Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.3%

Time: 5.3s

Alternatives: 14

Speedup: 1.6×

• 47.1% 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: 84.8% 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, 0.8× speedup

Math

\[\begin{array}{l} t_0 := w \cdot \left|r\right|\\ t_1 := \frac{2}{\left|r\right| \cdot \left|r\right|}\\ \mathbf{if}\;\left|r\right| \leq 3 \cdot 10^{+207}:\\ \;\;\;\;\left(\left(3 + t\_1\right) - \left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(t\_0 \cdot \left|r\right|\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-\left(1.5 - \left(t\_1 - \left(\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot w\right) \cdot t\_0\right) \cdot \frac{\left|r\right|}{1 - v}\right)\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* w (fabs r))) (t_1 (/ 2.0 (* (fabs r) (fabs r)))))
  (if (<= (fabs r) 3e+207)
    (-
     (-
      (+ 3.0 t_1)
      (*
       (* (/ (fma (+ v v) 0.125 -0.375) (- v 1.0)) w)
       (* t_0 (fabs r))))
     4.5)
    (-
     (-
      1.5
      (-
       t_1
       (*
        (* (* (* (fma -2.0 v 3.0) 0.125) w) t_0)
        (/ (fabs r) (- 1.0 v)))))))))

C

double code(double v, double w, double r) {
	double t_0 = w * fabs(r);
	double t_1 = 2.0 / (fabs(r) * fabs(r));
	double tmp;
	if (fabs(r) <= 3e+207) {
		tmp = ((3.0 + t_1) - (((fma((v + v), 0.125, -0.375) / (v - 1.0)) * w) * (t_0 * fabs(r)))) - 4.5;
	} else {
		tmp = -(1.5 - (t_1 - ((((fma(-2.0, v, 3.0) * 0.125) * w) * t_0) * (fabs(r) / (1.0 - v)))));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(w * abs(r))
	t_1 = Float64(2.0 / Float64(abs(r) * abs(r)))
	tmp = 0.0
	if (abs(r) <= 3e+207)
		tmp = Float64(Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(fma(Float64(v + v), 0.125, -0.375) / Float64(v - 1.0)) * w) * Float64(t_0 * abs(r)))) - 4.5);
	else
		tmp = Float64(-Float64(1.5 - Float64(t_1 - Float64(Float64(Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * w) * t_0) * Float64(abs(r) / Float64(1.0 - v))))));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[Abs[r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 3e+207], N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(N[(N[(v + v), $MachinePrecision] * 0.125 + -0.375), $MachinePrecision] / N[(v - 1.0), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(t$95$0 * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-N[(1.5 - N[(t$95$1 - N[(N[(N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * w), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[r], $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.9999999999999998e207

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   if 2.9999999999999998e207 < r

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 84.9%

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

   3. Applied rewrites 91.5%

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

Alternative 2: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - -3\\ t_2 := \left(w \cdot r\right) \cdot r\\ \mathbf{if}\;v \leq -4 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \left(-w\right) \cdot 0.25, t\_1\right) - 4.5\\ \mathbf{elif}\;v \leq 4.05 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{\left(0.375 + -0.25 \cdot v\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(0.25 \cdot w, t\_2, 4.5\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (/ 2.0 (* r r))) (t_1 (- t_0 -3.0)) (t_2 (* (* w r) r)))
  (if (<= v -4e+43)
    (- (fma t_2 (* (- w) 0.25) t_1) 4.5)
    (if (<= v 4.05e-20)
      (-
       (-
        (+ 3.0 t_0)
        (/ (* (+ 0.375 (* -0.25 v)) (* (* w r) (* w r))) (- 1.0 v)))
       4.5)
      (- t_1 (fma (* 0.25 w) t_2 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - -3.0;
	double t_2 = (w * r) * r;
	double tmp;
	if (v <= -4e+43) {
		tmp = fma(t_2, (-w * 0.25), t_1) - 4.5;
	} else if (v <= 4.05e-20) {
		tmp = ((3.0 + t_0) - (((0.375 + (-0.25 * v)) * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1 - fma((0.25 * w), t_2, 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - -3.0)
	t_2 = Float64(Float64(w * r) * r)
	tmp = 0.0
	if (v <= -4e+43)
		tmp = Float64(fma(t_2, Float64(Float64(-w) * 0.25), t_1) - 4.5);
	elseif (v <= 4.05e-20)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.375 + Float64(-0.25 * v)) * Float64(Float64(w * r) * Float64(w * r))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(t_1 - fma(Float64(0.25 * w), t_2, 4.5));
	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[(t$95$0 - -3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[v, -4e+43], N[(N[(t$95$2 * N[((-w) * 0.25), $MachinePrecision] + t$95$1), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 4.05e-20], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$1 - N[(N[(0.25 * w), $MachinePrecision] * t$95$2 + 4.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.0000000000000001e43

