Rosa's TurbineBenchmark

Percentage Accurate: 84.1% → 99.3%

Time: 5.3s

Alternatives: 16

Speedup: 1.1×

• 52.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

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

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

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

Initial Program: 84.1% accurate, 1.0× speedup

Math

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

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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} - 1.5\\ \mathbf{if}\;\left|w\right| \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{fma}\left(2, v, -3\right) \cdot \left|w\right|\right) \cdot \frac{r}{1 - v}\right) \cdot \left|w\right|\right) \cdot r, 0.125, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, v, -3\right) \cdot \left(\left(r \cdot r\right) \cdot \left|w\right|\right)}{1 - v} \cdot \left|w\right|, 0.125, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (* r r)) 1.5)))
   (if (<= (fabs w) 2.4e-40)
     (fma
      (* (* (* (* (fma 2.0 v -3.0) (fabs w)) (/ r (- 1.0 v))) (fabs w)) r)
      0.125
      t_0)
     (fma
      (* (/ (* (fma 2.0 v -3.0) (* (* r r) (fabs w))) (- 1.0 v)) (fabs w))
      0.125
      t_0))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / (r * r)) - 1.5;
	double tmp;
	if (fabs(w) <= 2.4e-40) {
		tmp = fma(((((fma(2.0, v, -3.0) * fabs(w)) * (r / (1.0 - v))) * fabs(w)) * r), 0.125, t_0);
	} else {
		tmp = fma((((fma(2.0, v, -3.0) * ((r * r) * fabs(w))) / (1.0 - v)) * fabs(w)), 0.125, t_0);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / Float64(r * r)) - 1.5)
	tmp = 0.0
	if (abs(w) <= 2.4e-40)
		tmp = fma(Float64(Float64(Float64(Float64(fma(2.0, v, -3.0) * abs(w)) * Float64(r / Float64(1.0 - v))) * abs(w)) * r), 0.125, t_0);
	else
		tmp = fma(Float64(Float64(Float64(fma(2.0, v, -3.0) * Float64(Float64(r * r) * abs(w))) / Float64(1.0 - v)) * abs(w)), 0.125, t_0);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]}, If[LessEqual[N[Abs[w], $MachinePrecision], 2.4e-40], N[(N[(N[(N[(N[(N[(2.0 * v + -3.0), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * 0.125 + t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * v + -3.0), $MachinePrecision] * N[(N[(r * r), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * 0.125 + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.39999999999999991e-40

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

   5. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 92.1%

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

   if 2.39999999999999991e-40 < w

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 97.7

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

   7. Applied rewrites 97.7%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-fma.f64 N/A

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

   9. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, v, -3\right) \cdot \left(\left(r \cdot r\right) \cdot w\right)}{1 - v} \cdot w, 0.125, \frac{2}{r \cdot r} - 1.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 92.5

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

3. Applied rewrites 92.5%

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

5. Applied rewrites 99.3%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left|w\right| \leq 2.5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{fma}\left(2, v, -3\right) \cdot \left|w\right|\right) \cdot \frac{r}{1 - v}\right) \cdot \left|w\right|\right) \cdot r, 0.125, t\_0 - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{0.375 \cdot \left(\left(\left(r \cdot r\right) \cdot \left|w\right|\right) \cdot \left|w\right|\right)}{1}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (fabs w) 2.5e+151)
     (fma
      (* (* (* (* (fma 2.0 v -3.0) (fabs w)) (/ r (- 1.0 v))) (fabs w)) r)
      0.125
      (- t_0 1.5))
     (-
      (- (+ 3.0 t_0) (/ (* 0.375 (* (* (* r r) (fabs w)) (fabs w))) 1.0))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (fabs(w) <= 2.5e+151) {
		tmp = fma(((((fma(2.0, v, -3.0) * fabs(w)) * (r / (1.0 - v))) * fabs(w)) * r), 0.125, (t_0 - 1.5));
	} else {
		tmp = ((3.0 + t_0) - ((0.375 * (((r * r) * fabs(w)) * fabs(w))) / 1.0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (abs(w) <= 2.5e+151)
		tmp = fma(Float64(Float64(Float64(Float64(fma(2.0, v, -3.0) * abs(w)) * Float64(r / Float64(1.0 - v))) * abs(w)) * r), 0.125, Float64(t_0 - 1.5));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(0.375 * Float64(Float64(Float64(r * r) * abs(w)) * abs(w))) / 1.0)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[w], $MachinePrecision], 2.5e+151], N[(N[(N[(N[(N[(N[(2.0 * v + -3.0), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * 0.125 + N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.375 * N[(N[(N[(r * r), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.5000000000000001e151

