Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 98.0%

Time: 5.9s

Alternatives: 14

Speedup: 0.5×

• 52.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    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: 84.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    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: 98.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 98.8

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot {\left(w \cdot r\right)}^{2}\right) - 4.5 \]

   5. Applied rewrites 98.8%

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

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 97.9

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

   4. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 97.9

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

   6. Applied rewrites 97.9%

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

Alternative 2: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-r\right) \cdot r\\ t_1 := \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.375\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (- r) r))
        (t_1
         (-
          (-
           (+ 3.0 (/ 2.0 (* r r)))
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (* (* (* w w) 0.25) t_0)
     (if (<= t_1 -1e+23)
       (* (* (* w w) 0.375) t_0)
       (if (<= t_1 -1.5) -1.5 (/ (fma (* -1.5 r) r 2.0) (* r r)))))))

C

double code(double v, double w, double r) {
	double t_0 = -r * r;
	double t_1 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((w * w) * 0.25) * t_0;
	} else if (t_1 <= -1e+23) {
		tmp = ((w * w) * 0.375) * t_0;
	} else if (t_1 <= -1.5) {
		tmp = -1.5;
	} else {
		tmp = fma((-1.5 * r), r, 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(-r) * r)
	t_1 = 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)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(w * w) * 0.25) * t_0);
	elseif (t_1 <= -1e+23)
		tmp = Float64(Float64(Float64(w * w) * 0.375) * t_0);
	elseif (t_1 <= -1.5)
		tmp = -1.5;
	else
		tmp = Float64(fma(Float64(-1.5 * r), r, 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[((-r) * r), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(w * w), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -1e+23], N[(N[(N[(w * w), $MachinePrecision] * 0.375), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -1.5], -1.5, N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 88.5

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

   8. Applied rewrites 88.5%

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

   9. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto -\left(\left(w \cdot w\right) \cdot \frac{1}{4}\right) \cdot \left(r \cdot r\right) \]

       4. lift-*.f64 88.5

          \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   11. Applied rewrites 88.5%

       \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -9.9999999999999992e22

   1. Initial program 99.3%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in w around inf

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

       1. associate-*r/ N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. metadata-eval N/A

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

       8. fp-cancel-sign-sub-inv N/A

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

       9. +-commutative N/A

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

       10. associate-*l/ N/A

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

       11. lower-/.f64 N/A

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

   11. Applied rewrites 72.6%

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

   12. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 60.7

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

   14. Applied rewrites 60.7%

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

   if -9.9999999999999992e22 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 87.6%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 87.6%

      \[\leadsto -1.5 \]

   if -1.5 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r \cdot r} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r \cdot r} \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(\left(-r\right) \cdot r\right)\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.375\right) \cdot \left(\left(-r\right) \cdot r\right)\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-r\right) \cdot r\\ t_1 := \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.375\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (- r) r))
        (t_1
         (-
          (-
           (+ 3.0 (/ 2.0 (* r r)))
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 (- INFINITY))
     (* (* (* w w) 0.25) t_0)
     (if (<= t_1 -1e+23)
       (* (* (* w w) 0.375) t_0)
       (if (<= t_1 -1.5) -1.5 (/ (fma -1.5 (* r r) 2.0) (* r r)))))))

C

double code(double v, double w, double r) {
	double t_0 = -r * r;
	double t_1 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((w * w) * 0.25) * t_0;
	} else if (t_1 <= -1e+23) {
		tmp = ((w * w) * 0.375) * t_0;
	} else if (t_1 <= -1.5) {
		tmp = -1.5;
	} else {
		tmp = fma(-1.5, (r * r), 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(-r) * r)
	t_1 = 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)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(w * w) * 0.25) * t_0);
	elseif (t_1 <= -1e+23)
		tmp = Float64(Float64(Float64(w * w) * 0.375) * t_0);
	elseif (t_1 <= -1.5)
		tmp = -1.5;
	else
		tmp = Float64(fma(-1.5, Float64(r * r), 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[((-r) * r), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(w * w), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -1e+23], N[(N[(N[(w * w), $MachinePrecision] * 0.375), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -1.5], -1.5, N[(N[(-1.5 * N[(r * r), $MachinePrecision] + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 88.5

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

   8. Applied rewrites 88.5%

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

   9. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto -\left(\left(w \cdot w\right) \cdot \frac{1}{4}\right) \cdot \left(r \cdot r\right) \]