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 91.7%

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

   if -4.0000000000000001e43 < v < 4.0500000000000002e-20

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   4. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 94.5%

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

   6. Applied rewrites 94.5%

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

   if 4.0500000000000002e-20 < v

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 91.7%

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

   10. Applied rewrites 91.7%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(w \cdot r\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := 3 + t\_1\\ \mathbf{if}\;v \leq -50:\\ \;\;\;\;\left(t\_2 - \left(t\_0 \cdot \left(0.25 - \frac{0.125}{v}\right)\right) \cdot w\right) - 4.5\\ \mathbf{elif}\;v \leq 4.05 \cdot 10^{-20}:\\ \;\;\;\;\left(t\_2 - \frac{0.375 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - -3\right) - \mathsf{fma}\left(0.25 \cdot w, t\_0, 4.5\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* (* w r) r)) (t_1 (/ 2.0 (* r r))) (t_2 (+ 3.0 t_1)))
  (if (<= v -50.0)
    (- (- t_2 (* (* t_0 (- 0.25 (/ 0.125 v))) w)) 4.5)
    (if (<= v 4.05e-20)
      (- (- t_2 (/ (* 0.375 (* (* w r) (* w r))) (- 1.0 v))) 4.5)
      (- (- t_1 -3.0) (fma (* 0.25 w) t_0 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = (w * r) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = 3.0 + t_1;
	double tmp;
	if (v <= -50.0) {
		tmp = (t_2 - ((t_0 * (0.25 - (0.125 / v))) * w)) - 4.5;
	} else if (v <= 4.05e-20) {
		tmp = (t_2 - ((0.375 * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_1 - -3.0) - fma((0.25 * w), t_0, 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * r) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(3.0 + t_1)
	tmp = 0.0
	if (v <= -50.0)
		tmp = Float64(Float64(t_2 - Float64(Float64(t_0 * Float64(0.25 - Float64(0.125 / v))) * w)) - 4.5);
	elseif (v <= 4.05e-20)
		tmp = Float64(Float64(t_2 - Float64(Float64(0.375 * Float64(Float64(w * r) * Float64(w * r))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - -3.0) - fma(Float64(0.25 * w), t_0, 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * r), $MachinePrecision] * r), $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, -50.0], N[(N[(t$95$2 - N[(N[(t$95$0 * N[(0.25 - N[(0.125 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 4.05e-20], N[(N[(t$95$2 - N[(N[(0.375 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - -3.0), $MachinePrecision] - N[(N[(0.25 * w), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -50

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{1}{4} - \frac{1}{8} \cdot \frac{1}{v}\right)} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - 4.5 \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 77.9%

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.25 - 0.125 \cdot \frac{1}{\color{blue}{v}}\right) \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - 4.5 \]

   8. Applied rewrites 77.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.25 - 0.125 \cdot \frac{1}{v}\right)} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - 4.5 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 79.8%

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot \left(0.25 - 0.125 \cdot \frac{1}{v}\right)\right)} \cdot w\right) - 4.5 \]