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

   5. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 92.1%

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

   if 2.5000000000000001e151 < w

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 91.0

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 86.5

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

   12. Applied rewrites 86.5%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{r}{1 - v}\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1600000:\\ \;\;\;\;\left(t\_1 - \left(\left(t\_0 \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \left(-0.25 \cdot v\right)\right) - 4.5\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(t\_1 - \frac{\left(w \cdot r\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \left(-0.25 \cdot v\right) \cdot \left(\left(\left(w \cdot t\_0\right) \cdot r\right) \cdot w\right)\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ r (- 1.0 v))) (t_1 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= v -1600000.0)
     (- (- t_1 (* (* (* t_0 w) (* w r)) (* -0.25 v))) 4.5)
     (if (<= v 1.5)
       (- (- t_1 (/ (* (* w r) (* (* 0.375 w) r)) (- 1.0 v))) 4.5)
       (- (- t_1 (* (* -0.25 v) (* (* (* w t_0) r) w))) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = r / (1.0 - v);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -1600000.0) {
		tmp = (t_1 - (((t_0 * w) * (w * r)) * (-0.25 * v))) - 4.5;
	} else if (v <= 1.5) {
		tmp = (t_1 - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_1 - ((-0.25 * v) * (((w * t_0) * r) * w))) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r / (1.0d0 - v)
    t_1 = 3.0d0 + (2.0d0 / (r * r))
    if (v <= (-1600000.0d0)) then
        tmp = (t_1 - (((t_0 * w) * (w * r)) * ((-0.25d0) * v))) - 4.5d0
    else if (v <= 1.5d0) then
        tmp = (t_1 - (((w * r) * ((0.375d0 * w) * r)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = (t_1 - (((-0.25d0) * v) * (((w * t_0) * r) * w))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r / (1.0 - v);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= -1600000.0) {
		tmp = (t_1 - (((t_0 * w) * (w * r)) * (-0.25 * v))) - 4.5;
	} else if (v <= 1.5) {
		tmp = (t_1 - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_1 - ((-0.25 * v) * (((w * t_0) * r) * w))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r / (1.0 - v)
	t_1 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if v <= -1600000.0:
		tmp = (t_1 - (((t_0 * w) * (w * r)) * (-0.25 * v))) - 4.5
	elif v <= 1.5:
		tmp = (t_1 - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5
	else:
		tmp = (t_1 - ((-0.25 * v) * (((w * t_0) * r) * w))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r / Float64(1.0 - v))
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (v <= -1600000.0)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(t_0 * w) * Float64(w * r)) * Float64(-0.25 * v))) - 4.5);
	elseif (v <= 1.5)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(w * r) * Float64(Float64(0.375 * w) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(-0.25 * v) * Float64(Float64(Float64(w * t_0) * r) * w))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r / (1.0 - v);
	t_1 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (v <= -1600000.0)
		tmp = (t_1 - (((t_0 * w) * (w * r)) * (-0.25 * v))) - 4.5;
	elseif (v <= 1.5)
		tmp = (t_1 - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	else
		tmp = (t_1 - ((-0.25 * v) * (((w * t_0) * r) * w))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1600000.0], N[(N[(t$95$1 - N[(N[(N[(t$95$0 * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 1.5], N[(N[(t$95$1 - N[(N[(N[(w * r), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(-0.25 * v), $MachinePrecision] * N[(N[(N[(w * t$95$0), $MachinePrecision] * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.6e6

