Rosa's TurbineBenchmark

Percentage Accurate: 85.5% → 98.6%

Time: 4.5s

Alternatives: 10

Speedup: 1.2×

• 53.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 85.5% 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: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 1.3 \cdot 10^{+14}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(w \cdot r\_m\right) \cdot r\_m\right)}{1 - v}, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - -3\right) - \mathsf{fma}\left(\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot w\right) \cdot \left(w \cdot r\_m\right), \frac{r\_m}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 1.3e+14)
     (-
      t_0
      (fma (/ (* (fma -0.25 v 0.375) (* (* w r_m) r_m)) (- 1.0 v)) w 1.5))
     (-
      (- t_0 -3.0)
      (fma
       (* (* (* (fma -2.0 v 3.0) 0.125) w) (* w r_m))
       (/ r_m (- 1.0 v))
       4.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 1.3e+14) {
		tmp = t_0 - fma(((fma(-0.25, v, 0.375) * ((w * r_m) * r_m)) / (1.0 - v)), w, 1.5);
	} else {
		tmp = (t_0 - -3.0) - fma((((fma(-2.0, v, 3.0) * 0.125) * w) * (w * r_m)), (r_m / (1.0 - v)), 4.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 1.3e+14)
		tmp = Float64(t_0 - fma(Float64(Float64(fma(-0.25, v, 0.375) * Float64(Float64(w * r_m) * r_m)) / Float64(1.0 - v)), w, 1.5));
	else
		tmp = Float64(Float64(t_0 - -3.0) - fma(Float64(Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * w) * Float64(w * r_m)), Float64(r_m / Float64(1.0 - v)), 4.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 1.3e+14], N[(t$95$0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(N[(N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.3e14

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 92.9%

      \[\leadsto \color{blue}{-\left(1.5 - \left(\frac{2}{r \cdot r} - \frac{\left(\left(w \cdot r\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v} \cdot w\right)\right)} \]
   9. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate N/A

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

      4. lift--.f64 N/A

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

   10. Applied rewrites 92.9%

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

   if 1.3e14 < r

   1. Initial program 85.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   3. Applied rewrites 92.1%

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

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 7.4 \cdot 10^{-13}:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot r\_m\right) \cdot \left(w \cdot r\_m\right)}{1 - v} \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(w \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot \left(r\_m \cdot \left(w \cdot \frac{r\_m}{1 - v}\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 7.4e-13)
     (-
      (-
       1.5
       (- t_0 (* (/ (* (* (fma -0.25 v 0.375) r_m) (* w r_m)) (- 1.0 v)) w))))
     (-
      (-
       (+ 3.0 t_0)
       (* (* w (fma -0.25 v 0.375)) (* r_m (* w (/ r_m (- 1.0 v))))))
      4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 7.4e-13) {
		tmp = -(1.5 - (t_0 - ((((fma(-0.25, v, 0.375) * r_m) * (w * r_m)) / (1.0 - v)) * w)));
	} else {
		tmp = ((3.0 + t_0) - ((w * fma(-0.25, v, 0.375)) * (r_m * (w * (r_m / (1.0 - v)))))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 7.4e-13)
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(Float64(Float64(Float64(fma(-0.25, v, 0.375) * r_m) * Float64(w * r_m)) / Float64(1.0 - v)) * w))));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(w * fma(-0.25, v, 0.375)) * Float64(r_m * Float64(w * Float64(r_m / Float64(1.0 - v)))))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 7.4e-13], (-N[(1.5 - N[(t$95$0 - N[(N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * r$95$m), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(w * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(w * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 7.39999999999999977e-13

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 92.9%

      \[\leadsto \color{blue}{-\left(1.5 - \left(\frac{2}{r \cdot r} - \frac{\left(\left(w \cdot r\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v} \cdot w\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if 7.39999999999999977e-13 < r

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 93.0

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

   8. Applied rewrites 93.0%

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

Alternative 3: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 10^{+42}:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} - \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(w \cdot r\_m\right) \cdot r\_m\right)}{1 - v}, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(w \cdot r\_m\right)\right) \cdot \left(w \cdot r\_m\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1e+42)
   (-
    (/ 2.0 (* r_m r_m))
    (fma (/ (* (fma -0.25 v 0.375) (* (* w r_m) r_m)) (- 1.0 v)) w 1.5))
   (-
    (- 3.0 (/ (* (* (fma -0.25 v 0.375) (* w r_m)) (* w r_m)) (- 1.0 v)))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1e+42) {
		tmp = (2.0 / (r_m * r_m)) - fma(((fma(-0.25, v, 0.375) * ((w * r_m) * r_m)) / (1.0 - v)), w, 1.5);
	} else {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * (w * r_m)) * (w * r_m)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1e+42)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) - fma(Float64(Float64(fma(-0.25, v, 0.375) * Float64(Float64(w * r_m) * r_m)) / Float64(1.0 - v)), w, 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * Float64(w * r_m)) * Float64(w * r_m)) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1e+42], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.00000000000000004e42