       4. lift-*.f64 88.5

          \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   11. Applied rewrites 88.5%

       \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -9.9999999999999992e22

   1. Initial program 99.3%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in w around inf

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

       1. associate-*r/ N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. metadata-eval N/A

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

       8. fp-cancel-sign-sub-inv N/A

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

       9. +-commutative N/A

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

       10. associate-*l/ N/A

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

       11. lower-/.f64 N/A

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

   11. Applied rewrites 72.6%

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

   12. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 60.7

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

   14. Applied rewrites 60.7%

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

   if -9.9999999999999992e22 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 87.6%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 87.6%

      \[\leadsto -1.5 \]

   if -1.5 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(\left(-r\right) \cdot r\right)\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.375\right) \cdot \left(\left(-r\right) \cdot r\right)\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-r\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(\left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.375\right) \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq -0.005:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (- r) r))
        (t_1 (/ 2.0 (* r r)))
        (t_2
         (-
          (-
           (+ 3.0 t_1)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_2 (- INFINITY))
     (* (* (* w w) 0.25) t_0)
     (if (<= t_2 -1e+23)
       (* (* (* w w) 0.375) t_0)
       (if (<= t_2 -0.005) -1.5 t_1)))))

C

double code(double v, double w, double r) {
	double t_0 = -r * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((w * w) * 0.25) * t_0;
	} else if (t_2 <= -1e+23) {
		tmp = ((w * w) * 0.375) * t_0;
	} else if (t_2 <= -0.005) {
		tmp = -1.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = -r * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = ((w * w) * 0.25) * t_0;
	} else if (t_2 <= -1e+23) {
		tmp = ((w * w) * 0.375) * t_0;
	} else if (t_2 <= -0.005) {
		tmp = -1.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = -r * r
	t_1 = 2.0 / (r * r)
	t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_2 <= -math.inf:
		tmp = ((w * w) * 0.25) * t_0
	elif t_2 <= -1e+23:
		tmp = ((w * w) * 0.375) * t_0
	elif t_2 <= -0.005:
		tmp = -1.5
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(-r) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(Float64(3.0 + t_1) - 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)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(w * w) * 0.25) * t_0);
	elseif (t_2 <= -1e+23)
		tmp = Float64(Float64(Float64(w * w) * 0.375) * t_0);
	elseif (t_2 <= -0.005)
		tmp = -1.5;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = -r * r;
	t_1 = 2.0 / (r * r);
	t_2 = ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = ((w * w) * 0.25) * t_0;
	elseif (t_2 <= -1e+23)
		tmp = ((w * w) * 0.375) * t_0;
	elseif (t_2 <= -0.005)
		tmp = -1.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[((-r) * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(3.0 + t$95$1), $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]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(w * w), $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -1e+23], N[(N[(N[(w * w), $MachinePrecision] * 0.375), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -0.005], -1.5, t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 88.5

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

   8. Applied rewrites 88.5%

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

   9. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto -\left(\left(w \cdot w\right) \cdot \frac{1}{4}\right) \cdot \left(r \cdot r\right) \]

       4. lift-*.f64 88.5

          \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   11. Applied rewrites 88.5%

       \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -9.9999999999999992e22

   1. Initial program 99.3%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in w around inf

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

       1. associate-*r/ N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. metadata-eval N/A

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

       8. fp-cancel-sign-sub-inv N/A

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

       9. +-commutative N/A

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

       10. associate-*l/ N/A

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

       11. lower-/.f64 N/A

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

   11. Applied rewrites 72.6%

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

   12. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 60.7

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

   14. Applied rewrites 60.7%

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

   if -9.9999999999999992e22 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -0.0050000000000000001