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.4

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

   6. Applied rewrites 84.5%

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

   if -1.6e6 < v < 1.5

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 1.5 < v

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

   8. Applied rewrites 84.4%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := w \cdot \frac{r}{1 - v}\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := 3 + t\_1\\ \mathbf{if}\;v \leq -1600000:\\ \;\;\;\;\left(t\_1 - -3\right) - \mathsf{fma}\left(\left(-0.25 \cdot v\right) \cdot \left(w \cdot r\right), t\_0, 4.5\right)\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(t\_2 - \frac{\left(w \cdot r\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 - \left(-0.25 \cdot v\right) \cdot \left(\left(t\_0 \cdot r\right) \cdot w\right)\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (/ r (- 1.0 v)))) (t_1 (/ 2.0 (* r r))) (t_2 (+ 3.0 t_1)))
   (if (<= v -1600000.0)
     (- (- t_1 -3.0) (fma (* (* -0.25 v) (* w r)) t_0 4.5))
     (if (<= v 1.5)
       (- (- t_2 (/ (* (* w r) (* (* 0.375 w) r)) (- 1.0 v))) 4.5)
       (- (- t_2 (* (* -0.25 v) (* (* t_0 r) w))) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r / (1.0 - v));
	double t_1 = 2.0 / (r * r);
	double t_2 = 3.0 + t_1;
	double tmp;
	if (v <= -1600000.0) {
		tmp = (t_1 - -3.0) - fma(((-0.25 * v) * (w * r)), t_0, 4.5);
	} else if (v <= 1.5) {
		tmp = (t_2 - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_2 - ((-0.25 * v) * ((t_0 * r) * w))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r / Float64(1.0 - v)))
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(3.0 + t_1)
	tmp = 0.0
	if (v <= -1600000.0)
		tmp = Float64(Float64(t_1 - -3.0) - fma(Float64(Float64(-0.25 * v) * Float64(w * r)), t_0, 4.5));
	elseif (v <= 1.5)
		tmp = Float64(Float64(t_2 - Float64(Float64(Float64(w * r) * Float64(Float64(0.375 * w) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_2 - Float64(Float64(-0.25 * v) * Float64(Float64(t_0 * r) * w))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + t$95$1), $MachinePrecision]}, If[LessEqual[v, -1600000.0], N[(N[(t$95$1 - -3.0), $MachinePrecision] - N[(N[(N[(-0.25 * v), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.5], N[(N[(t$95$2 - N[(N[(N[(w * r), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$2 - N[(N[(-0.25 * v), $MachinePrecision] * N[(N[(t$95$0 * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.6e6

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

   8. Applied rewrites 80.5%

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

   if -1.6e6 < v < 1.5

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 1.5 < v

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

   8. Applied rewrites 84.4%

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

Alternative 6: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - -3\right) - \mathsf{fma}\left(\left(-0.25 \cdot v\right) \cdot \left(w \cdot r\right), w \cdot \frac{r}{1 - v}, 4.5\right)\\ \mathbf{if}\;v \leq -1600000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{\left(w \cdot r\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (- t_0 -3.0)
          (fma (* (* -0.25 v) (* w r)) (* w (/ r (- 1.0 v))) 4.5))))
   (if (<= v -1600000.0)
     t_1
     (if (<= v 1.5)
       (- (- (+ 3.0 t_0) (/ (* (* w r) (* (* 0.375 w) r)) (- 1.0 v))) 4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 - -3.0) - fma(((-0.25 * v) * (w * r)), (w * (r / (1.0 - v))), 4.5);
	double tmp;
	if (v <= -1600000.0) {
		tmp = t_1;
	} else if (v <= 1.5) {
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 - -3.0) - fma(Float64(Float64(-0.25 * v) * Float64(w * r)), Float64(w * Float64(r / Float64(1.0 - v))), 4.5))
	tmp = 0.0
	if (v <= -1600000.0)
		tmp = t_1;
	elseif (v <= 1.5)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(w * r) * Float64(Float64(0.375 * w) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(N[(N[(-0.25 * v), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1600000.0], t$95$1, If[LessEqual[v, 1.5], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(w * r), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.6e6 or 1.5 < v