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 79.8%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot \left(0.25 - \frac{0.125}{\color{blue}{v}}\right)\right) \cdot w\right) - 4.5 \]

   10. Applied rewrites 79.8%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot \left(0.25 - \frac{0.125}{v}\right)\right) \cdot w}\right) - 4.5 \]

   if -50 < v < 4.0500000000000002e-20

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 84.4%

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

   if 4.0500000000000002e-20 < v

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 91.7%

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

   10. Applied rewrites 91.7%

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

Alternative 4: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|w\right| \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left|w\right| \leq 2.3 \cdot 10^{-78}:\\ \;\;\;\;-\left(1.5 - \left(t\_1 - \left(\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left|w\right|\right) \cdot t\_0\right) \cdot \frac{r}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_1\right) - \left|w\right| \cdot \left(t\_0 \cdot \frac{r \cdot \mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1}\right)\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* (fabs w) r)) (t_1 (/ 2.0 (* r r))))
  (if (<= (fabs w) 2.3e-78)
    (-
     (-
      1.5
      (-
       t_1
       (*
        (* (* (* (fma -2.0 v 3.0) 0.125) (fabs w)) t_0)
        (/ r (- 1.0 v))))))
    (-
     (-
      (+ 3.0 t_1)
      (*
       (fabs w)
       (* t_0 (/ (* r (fma (+ v v) 0.125 -0.375)) (- v 1.0)))))
     4.5))))

C

double code(double v, double w, double r) {
	double t_0 = fabs(w) * r;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (fabs(w) <= 2.3e-78) {
		tmp = -(1.5 - (t_1 - ((((fma(-2.0, v, 3.0) * 0.125) * fabs(w)) * t_0) * (r / (1.0 - v)))));
	} else {
		tmp = ((3.0 + t_1) - (fabs(w) * (t_0 * ((r * fma((v + v), 0.125, -0.375)) / (v - 1.0))))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(abs(w) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (abs(w) <= 2.3e-78)
		tmp = Float64(-Float64(1.5 - Float64(t_1 - Float64(Float64(Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * abs(w)) * t_0) * Float64(r / Float64(1.0 - v))))));
	else
		tmp = Float64(Float64(Float64(3.0 + t_1) - Float64(abs(w) * Float64(t_0 * Float64(Float64(r * fma(Float64(v + v), 0.125, -0.375)) / Float64(v - 1.0))))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[Abs[w], $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[w], $MachinePrecision], 2.3e-78], (-N[(1.5 - N[(t$95$1 - N[(N[(N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[Abs[w], $MachinePrecision] * N[(t$95$0 * N[(N[(r * N[(N[(v + v), $MachinePrecision] * 0.125 + -0.375), $MachinePrecision]), $MachinePrecision] / N[(v - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.3000000000000002e-78

