Rosa's TurbineBenchmark

Percentage Accurate: 85.5% → 99.1%

Time: 4.9s

Alternatives: 11

Speedup: 1.3×

• 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: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 1.32 \cdot 10^{+194}:\\ \;\;\;\;\left(t\_0 - \mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(\left(r\_m \cdot \left(w \cdot r\_m\right)\right) \cdot \frac{w}{1 - v}\right)\right) - 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - r\_m \cdot \frac{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\_m\right)}{1 - v}\right) - 1.5\\ \end{array} \end{array} \]

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 1.32e+194)
     (-
      (- t_0 (* (fma -0.25 v 0.375) (* (* r_m (* w r_m)) (/ w (- 1.0 v)))))
      1.5)
     (-
      (- t_0 (* r_m (/ (* (* (fma v -0.25 0.375) w) (* w r_m)) (- 1.0 v))))
      1.5))))

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 <= 1.32e+194) {
		tmp = (t_0 - (fma(-0.25, v, 0.375) * ((r_m * (w * r_m)) * (w / (1.0 - v))))) - 1.5;
	} else {
		tmp = (t_0 - (r_m * (((fma(v, -0.25, 0.375) * w) * (w * r_m)) / (1.0 - v)))) - 1.5;
	}
	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 <= 1.32e+194)
		tmp = Float64(Float64(t_0 - Float64(fma(-0.25, v, 0.375) * Float64(Float64(r_m * Float64(w * r_m)) * Float64(w / Float64(1.0 - v))))) - 1.5);
	else
		tmp = Float64(Float64(t_0 - Float64(r_m * Float64(Float64(Float64(fma(v, -0.25, 0.375) * w) * Float64(w * r_m)) / Float64(1.0 - v)))) - 1.5);
	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, 1.32e+194], N[(N[(t$95$0 - N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(r$95$m * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision], N[(N[(t$95$0 - N[(r$95$m * N[(N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.32e194

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

   5. Applied rewrites 95.8%

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

   if 1.32e194 < r

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} - r \cdot \frac{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 5.3 \cdot 10^{+241}:\\ \;\;\;\;\left(t\_0 - \mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \left(w \cdot \left(r\_m \cdot \left(\frac{w}{1 - v} \cdot r\_m\right)\right)\right)\right) - 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - r\_m \cdot \frac{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\_m\right)}{1 - v}\right) - 1.5\\ \end{array} \end{array} \]

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 5.3e+241)
     (-
      (- t_0 (* (fma -0.25 v 0.375) (* w (* r_m (* (/ w (- 1.0 v)) r_m)))))
      1.5)
     (-
      (- t_0 (* r_m (/ (* (* (fma v -0.25 0.375) w) (* w r_m)) (- 1.0 v))))
      1.5))))

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 <= 5.3e+241) {
		tmp = (t_0 - (fma(-0.25, v, 0.375) * (w * (r_m * ((w / (1.0 - v)) * r_m))))) - 1.5;
	} else {
		tmp = (t_0 - (r_m * (((fma(v, -0.25, 0.375) * w) * (w * r_m)) / (1.0 - v)))) - 1.5;
	}
	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 <= 5.3e+241)
		tmp = Float64(Float64(t_0 - Float64(fma(-0.25, v, 0.375) * Float64(w * Float64(r_m * Float64(Float64(w / Float64(1.0 - v)) * r_m))))) - 1.5);
	else
		tmp = Float64(Float64(t_0 - Float64(r_m * Float64(Float64(Float64(fma(v, -0.25, 0.375) * w) * Float64(w * r_m)) / Float64(1.0 - v)))) - 1.5);
	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, 5.3e+241], N[(N[(t$95$0 - N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(w * N[(r$95$m * N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision], N[(N[(t$95$0 - N[(r$95$m * N[(N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 5.3000000000000001e241

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   if 5.3000000000000001e241 < r