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

   8. Applied rewrites 80.5%

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

   if -1.6e6 < v < 1.5

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

Alternative 7: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - -3\right) - \mathsf{fma}\left(w, \left(r \cdot w\right) \cdot \left(\left(-0.25 \cdot v\right) \cdot \frac{r}{1 - v}\right), 4.5\right)\\ \mathbf{if}\;v \leq -1600000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{\left(w \cdot r\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (-
          (- t_0 -3.0)
          (fma w (* (* r w) (* (* -0.25 v) (/ r (- 1.0 v)))) 4.5))))
   (if (<= v -1600000.0)
     t_1
     (if (<= v 1.5)
       (- (- (+ 3.0 t_0) (/ (* (* w r) (* (* 0.375 w) r)) (- 1.0 v))) 4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 - -3.0) - fma(w, ((r * w) * ((-0.25 * v) * (r / (1.0 - v)))), 4.5);
	double tmp;
	if (v <= -1600000.0) {
		tmp = t_1;
	} else if (v <= 1.5) {
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 - -3.0) - fma(w, Float64(Float64(r * w) * Float64(Float64(-0.25 * v) * Float64(r / Float64(1.0 - v)))), 4.5))
	tmp = 0.0
	if (v <= -1600000.0)
		tmp = t_1;
	elseif (v <= 1.5)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(w * r) * Float64(Float64(0.375 * w) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(w * N[(N[(r * w), $MachinePrecision] * N[(N[(-0.25 * v), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1600000.0], t$95$1, If[LessEqual[v, 1.5], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(w * r), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.6e6 or 1.5 < v

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   6. Applied rewrites 73.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   8. Applied rewrites 81.4%

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

   if -1.6e6 < v < 1.5

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

Alternative 8: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -7500000000000:\\ \;\;\;\;\left(t\_0 - 1.5\right) - r \cdot \left(\left(w \cdot w\right) \cdot \left(\left(-0.25 \cdot v\right) \cdot \frac{r}{1 - v}\right)\right)\\ \mathbf{elif}\;v \leq 55000000:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{\left(w \cdot r\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - w \cdot \frac{\left(\left(r \cdot r\right) \cdot w\right) \cdot \left(-0.25 \cdot v\right)}{1 - v}\right)\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -7500000000000.0)
     (- (- t_0 1.5) (* r (* (* w w) (* (* -0.25 v) (/ r (- 1.0 v))))))
     (if (<= v 55000000.0)
       (- (- (+ 3.0 t_0) (/ (* (* w r) (* (* 0.375 w) r)) (- 1.0 v))) 4.5)
       (-
        (- 1.5 (- t_0 (* w (/ (* (* (* r r) w) (* -0.25 v)) (- 1.0 v))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -7500000000000.0) {
		tmp = (t_0 - 1.5) - (r * ((w * w) * ((-0.25 * v) * (r / (1.0 - v)))));
	} else if (v <= 55000000.0) {
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = -(1.5 - (t_0 - (w * ((((r * r) * w) * (-0.25 * v)) / (1.0 - v)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (v <= (-7500000000000.0d0)) then
        tmp = (t_0 - 1.5d0) - (r * ((w * w) * (((-0.25d0) * v) * (r / (1.0d0 - v)))))
    else if (v <= 55000000.0d0) then
        tmp = ((3.0d0 + t_0) - (((w * r) * ((0.375d0 * w) * r)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = -(1.5d0 - (t_0 - (w * ((((r * r) * w) * ((-0.25d0) * v)) / (1.0d0 - v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -7500000000000.0) {
		tmp = (t_0 - 1.5) - (r * ((w * w) * ((-0.25 * v) * (r / (1.0 - v)))));
	} else if (v <= 55000000.0) {
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = -(1.5 - (t_0 - (w * ((((r * r) * w) * (-0.25 * v)) / (1.0 - v)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -7500000000000.0:
		tmp = (t_0 - 1.5) - (r * ((w * w) * ((-0.25 * v) * (r / (1.0 - v)))))
	elif v <= 55000000.0:
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5
	else:
		tmp = -(1.5 - (t_0 - (w * ((((r * r) * w) * (-0.25 * v)) / (1.0 - v)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -7500000000000.0)
		tmp = Float64(Float64(t_0 - 1.5) - Float64(r * Float64(Float64(w * w) * Float64(Float64(-0.25 * v) * Float64(r / Float64(1.0 - v))))));
	elseif (v <= 55000000.0)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(w * r) * Float64(Float64(0.375 * w) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(w * Float64(Float64(Float64(Float64(r * r) * w) * Float64(-0.25 * v)) / Float64(1.0 - v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -7500000000000.0)
		tmp = (t_0 - 1.5) - (r * ((w * w) * ((-0.25 * v) * (r / (1.0 - v)))));
	elseif (v <= 55000000.0)
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	else
		tmp = -(1.5 - (t_0 - (w * ((((r * r) * w) * (-0.25 * v)) / (1.0 - v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -7500000000000.0], N[(N[(t$95$0 - 1.5), $MachinePrecision] - N[(r * N[(N[(w * w), $MachinePrecision] * N[(N[(-0.25 * v), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 55000000.0], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(w * r), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-N[(1.5 - N[(t$95$0 - N[(w * N[(N[(N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision] * N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if v < -7.5e12