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 84.9%

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

   3. Applied rewrites 91.5%

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

   if 2.3000000000000002e-78 < w

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 92.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{w \cdot \left(\left(w \cdot r\right) \cdot \frac{r \cdot \mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1}\right)}\right) - 4.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left(w \cdot r\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -760:\\ \;\;\;\;\left(\left(3 + t\_1\right) - \left(t\_0 \cdot \left(0.25 - \frac{0.125}{v}\right)\right) \cdot w\right) - 4.5\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-21}:\\ \;\;\;\;-\left(1.5 - \left(t\_1 - \left(\left(\left(3 \cdot 0.125\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - -3\right) - \mathsf{fma}\left(0.25 \cdot w, t\_0, 4.5\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* (* w r) r)) (t_1 (/ 2.0 (* r r))))
  (if (<= v -760.0)
    (- (- (+ 3.0 t_1) (* (* t_0 (- 0.25 (/ 0.125 v))) w)) 4.5)
    (if (<= v 1.6e-21)
      (- (- 1.5 (- t_1 (* (* (* (* 3.0 0.125) w) (* w r)) r))))
      (- (- t_1 -3.0) (fma (* 0.25 w) t_0 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = (w * r) * r;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -760.0) {
		tmp = ((3.0 + t_1) - ((t_0 * (0.25 - (0.125 / v))) * w)) - 4.5;
	} else if (v <= 1.6e-21) {
		tmp = -(1.5 - (t_1 - ((((3.0 * 0.125) * w) * (w * r)) * r)));
	} else {
		tmp = (t_1 - -3.0) - fma((0.25 * w), t_0, 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * r) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -760.0)
		tmp = Float64(Float64(Float64(3.0 + t_1) - Float64(Float64(t_0 * Float64(0.25 - Float64(0.125 / v))) * w)) - 4.5);
	elseif (v <= 1.6e-21)
		tmp = Float64(-Float64(1.5 - Float64(t_1 - Float64(Float64(Float64(Float64(3.0 * 0.125) * w) * Float64(w * r)) * r))));
	else
		tmp = Float64(Float64(t_1 - -3.0) - fma(Float64(0.25 * w), t_0, 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -760.0], N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(t$95$0 * N[(0.25 - N[(0.125 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 1.6e-21], (-N[(1.5 - N[(t$95$1 - N[(N[(N[(N[(3.0 * 0.125), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(t$95$1 - -3.0), $MachinePrecision] - N[(N[(0.25 * w), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -760

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{1}{4} - \frac{1}{8} \cdot \frac{1}{v}\right)} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - 4.5 \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 77.9%

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.25 - 0.125 \cdot \frac{1}{\color{blue}{v}}\right) \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - 4.5 \]

   8. Applied rewrites 77.9%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.25 - 0.125 \cdot \frac{1}{v}\right)} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - 4.5 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 79.8%

         \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot \left(0.25 - 0.125 \cdot \frac{1}{v}\right)\right)} \cdot w\right) - 4.5 \]

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 79.8%

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\left(w \cdot r\right) \cdot r\right) \cdot \left(0.25 - \frac{0.125}{\color{blue}{v}}\right)\right) \cdot w\right) - 4.5 \]

   10. Applied rewrites 79.8%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot \left(0.25 - \frac{0.125}{v}\right)\right) \cdot w}\right) - 4.5 \]

   if -760 < v < 1.6000000000000001e-21

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 84.9%

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

   3. Applied rewrites 91.5%

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

   4. Taylor expanded in v around 0

      \[\leadsto -\left(1.5 - \left(\frac{2}{r \cdot r} - \left(\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{r}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in v around 0

      \[\leadsto -\left(1.5 - \left(\frac{2}{r \cdot r} - \left(\left(\left(\color{blue}{3} \cdot 0.125\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot r\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 90.4%