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} - r \cdot \frac{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{w}{1 - v} \cdot r\_m\\ t_1 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 4 \cdot 10^{-116}:\\ \;\;\;\;\left(t\_1 - \left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r\_m \cdot t\_0\right)\right)\right) - 1.5\\ \mathbf{elif}\;r\_m \leq 6 \cdot 10^{+165}:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), t\_0 \cdot r\_m, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - r\_m \cdot \frac{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\_m\right)}{1 - v}\right) - 1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (/ w (- 1.0 v)) r_m)) (t_1 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 4e-116)
     (- (- t_1 (* (* v -0.25) (* w (* r_m t_0)))) 1.5)
     (if (<= r_m 6e+165)
       (- t_1 (fma (* w (fma v -0.25 0.375)) (* t_0 r_m) 1.5))
       (-
        (- t_1 (* r_m (/ (* (* (fma v -0.25 0.375) w) (* w r_m)) (- 1.0 v))))
        1.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (w / (1.0 - v)) * r_m;
	double t_1 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 4e-116) {
		tmp = (t_1 - ((v * -0.25) * (w * (r_m * t_0)))) - 1.5;
	} else if (r_m <= 6e+165) {
		tmp = t_1 - fma((w * fma(v, -0.25, 0.375)), (t_0 * r_m), 1.5);
	} else {
		tmp = (t_1 - (r_m * (((fma(v, -0.25, 0.375) * w) * (w * r_m)) / (1.0 - v)))) - 1.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(w / Float64(1.0 - v)) * r_m)
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 4e-116)
		tmp = Float64(Float64(t_1 - Float64(Float64(v * -0.25) * Float64(w * Float64(r_m * t_0)))) - 1.5);
	elseif (r_m <= 6e+165)
		tmp = Float64(t_1 - fma(Float64(w * fma(v, -0.25, 0.375)), Float64(t_0 * r_m), 1.5));
	else
		tmp = Float64(Float64(t_1 - Float64(r_m * Float64(Float64(Float64(fma(v, -0.25, 0.375) * w) * Float64(w * r_m)) / Float64(1.0 - v)))) - 1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 4e-116], N[(N[(t$95$1 - N[(N[(v * -0.25), $MachinePrecision] * N[(w * N[(r$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision], If[LessEqual[r$95$m, 6e+165], N[(t$95$1 - N[(N[(w * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * r$95$m), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[(r$95$m * N[(N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 4e-116

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 83.4%

       \[\leadsto \left(\frac{2}{r \cdot r} - \left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot r\right)\right)\right)\right) - 1.5 \]

   if 4e-116 < r < 5.99999999999999981e165

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.4

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 91.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.4

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

   7. Applied rewrites 91.4%

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

   if 5.99999999999999981e165 < r

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} - r \cdot \frac{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 1.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{w}{1 - v} \cdot r\_m\\ t_1 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 4 \cdot 10^{-116}:\\ \;\;\;\;\left(t\_1 - \left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r\_m \cdot t\_0\right)\right)\right) - 1.5\\ \mathbf{elif}\;r\_m \leq 7 \cdot 10^{+155}:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), t\_0 \cdot r\_m, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(\frac{\left(\left(w \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot r\_m\right) \cdot w}{1 - v}, r\_m, 1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (/ w (- 1.0 v)) r_m)) (t_1 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 4e-116)
     (- (- t_1 (* (* v -0.25) (* w (* r_m t_0)))) 1.5)
     (if (<= r_m 7e+155)
       (- t_1 (fma (* w (fma v -0.25 0.375)) (* t_0 r_m) 1.5))
       (-
        t_1
        (fma
         (/ (* (* (* w (fma -0.25 v 0.375)) r_m) w) (- 1.0 v))
         r_m
         1.5))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (w / (1.0 - v)) * r_m;
	double t_1 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 4e-116) {
		tmp = (t_1 - ((v * -0.25) * (w * (r_m * t_0)))) - 1.5;
	} else if (r_m <= 7e+155) {
		tmp = t_1 - fma((w * fma(v, -0.25, 0.375)), (t_0 * r_m), 1.5);
	} else {
		tmp = t_1 - fma(((((w * fma(-0.25, v, 0.375)) * r_m) * w) / (1.0 - v)), r_m, 1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(w / Float64(1.0 - v)) * r_m)
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 4e-116)
		tmp = Float64(Float64(t_1 - Float64(Float64(v * -0.25) * Float64(w * Float64(r_m * t_0)))) - 1.5);
	elseif (r_m <= 7e+155)
		tmp = Float64(t_1 - fma(Float64(w * fma(v, -0.25, 0.375)), Float64(t_0 * r_m), 1.5));
	else
		tmp = Float64(t_1 - fma(Float64(Float64(Float64(Float64(w * fma(-0.25, v, 0.375)) * r_m) * w) / Float64(1.0 - v)), r_m, 1.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 4e-116], N[(N[(t$95$1 - N[(N[(v * -0.25), $MachinePrecision] * N[(w * N[(r$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision], If[LessEqual[r$95$m, 7e+155], N[(t$95$1 - N[(N[(w * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * r$95$m), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(N[(N[(N[(N[(w * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision] * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r$95$m + 1.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 4e-116