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   6. Applied rewrites 73.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

   8. Applied rewrites 73.8%

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

   if -7.5e12 < v < 5.5e7

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

   if 5.5e7 < v

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 76.2%

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

Alternative 9: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - 1.5\right) - r \cdot \left(\left(w \cdot w\right) \cdot \left(\left(-0.25 \cdot v\right) \cdot \frac{r}{1 - v}\right)\right)\\ \mathbf{if}\;v \leq -7500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.55:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{\left(w \cdot r\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (- (- t_0 1.5) (* r (* (* w w) (* (* -0.25 v) (/ r (- 1.0 v))))))))
   (if (<= v -7500000000000.0)
     t_1
     (if (<= v 1.55)
       (- (- (+ 3.0 t_0) (/ (* (* w r) (* (* 0.375 w) r)) (- 1.0 v))) 4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 - 1.5) - (r * ((w * w) * ((-0.25 * v) * (r / (1.0 - v)))));
	double tmp;
	if (v <= -7500000000000.0) {
		tmp = t_1;
	} else if (v <= 1.55) {
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (t_0 - 1.5d0) - (r * ((w * w) * (((-0.25d0) * v) * (r / (1.0d0 - v)))))
    if (v <= (-7500000000000.0d0)) then
        tmp = t_1
    else if (v <= 1.55d0) then
        tmp = ((3.0d0 + t_0) - (((w * r) * ((0.375d0 * w) * r)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (t_0 - 1.5) - (r * ((w * w) * ((-0.25 * v) * (r / (1.0 - v)))));
	double tmp;
	if (v <= -7500000000000.0) {
		tmp = t_1;
	} else if (v <= 1.55) {
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (t_0 - 1.5) - (r * ((w * w) * ((-0.25 * v) * (r / (1.0 - v)))))
	tmp = 0
	if v <= -7500000000000.0:
		tmp = t_1
	elif v <= 1.55:
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(t_0 - 1.5) - Float64(r * Float64(Float64(w * w) * Float64(Float64(-0.25 * v) * Float64(r / Float64(1.0 - v))))))
	tmp = 0.0
	if (v <= -7500000000000.0)
		tmp = t_1;
	elseif (v <= 1.55)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(w * r) * Float64(Float64(0.375 * w) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (t_0 - 1.5) - (r * ((w * w) * ((-0.25 * v) * (r / (1.0 - v)))));
	tmp = 0.0;
	if (v <= -7500000000000.0)
		tmp = t_1;
	elseif (v <= 1.55)
		tmp = ((3.0 + t_0) - (((w * r) * ((0.375 * w) * r)) / (1.0 - v))) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 1.5), $MachinePrecision] - N[(r * N[(N[(w * w), $MachinePrecision] * N[(N[(-0.25 * v), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -7500000000000.0], t$95$1, If[LessEqual[v, 1.55], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(w * r), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -7.5e12 or 1.55000000000000004 < v