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

   if 1.6000000000000001e-21 < v

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 91.7%

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

   10. Applied rewrites 91.7%

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

Alternative 6: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 - -3\\ t_2 := \left(w \cdot r\right) \cdot r\\ \mathbf{if}\;v \leq -390:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \left(-w\right) \cdot 0.25, t\_1\right) - 4.5\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-21}:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - \left(\left(\left(3 \cdot 0.125\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(0.25 \cdot w, t\_2, 4.5\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (/ 2.0 (* r r))) (t_1 (- t_0 -3.0)) (t_2 (* (* w r) r)))
  (if (<= v -390.0)
    (- (fma t_2 (* (- w) 0.25) t_1) 4.5)
    (if (<= v 1.6e-21)
      (- (- 1.5 (- t_0 (* (* (* (* 3.0 0.125) w) (* w r)) r))))
      (- t_1 (fma (* 0.25 w) t_2 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 - -3.0;
	double t_2 = (w * r) * r;
	double tmp;
	if (v <= -390.0) {
		tmp = fma(t_2, (-w * 0.25), t_1) - 4.5;
	} else if (v <= 1.6e-21) {
		tmp = -(1.5 - (t_0 - ((((3.0 * 0.125) * w) * (w * r)) * r)));
	} else {
		tmp = t_1 - fma((0.25 * w), t_2, 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 - -3.0)
	t_2 = Float64(Float64(w * r) * r)
	tmp = 0.0
	if (v <= -390.0)
		tmp = Float64(fma(t_2, Float64(Float64(-w) * 0.25), t_1) - 4.5);
	elseif (v <= 1.6e-21)
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(Float64(Float64(Float64(3.0 * 0.125) * w) * Float64(w * r)) * r))));
	else
		tmp = Float64(t_1 - fma(Float64(0.25 * w), t_2, 4.5));
	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[(t$95$0 - -3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[v, -390.0], N[(N[(t$95$2 * N[((-w) * 0.25), $MachinePrecision] + t$95$1), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 1.6e-21], (-N[(1.5 - N[(t$95$0 - N[(N[(N[(N[(3.0 * 0.125), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(t$95$1 - N[(N[(0.25 * w), $MachinePrecision] * t$95$2 + 4.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -390

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 91.7%

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

   if -390 < v < 1.6000000000000001e-21

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 84.9%

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

   3. Applied rewrites 91.5%

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

   4. Taylor expanded in v around 0

      \[\leadsto -\left(1.5 - \left(\frac{2}{r \cdot r} - \left(\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{r}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in v around 0

      \[\leadsto -\left(1.5 - \left(\frac{2}{r \cdot r} - \left(\left(\left(\color{blue}{3} \cdot 0.125\right) \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot r\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 90.4%

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

   if 1.6000000000000001e-21 < v

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 91.7%

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

   10. Applied rewrites 91.7%

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

Alternative 7: 96.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 3 + t\_0\\ t_2 := \left(w \cdot r\right) \cdot r\\ \mathbf{if}\;v \leq -1.65 \cdot 10^{-141}:\\ \;\;\;\;\left(t\_1 - \left(\left(0.25 \cdot w\right) \cdot r\right) \cdot \left(w \cdot r\right)\right) - 4.5\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\left(t\_1 - \left(0.375 \cdot w\right) \cdot t\_2\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - -3\right) - \mathsf{fma}\left(0.25 \cdot w, t\_2, 4.5\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ 3.0 t_0)) (t_2 (* (* w r) r)))
  (if (<= v -1.65e-141)
    (- (- t_1 (* (* (* 0.25 w) r) (* w r))) 4.5)
    (if (<= v 4e-20)
      (- (- t_1 (* (* 0.375 w) t_2)) 4.5)
      (- (- t_0 -3.0) (fma (* 0.25 w) t_2 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double t_2 = (w * r) * r;
	double tmp;
	if (v <= -1.65e-141) {
		tmp = (t_1 - (((0.25 * w) * r) * (w * r))) - 4.5;
	} else if (v <= 4e-20) {
		tmp = (t_1 - ((0.375 * w) * t_2)) - 4.5;
	} else {
		tmp = (t_0 - -3.0) - fma((0.25 * w), t_2, 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(3.0 + t_0)
	t_2 = Float64(Float64(w * r) * r)
	tmp = 0.0
	if (v <= -1.65e-141)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(0.25 * w) * r) * Float64(w * r))) - 4.5);
	elseif (v <= 4e-20)
		tmp = Float64(Float64(t_1 - Float64(Float64(0.375 * w) * t_2)) - 4.5);
	else
		tmp = Float64(Float64(t_0 - -3.0) - fma(Float64(0.25 * w), t_2, 4.5));
	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[(3.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[v, -1.65e-141], N[(N[(t$95$1 - N[(N[(N[(0.25 * w), $MachinePrecision] * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[v, 4e-20], N[(N[(t$95$1 - N[(N[(0.375 * w), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(N[(0.25 * w), $MachinePrecision] * t$95$2 + 4.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.65e-141

