Rosa's TurbineBenchmark

Percentage Accurate: 84.1% → 98.8%

Time: 9.5s

Alternatives: 11

Speedup: 1.8×

• 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: 84.1% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.2 \cdot 10^{+18}:\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \left(\left(-w\right) \cdot \frac{r}{v}\right), -0.25 \cdot v, 4.5\right)\right)\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \left(\mathsf{fma}\left(w, v, w\right) \cdot r\right), \mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, 4.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{1 - v}, -0.25 \cdot v, 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -1.2e+18)
     (+ t_0 (- 3.0 (fma (* (* r w) (* (- w) (/ r v))) (* -0.25 v) 4.5)))
     (if (<= v 1.5)
       (+
        t_0
        (-
         3.0
         (fma (* (* r w) (* (fma w v w) r)) (* (fma -2.0 v 3.0) 0.125) 4.5)))
       (+
        t_0
        (- 3.0 (fma (* (* r w) (/ (* r w) (- 1.0 v))) (* -0.25 v) 4.5)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -1.2e+18) {
		tmp = t_0 + (3.0 - fma(((r * w) * (-w * (r / v))), (-0.25 * v), 4.5));
	} else if (v <= 1.5) {
		tmp = t_0 + (3.0 - fma(((r * w) * (fma(w, v, w) * r)), (fma(-2.0, v, 3.0) * 0.125), 4.5));
	} else {
		tmp = t_0 + (3.0 - fma(((r * w) * ((r * w) / (1.0 - v))), (-0.25 * v), 4.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -1.2e+18)
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(Float64(r * w) * Float64(Float64(-w) * Float64(r / v))), Float64(-0.25 * v), 4.5)));
	elseif (v <= 1.5)
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(Float64(r * w) * Float64(fma(w, v, w) * r)), Float64(fma(-2.0, v, 3.0) * 0.125), 4.5)));
	else
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))), Float64(-0.25 * v), 4.5)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1.2e+18], N[(t$95$0 + N[(3.0 - N[(N[(N[(r * w), $MachinePrecision] * N[((-w) * N[(r / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.5], N[(t$95$0 + N[(3.0 - N[(N[(N[(r * w), $MachinePrecision] * N[(N[(w * v + w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(3.0 - N[(N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.2e18

   1. Initial program 80.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in v around inf

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

       1. lower-*.f64 99.8

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

   12. Applied rewrites 99.8%

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

   if -1.2e18 < v < 1.5

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in v around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 1.5 < v

   1. Initial program 76.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in v around inf

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

      1. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

Alternative 2: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(\frac{w}{1 - v} \cdot \left(\mathsf{fma}\left(-2, v, 3\right) \cdot w\right)\right) \cdot \left(-0.125 \cdot r\right)\right) \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), 0.375, 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -2e+45)
     (* (* (* (/ w (- 1.0 v)) (* (fma -2.0 v 3.0) w)) (* -0.125 r)) r)
     (+ t_0 (- 3.0 (fma (* (* w r) (* w r)) 0.375 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -2e+45) {
		tmp = (((w / (1.0 - v)) * (fma(-2.0, v, 3.0) * w)) * (-0.125 * r)) * r;
	} else {
		tmp = t_0 + (3.0 - fma(((w * r) * (w * r)), 0.375, 4.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= -2e+45)
		tmp = Float64(Float64(Float64(Float64(w / Float64(1.0 - v)) * Float64(fma(-2.0, v, 3.0) * w)) * Float64(-0.125 * r)) * r);
	else
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(Float64(w * r) * Float64(w * r)), 0.375, 4.5)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+45], N[(N[(N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * v + 3.0), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], N[(t$95$0 + N[(3.0 - N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * 0.375 + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -1.9999999999999999e45

   1. Initial program 90.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 96.3%

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

   if -1.9999999999999999e45 < (-.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 82.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 92.3

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 98.7%

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

Alternative 3: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -5 \cdot 10^{+48}:\\ \;\;\;\;\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -5e+48)
     (* (* (* -0.25 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -5e+48) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-5d+48)) then
        tmp = (((-0.25d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -5e+48) {
		tmp = ((-0.25 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -5e+48:
		tmp = ((-0.25 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= -5e+48)
		tmp = Float64(Float64(Float64(-0.25 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -5e+48)
		tmp = ((-0.25 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+48], N[(N[(N[(-0.25 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -4.99999999999999973e48