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   6. Applied rewrites 73.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

   8. Applied rewrites 73.8%

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

   if -7.5e12 < v < 1.55000000000000004

   1. Initial program 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   3. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

Alternative 10: 92.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;\left|w\right| \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\left(t\_0 - \frac{0.375 \cdot \left(\left(\left(r \cdot \left|w\right|\right) \cdot \left|w\right|\right) \cdot r\right)}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \frac{0.375 \cdot \left(\left(\left(r \cdot r\right) \cdot \left|w\right|\right) \cdot \left|w\right|\right)}{1}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= (fabs w) 4e+68)
     (- (- t_0 (/ (* 0.375 (* (* (* r (fabs w)) (fabs w)) r)) 1.0)) 4.5)
     (- (- t_0 (/ (* 0.375 (* (* (* r r) (fabs w)) (fabs w))) 1.0)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (fabs(w) <= 4e+68) {
		tmp = (t_0 - ((0.375 * (((r * fabs(w)) * fabs(w)) * r)) / 1.0)) - 4.5;
	} else {
		tmp = (t_0 - ((0.375 * (((r * r) * fabs(w)) * fabs(w))) / 1.0)) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    if (abs(w) <= 4d+68) then
        tmp = (t_0 - ((0.375d0 * (((r * abs(w)) * abs(w)) * r)) / 1.0d0)) - 4.5d0
    else
        tmp = (t_0 - ((0.375d0 * (((r * r) * abs(w)) * abs(w))) / 1.0d0)) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (Math.abs(w) <= 4e+68) {
		tmp = (t_0 - ((0.375 * (((r * Math.abs(w)) * Math.abs(w)) * r)) / 1.0)) - 4.5;
	} else {
		tmp = (t_0 - ((0.375 * (((r * r) * Math.abs(w)) * Math.abs(w))) / 1.0)) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if math.fabs(w) <= 4e+68:
		tmp = (t_0 - ((0.375 * (((r * math.fabs(w)) * math.fabs(w)) * r)) / 1.0)) - 4.5
	else:
		tmp = (t_0 - ((0.375 * (((r * r) * math.fabs(w)) * math.fabs(w))) / 1.0)) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (abs(w) <= 4e+68)
		tmp = Float64(Float64(t_0 - Float64(Float64(0.375 * Float64(Float64(Float64(r * abs(w)) * abs(w)) * r)) / 1.0)) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(0.375 * Float64(Float64(Float64(r * r) * abs(w)) * abs(w))) / 1.0)) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (abs(w) <= 4e+68)
		tmp = (t_0 - ((0.375 * (((r * abs(w)) * abs(w)) * r)) / 1.0)) - 4.5;
	else
		tmp = (t_0 - ((0.375 * (((r * r) * abs(w)) * abs(w))) / 1.0)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[w], $MachinePrecision], 4e+68], N[(N[(t$95$0 - N[(N[(0.375 * N[(N[(N[(r * N[Abs[w], $MachinePrecision]), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(0.375 * N[(N[(N[(r * r), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 3.99999999999999981e68

