r * N[(r * N[(0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $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))) < -2e24

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 75.1

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

   10. Applied rewrites 75.1%

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

   if -2e24 < (-.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 86.2%

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

   3. Taylor expanded in w around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ t_0 3.0)
         (/ (* (* 0.125 (- 3.0 (* v 2.0))) (* r (* r (* w w)))) (+ v -1.0)))
        -2e+24)
     (* (* r r) (* (* w w) -0.375))
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2e+24) {
		tmp = (r * r) * ((w * w) * -0.375);
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((t_0 + 3.0d0) + (((0.125d0 * (3.0d0 - (v * 2.0d0))) * (r * (r * (w * w)))) / (v + (-1.0d0)))) <= (-2d+24)) then
        tmp = (r * r) * ((w * w) * (-0.375d0))
    else
        tmp = t_0 + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2e+24) {
		tmp = (r * r) * ((w * w) * -0.375);
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2e+24:
		tmp = (r * r) * ((w * w) * -0.375)
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v * 2.0))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(v + -1.0))) <= -2e+24)
		tmp = Float64(Float64(r * r) * Float64(Float64(w * w) * -0.375));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((t_0 + 3.0) + (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (v + -1.0))) <= -2e+24)
		tmp = (r * r) * ((w * w) * -0.375);
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+24], N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2e24

   1. Initial program 81.0%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 70.9%

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

   if -2e24 < (-.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 86.2%

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

   3. Taylor expanded in w around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-155}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), 0.375, 1.5\right)\\ \mathbf{elif}\;w \cdot w \leq 10^{+68}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(r \cdot \left(w \cdot r\right)\right) \cdot \left(0.375 \cdot w\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e-155)
     (- t_0 (fma (* r (* r (* w w))) 0.375 1.5))
     (if (<= (* w w) 1e+68)
       (+ -1.5 (fma (* w (* -0.25 (* r r))) w t_0))
       (- t_0 (* (* r (* w r)) (* 0.375 w)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e-155) {
		tmp = t_0 - fma((r * (r * (w * w))), 0.375, 1.5);
	} else if ((w * w) <= 1e+68) {
		tmp = -1.5 + fma((w * (-0.25 * (r * r))), w, t_0);
	} else {
		tmp = t_0 - ((r * (w * r)) * (0.375 * w));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e-155)
		tmp = Float64(t_0 - fma(Float64(r * Float64(r * Float64(w * w))), 0.375, 1.5));
	elseif (Float64(w * w) <= 1e+68)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r * r))), w, t_0));
	else
		tmp = Float64(t_0 - Float64(Float64(r * Float64(w * r)) * Float64(0.375 * w)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e-155], N[(t$95$0 - N[(N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(w * w), $MachinePrecision], 1e+68], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(r * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(0.375 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 w w) < 2.00000000000000003e-155

   1. Initial program 91.6%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 89.2%

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

   if 2.00000000000000003e-155 < (*.f64 w w) < 9.99999999999999953e67

   1. Initial program 92.2%

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

   3. Taylor expanded in v around inf

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

      1. 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. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. associate-*r/ N/A

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

   5. Applied rewrites 96.6%

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

   if 9.99999999999999953e67 < (*.f64 w w)

   1. Initial program 73.7%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 72.8%

      \[\leadsto \frac{2}{r \cdot r} - r \cdot \color{blue}{\left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.2%