   1. Initial program 90.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 92.6%

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

   if -4.99999999999999973e48 < (-.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 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 w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.9

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

   5. Applied rewrites 94.9%

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

Alternative 4: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq -5 \cdot 10^{+48}:\\ \;\;\;\;\left(\left(-0.375 \cdot \left(r \cdot r\right)\right) \cdot w\right) \cdot w\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        -5e+48)
     (* (* (* -0.375 (* r r)) w) w)
     (- t_0 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -5e+48) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) <= (-5d+48)) then
        tmp = (((-0.375d0) * (r * r)) * w) * w
    else
        tmp = t_0 - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -5e+48) {
		tmp = ((-0.375 * (r * r)) * w) * w;
	} else {
		tmp = t_0 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -5e+48:
		tmp = ((-0.375 * (r * r)) * w) * w
	else:
		tmp = t_0 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= -5e+48)
		tmp = Float64(Float64(Float64(-0.375 * Float64(r * r)) * w) * w);
	else
		tmp = Float64(t_0 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= -5e+48)
		tmp = ((-0.375 * (r * r)) * w) * w;
	else
		tmp = t_0 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+48], N[(N[(N[(-0.375 * N[(r * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision], N[(t$95$0 - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -4.99999999999999973e48

   1. Initial program 90.3%

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

   3. Taylor expanded in w around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 85.1%

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

   if -4.99999999999999973e48 < (-.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 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 w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.9

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

   5. Applied rewrites 94.9%

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

Alternative 5: 55.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} \leq 5:\\ \;\;\;\;3 - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
        5.0)
     (- 3.0 4.5)
     t_0)))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= 5.0) {
		tmp = 3.0 - 4.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) <= 5.0d0) then
        tmp = 3.0d0 - 4.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= 5.0) {
		tmp = 3.0 - 4.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= 5.0:
		tmp = 3.0 - 4.5
	else:
		tmp = t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) <= 5.0)
		tmp = Float64(3.0 - 4.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) <= 5.0)
		tmp = 3.0 - 4.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5.0], N[(3.0 - 4.5), $MachinePrecision], t$95$0]]

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))) < 5

   1. Initial program 89.1%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + 3 \cdot {r}^{2}}{{r}^{2}}} - \frac{9}{2} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{2 + 3 \cdot {r}^{2}}{\color{blue}{r \cdot r}} - \frac{9}{2} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}} - \frac{9}{2} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}} - \frac{9}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2 + 3 \cdot {r}^{2}}{r}}}{r} - \frac{9}{2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{3 \cdot {r}^{2} + 2}}{r}}{r} - \frac{9}{2} \]

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 22.7

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

   5. Applied rewrites 22.7%

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

   6. Taylor expanded in r around inf

      \[\leadsto 3 - \frac{9}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.5%

      \[\leadsto 3 - 4.5 \]

   if 5 < (-.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 81.3%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 6: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.2 \cdot 10^{+18} \lor \neg \left(v \leq 7.5 \cdot 10^{-13}\right):\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \left(\left(-w\right) \cdot \frac{r}{v}\right), -0.25 \cdot v, 4.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \left(\mathsf{fma}\left(w, v, w\right) \cdot r\right), \mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1.2e+18) (not (<= v 7.5e-13)))
     (+ t_0 (- 3.0 (fma (* (* r w) (* (- w) (/ r v))) (* -0.25 v) 4.5)))
     (+
      t_0
      (-
       3.0
       (fma (* (* r w) (* (fma w v w) r)) (* (fma -2.0 v 3.0) 0.125) 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.2e+18) || !(v <= 7.5e-13)) {
		tmp = t_0 + (3.0 - fma(((r * w) * (-w * (r / v))), (-0.25 * v), 4.5));
	} else {
		tmp = t_0 + (3.0 - fma(((r * w) * (fma(w, v, w) * r)), (fma(-2.0, v, 3.0) * 0.125), 4.5));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.2e+18) || !(v <= 7.5e-13))
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(Float64(r * w) * Float64(Float64(-w) * Float64(r / v))), Float64(-0.25 * v), 4.5)));
	else
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(Float64(r * w) * Float64(fma(w, v, w) * r)), Float64(fma(-2.0, v, 3.0) * 0.125), 4.5)));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.2e+18], N[Not[LessEqual[v, 7.5e-13]], $MachinePrecision]], N[(t$95$0 + N[(3.0 - N[(N[(N[(r * w), $MachinePrecision] * N[((-w) * N[(r / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(3.0 - N[(N[(N[(r * w), $MachinePrecision] * N[(N[(w * v + w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.2e18 or 7.5000000000000004e-13 < v