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 89.7

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 89.7

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

   12. Applied rewrites 89.7%

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

   if 3.99999999999999981e68 < w

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 91.0

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 86.5

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

   12. Applied rewrites 86.5%

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

Alternative 11: 91.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left|w\right| \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\left(t\_0 - -3\right) - \mathsf{fma}\left(\left(0.375 \cdot r\right) \cdot \left(\left|w\right| \cdot \left|w\right|\right), \frac{r}{1}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{0.375 \cdot \left(\left(\left(r \cdot r\right) \cdot \left|w\right|\right) \cdot \left|w\right|\right)}{1}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (fabs w) 2e-46)
     (- (- t_0 -3.0) (fma (* (* 0.375 r) (* (fabs w) (fabs w))) (/ r 1.0) 4.5))
     (-
      (- (+ 3.0 t_0) (/ (* 0.375 (* (* (* r r) (fabs w)) (fabs w))) 1.0))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (fabs(w) <= 2e-46) {
		tmp = (t_0 - -3.0) - fma(((0.375 * r) * (fabs(w) * fabs(w))), (r / 1.0), 4.5);
	} else {
		tmp = ((3.0 + t_0) - ((0.375 * (((r * r) * fabs(w)) * fabs(w))) / 1.0)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (abs(w) <= 2e-46)
		tmp = Float64(Float64(t_0 - -3.0) - fma(Float64(Float64(0.375 * r) * Float64(abs(w) * abs(w))), Float64(r / 1.0), 4.5));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(0.375 * Float64(Float64(Float64(r * r) * abs(w)) * abs(w))) / 1.0)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[w], $MachinePrecision], 2e-46], N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(N[(N[(0.375 * r), $MachinePrecision] * N[(N[Abs[w], $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r / 1.0), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.375 * N[(N[(N[(r * r), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.00000000000000005e-46

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 82.3%

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

       1. lift--.f64 N/A

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

       2. lift--.f64 N/A

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

       3. associate--l- N/A

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

       4. lower--.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

       8. sub-flip N/A

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

       9. lift--.f64 N/A

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

   12. Applied rewrites 82.3%

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

   if 2.00000000000000005e-46 < w

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 91.0

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*l* N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 86.5

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

   12. Applied rewrites 86.5%

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

Alternative 12: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|r\right| \leq 3.4 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{2}{\left|r\right|}}{\left|r\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\left|r\right| \cdot \left|r\right|} - -3\right) - \mathsf{fma}\left(\left(0.375 \cdot \left|r\right|\right) \cdot \left(w \cdot w\right), \frac{\left|r\right|}{1}, 4.5\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= (fabs r) 3.4e-125)
   (/ (/ 2.0 (fabs r)) (fabs r))
   (-
    (- (/ 2.0 (* (fabs r) (fabs r))) -3.0)
    (fma (* (* 0.375 (fabs r)) (* w w)) (/ (fabs r) 1.0) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (fabs(r) <= 3.4e-125) {
		tmp = (2.0 / fabs(r)) / fabs(r);
	} else {
		tmp = ((2.0 / (fabs(r) * fabs(r))) - -3.0) - fma(((0.375 * fabs(r)) * (w * w)), (fabs(r) / 1.0), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (abs(r) <= 3.4e-125)
		tmp = Float64(Float64(2.0 / abs(r)) / abs(r));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(abs(r) * abs(r))) - -3.0) - fma(Float64(Float64(0.375 * abs(r)) * Float64(w * w)), Float64(abs(r) / 1.0), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[Abs[r], $MachinePrecision], 3.4e-125], N[(N[(2.0 / N[Abs[r], $MachinePrecision]), $MachinePrecision] / N[Abs[r], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision] - N[(N[(N[(0.375 * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[r], $MachinePrecision] / 1.0), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.39999999999999975e-125

   1. Initial program 84.1%

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

   2. 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 44.7

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

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   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 44.7

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

   6. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   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 44.7