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 93.4%

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

   10. Applied rewrites 93.4%

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

   if -1.65e-141 < v < 3.9999999999999998e-20

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{3}{8}} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - 4.5 \]
   7. Step-by-step derivation

   8. Applied rewrites 91.0%

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

   if 3.9999999999999998e-20 < v

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 91.7%

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

   10. Applied rewrites 91.7%

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

Alternative 8: 95.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(w \cdot r\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := \left(t\_1 - -3\right) - \mathsf{fma}\left(0.25 \cdot w, t\_0, 4.5\right)\\ \mathbf{if}\;v \leq -1 \cdot 10^{-60}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(3 + t\_1\right) - \left(0.375 \cdot w\right) \cdot t\_0\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* (* w r) r))
       (t_1 (/ 2.0 (* r r)))
       (t_2 (- (- t_1 -3.0) (fma (* 0.25 w) t_0 4.5))))
  (if (<= v -1e-60)
    t_2
    (if (<= v 4e-20)
      (- (- (+ 3.0 t_1) (* (* 0.375 w) t_0)) 4.5)
      t_2))))

C

double code(double v, double w, double r) {
	double t_0 = (w * r) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = (t_1 - -3.0) - fma((0.25 * w), t_0, 4.5);
	double tmp;
	if (v <= -1e-60) {
		tmp = t_2;
	} else if (v <= 4e-20) {
		tmp = ((3.0 + t_1) - ((0.375 * w) * t_0)) - 4.5;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * r) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(Float64(t_1 - -3.0) - fma(Float64(0.25 * w), t_0, 4.5))
	tmp = 0.0
	if (v <= -1e-60)
		tmp = t_2;
	elseif (v <= 4e-20)
		tmp = Float64(Float64(Float64(3.0 + t_1) - Float64(Float64(0.375 * w) * t_0)) - 4.5);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - -3.0), $MachinePrecision] - N[(N[(0.25 * w), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1e-60], t$95$2, If[LessEqual[v, 4e-20], N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(0.375 * w), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -9.9999999999999997e-61 or 3.9999999999999998e-20 < v

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 91.7%

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

   10. Applied rewrites 91.7%

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

   if -9.9999999999999997e-61 < v < 3.9999999999999998e-20

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{3}{8}} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) - 4.5 \]
   7. Step-by-step derivation

   8. Applied rewrites 91.0%

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

Alternative 9: 91.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (v w r)
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (* (* 0.25 w) (* (* w r) r))) 4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.25 * w) * ((w * r) * r))) - 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))) - ((0.25d0 * w) * ((w * r) * r))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.25 * w) * ((w * r) * r))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - ((0.25 * w) * ((w * r) * r))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(0.25 * w) * Float64(Float64(w * r) * r))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - ((0.25 * w) * ((w * r) * r))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.25 * w), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.8%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. 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. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. unswap-sqr N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 94.4%

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

3. Applied rewrites 94.4%

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

5. Applied rewrites 96.6%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

6. Taylor expanded in v around inf

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

8. Applied rewrites 91.7%

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

Alternative 10: 91.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (v w r)
  :precision binary64
  (- (- (/ 2.0 (* r r)) -3.0) (fma (* 0.25 w) (* (* w r) r) 4.5)))

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) - -3.0) - fma((0.25 * w), ((w * r) * r), 4.5);
}

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / Float64(r * r)) - -3.0) - fma(Float64(0.25 * w), Float64(Float64(w * r) * r), 4.5))
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision] - N[(N[(0.25 * w), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. 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. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. unswap-sqr N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 94.4%

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

3. Applied rewrites 94.4%

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

5. Applied rewrites 96.6%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

6. Taylor expanded in v around inf

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

8. Applied rewrites 91.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lower--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. add-flip N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 91.7%