   1. Initial program 79.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 97.4

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

   9. Applied rewrites 97.4%

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

   10. Taylor expanded in v around inf

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

       1. lower-*.f64 99.0

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

   12. Applied rewrites 99.0%

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

   if -1.2e18 < v < 7.5000000000000004e-13

   1. Initial program 91.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in v around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.2 \cdot 10^{+18} \lor \neg \left(v \leq 7.5 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \left(\left(-w\right) \cdot \frac{r}{v}\right), -0.25 \cdot v, 4.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(3 - \mathsf{fma}\left(\left(r \cdot w\right) \cdot \left(\mathsf{fma}\left(w, v, w\right) \cdot r\right), \mathsf{fma}\left(-2, v, 3\right) \cdot 0.125, 4.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 99.8

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 8: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{-156}:\\ \;\;\;\;t\_0 + \left(3 - \mathsf{fma}\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right), 0.375, 4.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(r \cdot r\right) \cdot -0.25\right) \cdot w, w, t\_0 - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 4e-156)
     (+ t_0 (- 3.0 (fma (* (* w r) (* w r)) 0.375 4.5)))
     (fma (* (* (* r r) -0.25) w) w (- t_0 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 4e-156) {
		tmp = t_0 + (3.0 - fma(((w * r) * (w * r)), 0.375, 4.5));
	} else {
		tmp = fma((((r * r) * -0.25) * w), w, (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(w * w) <= 4e-156)
		tmp = Float64(t_0 + Float64(3.0 - fma(Float64(Float64(w * r) * Float64(w * r)), 0.375, 4.5)));
	else
		tmp = fma(Float64(Float64(Float64(r * r) * -0.25) * w), w, Float64(t_0 - 1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 4e-156], N[(t$95$0 + N[(3.0 - N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * 0.375 + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision] * w), $MachinePrecision] * w + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 4.00000000000000016e-156

   1. Initial program 90.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 94.6%

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

   if 4.00000000000000016e-156 < (*.f64 w w)

   1. Initial program 83.1%

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

   3. Taylor expanded in v around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 97.9%

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

Alternative 9: 87.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in v around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. associate-+l+ N/A

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

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

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. +-commutative N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. lower-fma.f64 N/A

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

5. Applied rewrites 93.1%

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

Alternative 10: 57.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 56.9

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

5. Applied rewrites 56.9%

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

Alternative 11: 13.7% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ 3 - 4.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (- 3.0 4.5))

C

double code(double v, double w, double r) {
	return 3.0 - 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 - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return 3.0 - 4.5;
}

Python

def code(v, w, r):
	return 3.0 - 4.5

Julia

function code(v, w, r)
	return Float64(3.0 - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = 3.0 - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(3.0 - 4.5), $MachinePrecision]

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{2 + 3 \cdot {r}^{2}}{{r}^{2}}} - \frac{9}{2} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{2 + 3 \cdot {r}^{2}}{\color{blue}{r \cdot r}} - \frac{9}{2} \]

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}} - \frac{9}{2} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}} - \frac{9}{2} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{2 + 3 \cdot {r}^{2}}{r}}}{r} - \frac{9}{2} \]

   5. +-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{3 \cdot {r}^{2} + 2}}{r}}{r} - \frac{9}{2} \]

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 55.2

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

5. Applied rewrites 55.2%

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

6. Taylor expanded in r around inf

   \[\leadsto 3 - \frac{9}{2} \]
7. Step-by-step derivation

8. Applied rewrites 15.1%

   \[\leadsto 3 - 4.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(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))