   1. Initial program 89.3%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 81.5%

      \[\leadsto -1.5 \]

   if -0.0050000000000000001 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 83.4%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 99.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(\left(-r\right) \cdot r\right)\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.375\right) \cdot \left(\left(-r\right) \cdot r\right)\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -0.005:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right) \cdot w\right) \cdot \frac{0.125}{1 - v}\right) \cdot \left(-r\right)\right) \cdot r\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(t\_0 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1
         (-
          (-
           t_0
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 -2e+278)
     (* (* (* (* (* (fma v -2.0 3.0) w) w) (/ 0.125 (- 1.0 v))) (- r)) r)
     (if (<= t_1 -1.5)
       (-
        (- t_0 (/ (* (* (fma -0.25 v 0.375) (* w (* w r))) r) (- 1.0 v)))
        4.5)
       (/ (fma (* -1.5 r) r 2.0) (* r r))))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -2e+278) {
		tmp = ((((fma(v, -2.0, 3.0) * w) * w) * (0.125 / (1.0 - v))) * -r) * r;
	} else if (t_1 <= -1.5) {
		tmp = (t_0 - (((fma(-0.25, v, 0.375) * (w * (w * r))) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = fma((-1.5 * r), r, 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(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))) - 4.5)
	tmp = 0.0
	if (t_1 <= -2e+278)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * w) * w) * Float64(0.125 / Float64(1.0 - v))) * Float64(-r)) * r);
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * Float64(w * Float64(w * r))) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(fma(Float64(-1.5 * r), r, 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 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]}, If[LessEqual[t$95$1, -2e+278], N[(N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] * N[(0.125 / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-r)), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(t$95$0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.99999999999999993e278

   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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 85.3

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in w around inf

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

       1. associate-*r/ N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. metadata-eval N/A

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

       8. fp-cancel-sign-sub-inv N/A

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

       9. +-commutative N/A

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

       10. associate-*l/ N/A

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

       11. lower-/.f64 N/A

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

   11. Applied rewrites 86.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

   13. Applied rewrites 92.2%

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

   if -1.99999999999999993e278 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 92.1%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 92.1

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

   5. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 92.1

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

   7. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if -1.5 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r \cdot r} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r \cdot r} \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right) \cdot w\right) \cdot \frac{0.125}{1 - v}\right) \cdot \left(-r\right)\right) \cdot r\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right) \cdot w\right) \cdot \frac{0.125}{1 - v}\right) \cdot \left(-r\right)\right) \cdot r\\ \mathbf{elif}\;t\_0 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0
         (-
          (-
           (+ 3.0 (/ 2.0 (* r r)))
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_0 -1e+235)
     (* (* (* (* (* (fma v -2.0 3.0) w) w) (/ 0.125 (- 1.0 v))) (- r)) r)
     (if (<= t_0 -1.5)
       (-
        (-
         3.0
         (/ (* (* (* (fma -2.0 v 3.0) 0.125) (* w r)) (* w r)) (- 1.0 v)))
        4.5)
       (/ (fma (* -1.5 r) r 2.0) (* r r))))))

C

double code(double v, double w, double r) {
	double t_0 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_0 <= -1e+235) {
		tmp = ((((fma(v, -2.0, 3.0) * w) * w) * (0.125 / (1.0 - v))) * -r) * r;
	} else if (t_0 <= -1.5) {
		tmp = (3.0 - ((((fma(-2.0, v, 3.0) * 0.125) * (w * r)) * (w * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = fma((-1.5 * r), r, 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = 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)
	tmp = 0.0
	if (t_0 <= -1e+235)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * w) * w) * Float64(0.125 / Float64(1.0 - v))) * Float64(-r)) * r);
	elseif (t_0 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * Float64(w * r)) * Float64(w * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(fma(Float64(-1.5 * r), r, 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -1e+235], N[(N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] * N[(0.125 / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-r)), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$0, -1.5], N[(N[(3.0 - N[(N[(N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.0000000000000001e235

   1. Initial program 82.1%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in w around inf

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

       1. associate-*r/ N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. metadata-eval N/A

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

       8. fp-cancel-sign-sub-inv N/A

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

       9. +-commutative N/A

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

       10. associate-*l/ N/A

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

       11. lower-/.f64 N/A

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

   11. Applied rewrites 85.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

   13. Applied rewrites 92.5%

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

   if -1.0000000000000001e235 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 98.6%