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 83.4%

       \[\leadsto \left(\frac{2}{r \cdot r} - \left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot r\right)\right)\right)\right) - 1.5 \]

   if 4e-116 < r < 6.99999999999999969e155

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.4

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 91.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.4

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

   7. Applied rewrites 91.4%

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

   if 6.99999999999999969e155 < r

   1. Initial program 85.5%

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

   3. Applied rewrites 92.9%

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

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(\frac{\left(\left(w \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) \cdot r\right) \cdot w}{1 - v}, r, 1.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{w}{1 - v} \cdot r\_m\\ t_1 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;v \leq -2.25 \cdot 10^{+48}:\\ \;\;\;\;\left(t\_1 - \left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r\_m \cdot t\_0\right)\right)\right) - 1.5\\ \mathbf{elif}\;v \leq 1.12 \cdot 10^{-12}:\\ \;\;\;\;\left(t\_1 - \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(\left(w \cdot r\_m\right) \cdot r\_m\right)\right) \cdot w\right) - 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(\left(v \cdot -0.25\right) \cdot \left(w \cdot t\_0\right), r\_m, 1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (/ w (- 1.0 v)) r_m)) (t_1 (/ 2.0 (* r_m r_m))))
   (if (<= v -2.25e+48)
     (- (- t_1 (* (* v -0.25) (* w (* r_m t_0)))) 1.5)
     (if (<= v 1.12e-12)
       (- (- t_1 (* (* (fma v -0.25 0.375) (* (* w r_m) r_m)) w)) 1.5)
       (- t_1 (fma (* (* v -0.25) (* w t_0)) r_m 1.5))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (w / (1.0 - v)) * r_m;
	double t_1 = 2.0 / (r_m * r_m);
	double tmp;
	if (v <= -2.25e+48) {
		tmp = (t_1 - ((v * -0.25) * (w * (r_m * t_0)))) - 1.5;
	} else if (v <= 1.12e-12) {
		tmp = (t_1 - ((fma(v, -0.25, 0.375) * ((w * r_m) * r_m)) * w)) - 1.5;
	} else {
		tmp = t_1 - fma(((v * -0.25) * (w * t_0)), r_m, 1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(w / Float64(1.0 - v)) * r_m)
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (v <= -2.25e+48)
		tmp = Float64(Float64(t_1 - Float64(Float64(v * -0.25) * Float64(w * Float64(r_m * t_0)))) - 1.5);
	elseif (v <= 1.12e-12)
		tmp = Float64(Float64(t_1 - Float64(Float64(fma(v, -0.25, 0.375) * Float64(Float64(w * r_m) * r_m)) * w)) - 1.5);
	else
		tmp = Float64(t_1 - fma(Float64(Float64(v * -0.25) * Float64(w * t_0)), r_m, 1.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.25e+48], N[(N[(t$95$1 - N[(N[(v * -0.25), $MachinePrecision] * N[(w * N[(r$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision], If[LessEqual[v, 1.12e-12], N[(N[(t$95$1 - N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision], N[(t$95$1 - N[(N[(N[(v * -0.25), $MachinePrecision] * N[(w * t$95$0), $MachinePrecision]), $MachinePrecision] * r$95$m + 1.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.24999999999999998e48

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 83.4%

       \[\leadsto \left(\frac{2}{r \cdot r} - \left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot r\right)\right)\right)\right) - 1.5 \]

   if -2.24999999999999998e48 < v < 1.1200000000000001e-12

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 79.5

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 79.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.5