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

   8. Applied rewrites 44.7%

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

   if 3.39999999999999975e-125 < r

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 82.3%

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

       1. lift--.f64 N/A

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

       2. lift--.f64 N/A

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

       3. associate--l- N/A

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

       4. lower--.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

       8. sub-flip N/A

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

       9. lift--.f64 N/A

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

   12. Applied rewrites 82.3%

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

Alternative 13: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|r\right| \leq 3.4 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{2}{\left|r\right|}}{\left|r\right|}\\ \mathbf{else}:\\ \;\;\;\;-\left(1.5 - \left(\frac{2}{\left|r\right| \cdot \left|r\right|} - \frac{\left(0.375 \cdot \left|r\right|\right) \cdot \left(\left(w \cdot w\right) \cdot \left|r\right|\right)}{1}\right)\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= (fabs r) 3.4e-125)
   (/ (/ 2.0 (fabs r)) (fabs r))
   (-
    (-
     1.5
     (-
      (/ 2.0 (* (fabs r) (fabs r)))
      (/ (* (* 0.375 (fabs r)) (* (* w w) (fabs r))) 1.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (fabs(r) <= 3.4e-125) {
		tmp = (2.0 / fabs(r)) / fabs(r);
	} else {
		tmp = -(1.5 - ((2.0 / (fabs(r) * fabs(r))) - (((0.375 * fabs(r)) * ((w * w) * fabs(r))) / 1.0)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (abs(r) <= 3.4d-125) then
        tmp = (2.0d0 / abs(r)) / abs(r)
    else
        tmp = -(1.5d0 - ((2.0d0 / (abs(r) * abs(r))) - (((0.375d0 * abs(r)) * ((w * w) * abs(r))) / 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (Math.abs(r) <= 3.4e-125) {
		tmp = (2.0 / Math.abs(r)) / Math.abs(r);
	} else {
		tmp = -(1.5 - ((2.0 / (Math.abs(r) * Math.abs(r))) - (((0.375 * Math.abs(r)) * ((w * w) * Math.abs(r))) / 1.0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if math.fabs(r) <= 3.4e-125:
		tmp = (2.0 / math.fabs(r)) / math.fabs(r)
	else:
		tmp = -(1.5 - ((2.0 / (math.fabs(r) * math.fabs(r))) - (((0.375 * math.fabs(r)) * ((w * w) * math.fabs(r))) / 1.0)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (abs(r) <= 3.4e-125)
		tmp = Float64(Float64(2.0 / abs(r)) / abs(r));
	else
		tmp = Float64(-Float64(1.5 - Float64(Float64(2.0 / Float64(abs(r) * abs(r))) - Float64(Float64(Float64(0.375 * abs(r)) * Float64(Float64(w * w) * abs(r))) / 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (abs(r) <= 3.4e-125)
		tmp = (2.0 / abs(r)) / abs(r);
	else
		tmp = -(1.5 - ((2.0 / (abs(r) * abs(r))) - (((0.375 * abs(r)) * ((w * w) * abs(r))) / 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[Abs[r], $MachinePrecision], 3.4e-125], N[(N[(2.0 / N[Abs[r], $MachinePrecision]), $MachinePrecision] / N[Abs[r], $MachinePrecision]), $MachinePrecision], (-N[(1.5 - N[(N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.375 * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if r < 3.39999999999999975e-125

   1. Initial program 84.1%

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

   2. 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 44.7

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

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   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 44.7

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

   6. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   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 44.7

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

   8. Applied rewrites 44.7%

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

   if 3.39999999999999975e-125 < r

   1. Initial program 84.1%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in v around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 82.3%

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

       1. lift--.f64 N/A

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

       2. sub-negate-rev N/A

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

       3. lower-neg.f64 N/A

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

       4. lift--.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. associate--l+ N/A

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

       7. associate--r+ N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

   12. Applied rewrites 82.3%

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

Alternative 14: 57.1% accurate, 3.3× speedup

Math

\[\left(\frac{2}{r \cdot r} - -3\right) - 4.5 \]

FPCore

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

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) - -3.0) - 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 = ((2.0d0 / (r * r)) - (-3.0d0)) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / (r * r)) - -3.0) - 4.5;
}

Python

def code(v, w, r):
	return ((2.0 / (r * r)) - -3.0) - 4.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in v around inf

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

   1. lower-*.f64 73.4

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

4. Applied rewrites 73.4%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lower--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. add-flip N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

6. Applied rewrites 73.5%

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

7. Taylor expanded in w around 0

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

9. Applied rewrites 57.1%

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

Alternative 15: 44.7% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (v w r) :precision binary64 (/ (/ 2.0 r) r))

C

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

Fortran

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 = (2.0d0 / r) / r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. 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 44.7

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

4. Applied rewrites 44.7%

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
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 44.7

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

6. Applied rewrites 44.7%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
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 44.7

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

8. Applied rewrites 44.7%

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

Alternative 16: 44.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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 = 2.0d0 / (r * r)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. 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 44.7

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

4. Applied rewrites 44.7%

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
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 44.7

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

6. Applied rewrites 44.7%

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

Reproduce

herbie shell --seed 2025167 
(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))