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

   if -1.5 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r \cdot r} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r \cdot r} \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right) \cdot w\right) \cdot \frac{0.125}{1 - v}\right) \cdot \left(-r\right)\right) \cdot r\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot r\right) \cdot r\\ t_1 := \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot t\_0}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right) \cdot w\right) \cdot \frac{0.125}{1 - v}\right) \cdot \left(-r\right)\right) \cdot r\\ \mathbf{elif}\;t\_1 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot t\_0}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* (* w w) r) r))
        (t_1
         (-
          (-
           (+ 3.0 (/ 2.0 (* r r)))
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) t_0) (- 1.0 v)))
          4.5)))
   (if (<= t_1 -1e+235)
     (* (* (* (* (* (fma v -2.0 3.0) w) w) (/ 0.125 (- 1.0 v))) (- r)) r)
     (if (<= t_1 -1.5)
       (- (- 3.0 (/ (* (fma -0.25 v 0.375) t_0) (- 1.0 v))) 4.5)
       (/ (fma (* -1.5 r) r 2.0) (* r r))))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -1e+235) {
		tmp = ((((fma(v, -2.0, 3.0) * w) * w) * (0.125 / (1.0 - v))) * -r) * r;
	} else if (t_1 <= -1.5) {
		tmp = (3.0 - ((fma(-0.25, v, 0.375) * t_0) / (1.0 - v))) - 4.5;
	} else {
		tmp = fma((-1.5 * r), r, 2.0) / (r * r);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * w) * r) * r)
	t_1 = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * t_0) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_1 <= -1e+235)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * w) * w) * Float64(0.125 / Float64(1.0 - v))) * Float64(-r)) * r);
	elseif (t_1 <= -1.5)
		tmp = Float64(Float64(3.0 - Float64(Float64(fma(-0.25, v, 0.375) * t_0) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(fma(Float64(-1.5 * r), r, 2.0) / Float64(r * r));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = 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] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+235], N[(N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision] * N[(0.125 / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-r)), $MachinePrecision] * r), $MachinePrecision], If[LessEqual[t$95$1, -1.5], N[(N[(3.0 - N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(-1.5 * r), $MachinePrecision] * r + 2.0), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.0000000000000001e235

   1. Initial program 82.1%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in w around inf

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

       1. associate-*r/ N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. metadata-eval N/A

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

       8. fp-cancel-sign-sub-inv N/A

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

       9. +-commutative N/A

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

       10. associate-*l/ N/A

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

       11. lower-/.f64 N/A

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

   11. Applied rewrites 85.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

   13. Applied rewrites 92.5%

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

   if -1.0000000000000001e235 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 91.5%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 90.4%

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

   if -1.5 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r \cdot r} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r \cdot r} \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot w\right) \cdot w\right) \cdot \frac{0.125}{1 - v}\right) \cdot \left(-r\right)\right) \cdot r\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.5:\\ \;\;\;\;\left(3 - \frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 80.4%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 88.5

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

   8. Applied rewrites 88.5%

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

   9. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto -\left(\left(w \cdot w\right) \cdot \frac{1}{4}\right) \cdot \left(r \cdot r\right) \]

       4. lift-*.f64 88.5

          \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   11. Applied rewrites 88.5%

       \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   if -inf.0 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 92.6%

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

   3. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 91.8%

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

   if -1.5 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r \cdot r} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r \cdot r} \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 94.6%

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

Alternative 9: 96.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 96.3

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

   8. Applied rewrites 96.3%

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

   if -1.5 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   3. Taylor expanded in 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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

         \[\leadsto {r}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot 2 - \frac{3}{2} \]

      5. metadata-eval N/A

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

      6. lower-pow.f64 99.9

         \[\leadsto {r}^{-2} \cdot 2 - 1.5 \]

   5. Applied rewrites 99.9%

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

Alternative 10: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq 20000:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot w\right) \cdot r\right) \cdot \frac{w \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;{r}^{-2} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<=
        (-
         (-
          t_0
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
         4.5)
        20000.0)
     (-
      (- t_0 (* (* (* (* (fma v -2.0 3.0) 0.125) w) r) (/ (* w r) (- 1.0 v))))
      4.5)
     (* (pow r -2.0) 2.0))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= 20000.0) {
		tmp = (t_0 - ((((fma(v, -2.0, 3.0) * 0.125) * w) * r) * ((w * r) / (1.0 - v)))) - 4.5;
	} else {
		tmp = pow(r, -2.0) * 2.0;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (Float64(Float64(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))) - 4.5) <= 20000.0)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * w) * r) * Float64(Float64(w * r) / Float64(1.0 - v)))) - 4.5);
	else
		tmp = Float64((r ^ -2.0) * 2.0);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 - 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], 20000.0], N[(N[(t$95$0 - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[Power[r, -2.0], $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < 2e4