       \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot 0.375\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot r\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), 0.375, 1.5\right)\\ \mathbf{elif}\;w \cdot w \leq 10^{+68}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(r \cdot \left(w \cdot r\right)\right) \cdot \left(0.375 \cdot w\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-155}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(0.375 \cdot r, r \cdot \left(w \cdot w\right), 1.5\right)\\ \mathbf{elif}\;w \cdot w \leq 10^{+68}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(r \cdot \left(w \cdot r\right)\right) \cdot \left(0.375 \cdot w\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e-155)
     (- t_0 (fma (* 0.375 r) (* r (* w w)) 1.5))
     (if (<= (* w w) 1e+68)
       (+ -1.5 (fma (* w (* -0.25 (* r r))) w t_0))
       (- t_0 (* (* r (* w r)) (* 0.375 w)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e-155) {
		tmp = t_0 - fma((0.375 * r), (r * (w * w)), 1.5);
	} else if ((w * w) <= 1e+68) {
		tmp = -1.5 + fma((w * (-0.25 * (r * r))), w, t_0);
	} else {
		tmp = t_0 - ((r * (w * r)) * (0.375 * w));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e-155)
		tmp = Float64(t_0 - fma(Float64(0.375 * r), Float64(r * Float64(w * w)), 1.5));
	elseif (Float64(w * w) <= 1e+68)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r * r))), w, t_0));
	else
		tmp = Float64(t_0 - Float64(Float64(r * Float64(w * r)) * Float64(0.375 * w)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e-155], N[(t$95$0 - N[(N[(0.375 * r), $MachinePrecision] * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(w * w), $MachinePrecision], 1e+68], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(r * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(0.375 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 w w) < 2.00000000000000003e-155

   1. Initial program 91.6%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 89.2%

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

   if 2.00000000000000003e-155 < (*.f64 w w) < 9.99999999999999953e67

   1. Initial program 92.2%

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

   3. Taylor expanded in v around inf

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

      1. 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. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. associate-*r/ N/A

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

   5. Applied rewrites 96.6%

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

   if 9.99999999999999953e67 < (*.f64 w w)

   1. Initial program 73.7%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 72.8%

      \[\leadsto \frac{2}{r \cdot r} - r \cdot \color{blue}{\left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.2%

       \[\leadsto \frac{2}{r \cdot r} - \left(w \cdot 0.375\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot r\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(0.375 \cdot r, r \cdot \left(w \cdot w\right), 1.5\right)\\ \mathbf{elif}\;w \cdot w \leq 10^{+68}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(r \cdot \left(w \cdot r\right)\right) \cdot \left(0.375 \cdot w\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 40000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(0.375 \cdot \left(r \cdot \left(w \cdot r\right)\right), w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot \frac{0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 40000.0)
   (- (/ 2.0 (* r r)) (fma (* 0.375 (* r (* w r))) w 1.5))
   (-
    (+ 3.0 (* (* r (* w (* w r))) (/ (* 0.125 (fma v -2.0 3.0)) (+ v -1.0))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 40000.0) {
		tmp = (2.0 / (r * r)) - fma((0.375 * (r * (w * r))), w, 1.5);
	} else {
		tmp = (3.0 + ((r * (w * (w * r))) * ((0.125 * fma(v, -2.0, 3.0)) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 40000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(0.375 * Float64(r * Float64(w * r))), w, 1.5));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(r * Float64(w * Float64(w * r))) * Float64(Float64(0.125 * fma(v, -2.0, 3.0)) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 40000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 * N[(r * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4e4

   1. Initial program 82.7%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 88.4%

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

   9. Applied rewrites 90.8%

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

   if 4e4 < r

   1. Initial program 87.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 40000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(0.375 \cdot \left(r \cdot \left(w \cdot r\right)\right), w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot \frac{0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(w \cdot r\right)\\ \mathbf{if}\;r \leq 40000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(0.375 \cdot t\_0, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), \frac{w \cdot t\_0}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* w r))))
   (if (<= r 40000.0)
     (- (/ 2.0 (* r r)) (fma (* 0.375 t_0) w 1.5))
     (- 3.0 (fma (fma v -0.25 0.375) (/ (* w t_0) (- 1.0 v)) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = r * (w * r);
	double tmp;
	if (r <= 40000.0) {
		tmp = (2.0 / (r * r)) - fma((0.375 * t_0), w, 1.5);
	} else {
		tmp = 3.0 - fma(fma(v, -0.25, 0.375), ((w * t_0) / (1.0 - v)), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(w * r))
	tmp = 0.0
	if (r <= 40000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(0.375 * t_0), w, 1.5));
	else
		tmp = Float64(3.0 - fma(fma(v, -0.25, 0.375), Float64(Float64(w * t_0) / Float64(1.0 - v)), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(w * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 40000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 * t$95$0), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(N[(w * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4e4