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

10. Applied rewrites 91.7%

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

Alternative 11: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1.500000000010223:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - \left(\left(0.25 \cdot w\right) \cdot w\right) \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {r}^{-2} - 1.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (/ 2.0 (* r r))))
  (if (<=
       (-
        (-
         (+ 3.0 t_0)
         (/
          (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r))
          (- 1.0 v)))
        4.5)
       -1.500000000010223)
    (- (- 1.5 (- t_0 (* (* (* 0.25 w) w) (* r r)))))
    (- (* 2.0 (pow r -2.0)) 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.500000000010223) {
		tmp = -(1.5 - (t_0 - (((0.25 * w) * w) * (r * r))));
	} else {
		tmp = (2.0 * pow(r, -2.0)) - 1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-1.500000000010223d0)) then
        tmp = -(1.5d0 - (t_0 - (((0.25d0 * w) * w) * (r * r))))
    else
        tmp = (2.0d0 * (r ** (-2.0d0))) - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.500000000010223) {
		tmp = -(1.5 - (t_0 - (((0.25 * w) * w) * (r * r))));
	} else {
		tmp = (2.0 * Math.pow(r, -2.0)) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.500000000010223:
		tmp = -(1.5 - (t_0 - (((0.25 * w) * w) * (r * r))))
	else:
		tmp = (2.0 * math.pow(r, -2.0)) - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5) <= -1.500000000010223)
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(Float64(Float64(0.25 * w) * w) * Float64(r * r)))));
	else
		tmp = Float64(Float64(2.0 * (r ^ -2.0)) - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.500000000010223)
		tmp = -(1.5 - (t_0 - (((0.25 * w) * w) * (r * r))));
	else
		tmp = (2.0 * (r ^ -2.0)) - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.500000000010223], (-N[(1.5 - N[(t$95$0 - N[(N[(N[(0.25 * w), $MachinePrecision] * w), $MachinePrecision] * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5000000000102229

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.4%

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

   3. Applied rewrites 94.4%

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

   5. Applied rewrites 96.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\mathsf{fma}\left(v + v, 0.125, -0.375\right)}{v - 1} \cdot w\right) \cdot \left(\left(w \cdot r\right) \cdot r\right)}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

   8. Applied rewrites 91.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

   10. Applied rewrites 78.3%

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

   if -1.5000000000102229 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.8%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 57.2%

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - 1.5 \]

   4. Applied rewrites 57.2%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 57.2%

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

   6. Applied rewrites 57.2%

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

Alternative 12: 57.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(v, w, r):
	return (2.0 * math.pow(r, -2.0)) - 1.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 57.2%

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - 1.5 \]

4. Applied rewrites 57.2%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. lower-pow.f64 57.2%

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

6. Applied rewrites 57.2%

   \[\leadsto 2 \cdot {r}^{-2} - 1.5 \]
7. Add Preprocessing

Alternative 13: 57.2% accurate, 4.1× speedup

Math

\[\frac{2}{r \cdot r} - 1.5 \]

FPCore

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

C

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

Fortran

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)) - 1.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 57.2%

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - 1.5 \]

4. Applied rewrites 57.2%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. pow-to-exp N/A

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

   6. lower-unsound-exp.f64 N/A

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

   7. lower-unsound-*.f64 N/A

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

   8. lower-unsound-log.f64 27.1%

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

6. Applied rewrites 27.1%

   \[\leadsto 2 \cdot e^{\log r \cdot -2} - 1.5 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. exp-to-pow N/A

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

   6. metadata-eval N/A

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

   7. pow-flip N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. mult-flip-rev N/A

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

   11. lift-/.f64 57.2%

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

8. Applied rewrites 57.2%

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

Alternative 14: 13.5% accurate, 42.6× speedup

Math

\[-1.5 \]

FPCore

(FPCore (v w r)
  :precision binary64
  -1.5)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

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

Python

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

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

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

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 84.8%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 57.2%

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} - 1.5 \]

4. Applied rewrites 57.2%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]

5. Taylor expanded in r around inf

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

7. Applied rewrites 13.5%

   \[\leadsto -1.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))