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

   10. Applied rewrites 79.5%

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

   if 1.1200000000000001e-12 < v

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

   4. Taylor expanded in v around inf

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

      1. lower-*.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 77.3

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

   8. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(\frac{w}{1 - v} \cdot r\right)\right), r, 1.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{w}{1 - v} \cdot r\_m\\ t_1 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 4 \cdot 10^{-116}:\\ \;\;\;\;\left(t\_1 - \left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r\_m \cdot t\_0\right)\right)\right) - 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), t\_0 \cdot r\_m, 1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (/ w (- 1.0 v)) r_m)) (t_1 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 4e-116)
     (- (- t_1 (* (* v -0.25) (* w (* r_m t_0)))) 1.5)
     (- t_1 (fma (* w (fma v -0.25 0.375)) (* t_0 r_m) 1.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (w / (1.0 - v)) * r_m;
	double t_1 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 4e-116) {
		tmp = (t_1 - ((v * -0.25) * (w * (r_m * t_0)))) - 1.5;
	} else {
		tmp = t_1 - fma((w * fma(v, -0.25, 0.375)), (t_0 * r_m), 1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(w / Float64(1.0 - v)) * r_m)
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 4e-116)
		tmp = Float64(Float64(t_1 - Float64(Float64(v * -0.25) * Float64(w * Float64(r_m * t_0)))) - 1.5);
	else
		tmp = Float64(t_1 - fma(Float64(w * fma(v, -0.25, 0.375)), Float64(t_0 * r_m), 1.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 4e-116], N[(N[(t$95$1 - N[(N[(v * -0.25), $MachinePrecision] * N[(w * N[(r$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision], N[(t$95$1 - N[(N[(w * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * r$95$m), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 4e-116

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in v around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 83.4%

       \[\leadsto \left(\frac{2}{r \cdot r} - \left(v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot r\right)\right)\right)\right) - 1.5 \]

   if 4e-116 < r

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.4

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 91.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.4

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

   7. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), \left(\frac{w}{1 - v} \cdot r\right) \cdot r, 1.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := t\_0 - \mathsf{fma}\left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(\frac{w}{1 - v} \cdot r\_m\right)\right), r\_m, 1.5\right)\\ \mathbf{if}\;v \leq -2.7 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.12 \cdot 10^{-12}:\\ \;\;\;\;\left(t\_0 - \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(\left(w \cdot r\_m\right) \cdot r\_m\right)\right) \cdot w\right) - 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1
         (- t_0 (fma (* (* v -0.25) (* w (* (/ w (- 1.0 v)) r_m))) r_m 1.5))))
   (if (<= v -2.7e+48)
     t_1
     (if (<= v 1.12e-12)
       (- (- t_0 (* (* (fma v -0.25 0.375) (* (* w r_m) r_m)) w)) 1.5)
       t_1))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = t_0 - fma(((v * -0.25) * (w * ((w / (1.0 - v)) * r_m))), r_m, 1.5);
	double tmp;
	if (v <= -2.7e+48) {
		tmp = t_1;
	} else if (v <= 1.12e-12) {
		tmp = (t_0 - ((fma(v, -0.25, 0.375) * ((w * r_m) * r_m)) * w)) - 1.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	t_1 = Float64(t_0 - fma(Float64(Float64(v * -0.25) * Float64(w * Float64(Float64(w / Float64(1.0 - v)) * r_m))), r_m, 1.5))
	tmp = 0.0
	if (v <= -2.7e+48)
		tmp = t_1;
	elseif (v <= 1.12e-12)
		tmp = Float64(Float64(t_0 - Float64(Float64(fma(v, -0.25, 0.375) * Float64(Float64(w * r_m) * r_m)) * w)) - 1.5);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(N[(v * -0.25), $MachinePrecision] * N[(w * N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r$95$m + 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.7e+48], t$95$1, If[LessEqual[v, 1.12e-12], N[(N[(t$95$0 - N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.70000000000000004e48 or 1.1200000000000001e-12 < v

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

   4. Taylor expanded in v around inf

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

      1. lower-*.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 77.3

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

   8. Applied rewrites 80.6%

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

   if -2.70000000000000004e48 < v < 1.1200000000000001e-12

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/l* N/A

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

      21. lift-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 79.5

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 79.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.5