   1. Initial program 82.7%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 88.4%

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

   9. Applied rewrites 90.8%

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

   if 4e4 < r

   1. Initial program 87.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

   9. Applied rewrites 96.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 40000:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(0.375 \cdot \left(r \cdot \left(w \cdot r\right)\right), w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), \frac{w \cdot \left(r \cdot \left(w \cdot r\right)\right)}{1 - v}, 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.9% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 5.0)
     (- t_0 (fma (* 0.375 (* r (* w r))) w 1.5))
     (+ -1.5 (fma (* w (* -0.25 (* r r))) w t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 5.0) {
		tmp = t_0 - fma((0.375 * (r * (w * r))), w, 1.5);
	} else {
		tmp = -1.5 + fma((w * (-0.25 * (r * r))), w, t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 5.0)
		tmp = Float64(t_0 - fma(Float64(0.375 * Float64(r * Float64(w * r))), w, 1.5));
	else
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r * r))), w, 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[v, 5.0], N[(t$95$0 - N[(N[(0.375 * N[(r * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 5

   1. Initial program 84.3%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 87.9%

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

   9. Applied rewrites 93.1%

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

   if 5 < v

   1. Initial program 82.7%

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

   3. Taylor expanded in 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. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. associate-*r/ N/A

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

   5. Applied rewrites 94.8%

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

4. Final simplification 93.6%

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

Alternative 12: 89.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 8.6 \cdot 10^{+141}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 8.6e+141)
   (+ -1.5 (fma (* w (* -0.25 (* r r))) w (/ 2.0 (* r r))))
   (- (- 3.0 (* r (* r (* 0.375 (* w w))))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 8.6e+141) {
		tmp = -1.5 + fma((w * (-0.25 * (r * r))), w, (2.0 / (r * r)));
	} else {
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 8.6e+141)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r * r))), w, Float64(2.0 / Float64(r * r))));
	else
		tmp = Float64(Float64(3.0 - Float64(r * Float64(r * Float64(0.375 * Float64(w * w))))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 8.6e+141], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(r * N[(r * N[(0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.5999999999999997e141

   1. Initial program 83.4%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. associate-*r/ N/A

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

   5. Applied rewrites 88.7%

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

   if 8.5999999999999997e141 < r

   1. Initial program 87.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 86.9

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

   10. Applied rewrites 86.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8.6 \cdot 10^{+141}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r} - w \cdot \left(0.375 \cdot \left(r \cdot \left(w \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.15)
   (- (/ 2.0 (* r r)) (* w (* 0.375 (* r (* w r)))))
   (- (- 3.0 (* r (* r (* 0.375 (* w w))))) 4.5)))

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = (2.0d0 / (r * r)) - (w * (0.375d0 * (r * (w * r))))
    else
        tmp = (3.0d0 - (r * (r * (0.375d0 * (w * w))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = (2.0 / (r * r)) - (w * (0.375 * (r * (w * r))));
	} else {
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = (2.0 / (r * r)) - (w * (0.375 * (r * (w * r))))
	else:
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = (2.0 / (r * r)) - (w * (0.375 * (r * (w * r))));
	else
		tmp = (3.0 - (r * (r * (0.375 * (w * w))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 82.7%

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

   3. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 73.1%

      \[\leadsto \frac{2}{r \cdot r} - r \cdot \color{blue}{\left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.4%

       \[\leadsto \frac{2}{r \cdot r} - \left(0.375 \cdot \left(r \cdot \left(w \cdot r\right)\right)\right) \cdot w \]

   if 1.1499999999999999 < r

   1. Initial program 87.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 86.7

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

   10. Applied rewrites 86.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r} - w \cdot \left(0.375 \cdot \left(r \cdot \left(w \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in w around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 55.4

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

5. Applied rewrites 55.4%

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

6. Final simplification 55.4%

   \[\leadsto \frac{2}{r \cdot r} + -1.5 \]
7. Add Preprocessing

Alternative 15: 44.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 43.6

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

5. Applied rewrites 43.6%

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

Reproduce

herbie shell --seed 2024231 
(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))