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 92.9%

      \[\leadsto \color{blue}{-\left(1.5 - \left(\frac{2}{r \cdot r} - \frac{\left(\left(w \cdot r\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v} \cdot w\right)\right)} \]
   9. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate N/A

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

      4. lift--.f64 N/A

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

   10. Applied rewrites 92.9%

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

   if 1.00000000000000004e42 < r

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 52.6%

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

Alternative 4: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 3.4:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \frac{w \cdot \left(\left(\left(w \cdot r\_m\right) \cdot r\_m\right) \cdot 0.375\right)}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(w \cdot r\_m\right)\right) \cdot \left(w \cdot r\_m\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 3.4)
   (-
    (- (+ 3.0 (/ 2.0 (* r_m r_m))) (/ (* w (* (* (* w r_m) r_m) 0.375)) 1.0))
    4.5)
   (-
    (- 3.0 (/ (* (* (fma -0.25 v 0.375) (* w r_m)) (* w r_m)) (- 1.0 v)))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 3.4) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((w * (((w * r_m) * r_m) * 0.375)) / 1.0)) - 4.5;
	} else {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * (w * r_m)) * (w * r_m)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 3.4)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(w * Float64(Float64(Float64(w * r_m) * r_m) * 0.375)) / 1.0)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * Float64(w * r_m)) * Float64(w * r_m)) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 3.4], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(w * N[(N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.39999999999999991

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. swap-sqr N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 91.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 91.0

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

   6. Applied rewrites 91.0%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 84.2%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 92.0%

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

   if 3.39999999999999991 < r

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 52.6%

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

Alternative 5: 94.3% accurate, 0.6× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r_m r_m)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r_m) r_m)) (- 1.0 v)))
       4.5)
      -1.0)
   (-
    (- 3.0 (/ (* (* (fma -0.25 v 0.375) (* w r_m)) (* w r_m)) (- 1.0 v)))
    4.5)
   (- (- 1.5 (/ 2.0 (pow r_m 2.0))))))

C

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

Julia

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $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$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.0], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-N[(1.5 - N[(2.0 / N[Power[r$95$m, 2.0], $MachinePrecision]), $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

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 52.6%

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

   if -1 < (-.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 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 92.9%

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

   9. Taylor expanded in w around 0

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

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 57.8

          \[\leadsto -\left(1.5 - \frac{2}{{r}^{\color{blue}{2}}}\right) \]

   11. Applied rewrites 57.8%

       \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.1% accurate, 0.6× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r_m r_m)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r_m) r_m)) (- 1.0 v)))
       4.5)
      -1.0)
   (-
    (- 3.0 (/ (* w (* (* (* w r_m) r_m) (fma -0.25 v 0.375))) (- 1.0 v)))
    4.5)
   (- (- 1.5 (/ 2.0 (pow r_m 2.0))))))

C

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

Julia

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $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$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.0], N[(N[(3.0 - N[(N[(w * N[(N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-N[(1.5 - N[(2.0 / N[Power[r$95$m, 2.0], $MachinePrecision]), $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

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. swap-sqr N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 91.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 91.0

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

   6. Applied rewrites 91.0%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 48.4%

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

   if -1 < (-.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 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 92.9%

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

   9. Taylor expanded in w around 0

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

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 57.8

          \[\leadsto -\left(1.5 - \frac{2}{{r}^{\color{blue}{2}}}\right) \]

   11. Applied rewrites 57.8%

       \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 82.5% accurate, 0.6× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r_m r_m)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r_m) r_m)) (- 1.0 v)))
       4.5)
      -1.0)
   (- (- 3.0 (/ (* w (* (* (* w r_m) r_m) 0.375)) (- 1.0 v))) 4.5)
   (- (- 1.5 (/ 2.0 (pow r_m 2.0))))))