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

   10. Applied rewrites 79.5%

       \[\leadsto \left(\frac{2}{r \cdot r} - \color{blue}{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) \cdot w}\right) - 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 4.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r\_m \cdot r\_m} - r\_m \cdot \frac{\left(0.375 \cdot w\right) \cdot \left(w \cdot r\_m\right)}{1}\right) - 1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 4.6e-101)
   (/ (/ 2.0 r_m) r_m)
   (- (- (/ 2.0 (* r_m r_m)) (* r_m (/ (* (* 0.375 w) (* w r_m)) 1.0))) 1.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 4.6e-101) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = ((2.0 / (r_m * r_m)) - (r_m * (((0.375 * w) * (w * r_m)) / 1.0))) - 1.5;
	}
	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 (r_m <= 4.6d-101) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = ((2.0d0 / (r_m * r_m)) - (r_m * (((0.375d0 * w) * (w * r_m)) / 1.0d0))) - 1.5d0
    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 (r_m <= 4.6e-101) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = ((2.0 / (r_m * r_m)) - (r_m * (((0.375 * w) * (w * r_m)) / 1.0))) - 1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 4.6e-101:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = ((2.0 / (r_m * r_m)) - (r_m * (((0.375 * w) * (w * r_m)) / 1.0))) - 1.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 4.6e-101)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(r_m * r_m)) - Float64(r_m * Float64(Float64(Float64(0.375 * w) * Float64(w * r_m)) / 1.0))) - 1.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 4.6e-101)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = ((2.0 / (r_m * r_m)) - (r_m * (((0.375 * w) * (w * r_m)) / 1.0))) - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 4.6e-101], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], N[(N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - N[(r$95$m * N[(N[(N[(0.375 * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.5999999999999999e-101

   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-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 44.1

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

   6. Applied rewrites 44.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 44.1

         \[\leadsto \frac{\frac{2}{r}}{r} \]

   8. Applied rewrites 44.1%

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

   if 4.5999999999999999e-101 < r

   1. Initial program 85.5%

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

   3. Applied rewrites 92.3%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 82.7%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 91.4%

      \[\leadsto \left(\frac{2}{r \cdot r} - r \cdot \frac{\left(0.375 \cdot w\right) \cdot \left(w \cdot r\right)}{\color{blue}{1}}\right) - 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\_m\right) \cdot r\_m\right)}{1 - v}\right) - 4.5 \leq -4000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.25, v, 0.375\right), \left(\frac{r\_m}{v - 1} \cdot r\_m\right) \cdot \left(w \cdot w\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - -3\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<=
        (-
         (-
          (+ 3.0 t_0)
          (/
           (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r_m) r_m))
           (- 1.0 v)))
         4.5)
        -4000000000000.0)
     (fma (fma -0.25 v 0.375) (* (* (/ r_m (- v 1.0)) r_m) (* w w)) -1.5)
     (- (- t_0 -3.0) 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 ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r_m) * r_m)) / (1.0 - v))) - 4.5) <= -4000000000000.0) {
		tmp = fma(fma(-0.25, v, 0.375), (((r_m / (v - 1.0)) * r_m) * (w * w)), -1.5);
	} else {
		tmp = (t_0 - -3.0) - 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 (Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r_m) * r_m)) / Float64(1.0 - v))) - 4.5) <= -4000000000000.0)
		tmp = fma(fma(-0.25, v, 0.375), Float64(Float64(Float64(r_m / Float64(v - 1.0)) * r_m) * Float64(w * w)), -1.5);
	else
		tmp = Float64(Float64(t_0 - -3.0) - 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[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -4000000000000.0], N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(N[(r$95$m / N[(v - 1.0), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(t$95$0 - -3.0), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -4e12

   1. Initial program 85.5%

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

   3. Applied rewrites 92.9%

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 47.5%

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

   if -4e12 < (-.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%

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

   3. Applied rewrites 92.9%

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

   4. Taylor expanded in w around 0

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

   6. Applied rewrites 57.8%

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

Alternative 10: 57.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \left(\frac{2}{r\_m \cdot r\_m} - -3\right) - 4.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (- (- (/ 2.0 (* r_m r_m)) -3.0) 4.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return ((2.0 / (r_m * r_m)) - -3.0) - 4.5;
}

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)) - (-3.0d0)) - 4.5d0
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)) - -3.0) - 4.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return ((2.0 / (r_m * r_m)) - -3.0) - 4.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(Float64(2.0 / Float64(r_m * r_m)) - -3.0) - 4.5)
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = ((2.0 / (r_m * r_m)) - -3.0) - 4.5;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

3. Applied rewrites 92.9%

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

4. Taylor expanded in w around 0

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

6. Applied rewrites 57.8%

   \[\leadsto \left(\frac{2}{r \cdot r} - \color{blue}{-3}\right) - 4.5 \]
7. Add Preprocessing

Alternative 11: 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-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 44.1

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

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))