   1. Initial program 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 91.1

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

   4. Applied rewrites 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 91.1

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

   6. Applied rewrites 91.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 96.3

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

   8. Applied rewrites 96.3%

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

   if 2e4 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 83.3%

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

   3. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow-flip N/A

         \[\leadsto {r}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot 2 \]

      6. metadata-eval N/A

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

      7. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 11: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(\left(-r\right) \cdot r\right)\\ \mathbf{elif}\;t\_1 \leq -0.005:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (-
           (+ 3.0 t_0)
           (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
          4.5)))
   (if (<= t_1 -1e+23)
     (* (* (* w w) 0.25) (* (- r) r))
     (if (<= t_1 -0.005) -1.5 t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -1e+23) {
		tmp = ((w * w) * 0.25) * (-r * r);
	} else if (t_1 <= -0.005) {
		tmp = -1.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = ((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
    if (t_1 <= (-1d+23)) then
        tmp = ((w * w) * 0.25d0) * (-r * r)
    else if (t_1 <= (-0.005d0)) then
        tmp = -1.5d0
    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 t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -1e+23) {
		tmp = ((w * w) * 0.25) * (-r * r);
	} else if (t_1 <= -0.005) {
		tmp = -1.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -1e+23:
		tmp = ((w * w) * 0.25) * (-r * r)
	elif t_1 <= -0.005:
		tmp = -1.5
	else:
		tmp = t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(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))) - 4.5)
	tmp = 0.0
	if (t_1 <= -1e+23)
		tmp = Float64(Float64(Float64(w * w) * 0.25) * Float64(Float64(-r) * r));
	elseif (t_1 <= -0.005)
		tmp = -1.5;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -1e+23)
		tmp = ((w * w) * 0.25) * (-r * r);
	elseif (t_1 <= -0.005)
		tmp = -1.5;
	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]}, Block[{t$95$1 = N[(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] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+23], N[(N[(N[(w * w), $MachinePrecision] * 0.25), $MachinePrecision] * N[((-r) * r), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.005], -1.5, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -9.9999999999999992e22

   1. Initial program 84.6%

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

   3. Taylor expanded in r around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in v around inf

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 75.1

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

   8. Applied rewrites 75.1%

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

   9. Taylor expanded in w around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

          \[\leadsto -\left(\left(w \cdot w\right) \cdot \frac{1}{4}\right) \cdot \left(r \cdot r\right) \]

       4. lift-*.f64 75.1

          \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   11. Applied rewrites 75.1%

       \[\leadsto -\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(r \cdot r\right) \]

   if -9.9999999999999992e22 < (-.f64 (-.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))) #s(literal 9/2 binary64)) < -0.0050000000000000001

   1. Initial program 89.3%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 81.5%

      \[\leadsto -1.5 \]

   if -0.0050000000000000001 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 83.4%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 99.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(w \cdot w\right) \cdot 0.25\right) \cdot \left(\left(-r\right) \cdot r\right)\\ \mathbf{elif}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -0.005:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.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))) #s(literal 9/2 binary64)) < -1.5

   1. Initial program 85.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 96.3

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

   8. Applied rewrites 96.3%

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

   if -1.5 < (-.f64 (-.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))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(r \cdot r\right) + 2}{r \cdot r} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-3}{2} \cdot r\right) \cdot r + 2}{r \cdot r} \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 13: 50.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.036:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.036) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.036) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.036d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.036) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.036:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.036)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.036)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.036], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 0.0359999999999999973

   1. Initial program 84.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 60.8%

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

   if 0.0359999999999999973 < r

   1. Initial program 86.4%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 23.2%

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

Alternative 14: 13.9% accurate, 73.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 84.7%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 + \frac{-3}{2} \cdot {r}^{2}}{\color{blue}{{r}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{-3}{2} \cdot {r}^{2} + 2}{{\color{blue}{r}}^{2}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, {r}^{2}, 2\right)}{{\color{blue}{r}}^{2}} \]

   4. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{{r}^{2}} \]

   6. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

   7. lift-*.f64 55.9

      \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot \color{blue}{r}} \]

5. Applied rewrites 55.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r}} \]

6. Taylor expanded in r around inf

   \[\leadsto \frac{-3}{2} \]
7. Step-by-step derivation

8. Applied rewrites 13.3%

   \[\leadsto -1.5 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025061 
(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))