C

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

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if ((((3.0d0 + (2.0d0 / (r_m * r_m))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r_m) * r_m)) / (1.0d0 - v))) - 4.5d0) <= (-1.0d0)) then
        tmp = (3.0d0 - ((w * (((w * r_m) * r_m) * 0.375d0)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = -(1.5d0 - (2.0d0 / (r_m ** 2.0d0)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if ((((3.0 + (2.0 / (r_m * r_m))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r_m) * r_m)) / (1.0 - v))) - 4.5) <= -1.0) {
		tmp = (3.0 - ((w * (((w * r_m) * r_m) * 0.375)) / (1.0 - v))) - 4.5;
	} else {
		tmp = -(1.5 - (2.0 / Math.pow(r_m, 2.0)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if (((3.0 + (2.0 / (r_m * r_m))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r_m) * r_m)) / (1.0 - v))) - 4.5) <= -1.0:
		tmp = (3.0 - ((w * (((w * r_m) * r_m) * 0.375)) / (1.0 - v))) - 4.5
	else:
		tmp = -(1.5 - (2.0 / math.pow(r_m, 2.0)))
	return tmp

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $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$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.0], N[(N[(3.0 - N[(N[(w * N[(N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-N[(1.5 - N[(2.0 / N[Power[r$95$m, 2.0], $MachinePrecision]), $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

   1. Initial program 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. swap-sqr N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 91.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 91.0

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

   6. Applied rewrites 91.0%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 84.2%

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

   10. Taylor expanded in r around inf

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

   12. Applied rewrites 40.7%

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

   if -1 < (-.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 85.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 92.9%

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

   9. Taylor expanded in w around 0

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

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 57.8

          \[\leadsto -\left(1.5 - \frac{2}{{r}^{\color{blue}{2}}}\right) \]

   11. Applied rewrites 57.8%

       \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ -\left(1.5 - \frac{2}{{r\_m}^{2}}\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (- (- 1.5 (/ 2.0 (pow r_m 2.0)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return -(1.5 - (2.0 / pow(r_m, 2.0)));
}

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = -(1.5d0 - (2.0d0 / (r_m ** 2.0d0)))
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return -(1.5 - (2.0 / Math.pow(r_m, 2.0)));
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -(1.5 - (2.0 / math.pow(r_m, 2.0)))

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(-Float64(1.5 - Float64(2.0 / (r_m ^ 2.0))))
end

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := (-N[(1.5 - N[(2.0 / N[Power[r$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 85.5%

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

2. Taylor expanded in v around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 85.5

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

4. Applied rewrites 85.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. swap-sqr N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 95.1

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 95.1

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

6. Applied rewrites 95.1%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

8. Applied rewrites 92.9%

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

9. Taylor expanded in w around 0

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

    1. lower-/.f64 N/A

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

    2. lower-pow.f64 57.8

       \[\leadsto -\left(1.5 - \frac{2}{{r}^{\color{blue}{2}}}\right) \]

11. Applied rewrites 57.8%

    \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
12. Add Preprocessing

Alternative 9: 57.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \mathsf{fma}\left(\frac{0.6666666666666666}{r\_m \cdot r\_m} - -1, 3, -4.5\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (fma (- (/ 0.6666666666666666 (* r_m r_m)) -1.0) 3.0 -4.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return fma(((0.6666666666666666 / (r_m * r_m)) - -1.0), 3.0, -4.5);
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	return fma(Float64(Float64(0.6666666666666666 / Float64(r_m * r_m)) - -1.0), 3.0, -4.5)
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(0.6666666666666666 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * 3.0 + -4.5), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

2. Taylor expanded in v around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 85.5

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

4. Applied rewrites 85.5%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. associate--l+ N/A

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

   5. lift-+.f64 N/A

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

   6. sum-to-mult N/A

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

   7. mult-flip N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f64 N/A

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

6. Applied rewrites 85.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.6666666666666666}{r \cdot r} - -1, 3, \frac{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)}{v - 1} - 4.5\right)} \]

7. Taylor expanded in w around 0

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

9. Applied rewrites 57.7%

   \[\leadsto \mathsf{fma}\left(\frac{0.6666666666666666}{r \cdot r} - -1, 3, \color{blue}{-4.5}\right) \]
10. Add Preprocessing

Alternative 10: 44.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{2}{r\_m \cdot r\_m} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (/ 2.0 (* r_m r_m)))

C

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

Fortran

r_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = 2.0d0 / (r_m * r_m)
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return 2.0 / (r_m * r_m);
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return 2.0 / (r_m * r_m)

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 44.1

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

4. Applied rewrites 44.1%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 44.1

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

6. Applied rewrites 44.1%

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

Reproduce

herbie shell --seed 2025156 
(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))