Rosa's TurbineBenchmark

Percentage Accurate: 85.2% → 99.3%

Time: 5.0s

Alternatives: 7

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 4.3 \cdot 10^{+176}:\\ \;\;\;\;-\mathsf{fma}\left(\mathsf{fma}\left(-0.25, v, 0.375\right), \left(\left(\frac{r\_m}{1 - v} \cdot w\right) \cdot r\_m\right) \cdot w, 1.5 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{1}{\frac{1 - v}{r\_m} \cdot \frac{1}{\left(\left(w \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot 0.125\right) \cdot \left(w \cdot r\_m\right)}}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 4.3e+176)
     (-
      (fma
       (fma -0.25 v 0.375)
       (* (* (* (/ r_m (- 1.0 v)) w) r_m) w)
       (- 1.5 t_0)))
     (-
      (-
       (+ 3.0 t_0)
       (/
        1.0
        (*
         (/ (- 1.0 v) r_m)
         (/ 1.0 (* (* (* w (fma -2.0 v 3.0)) 0.125) (* w r_m))))))
      4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 4.3e+176) {
		tmp = -fma(fma(-0.25, v, 0.375), ((((r_m / (1.0 - v)) * w) * r_m) * w), (1.5 - t_0));
	} else {
		tmp = ((3.0 + t_0) - (1.0 / (((1.0 - v) / r_m) * (1.0 / (((w * fma(-2.0, v, 3.0)) * 0.125) * (w * r_m)))))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 4.3e+176)
		tmp = Float64(-fma(fma(-0.25, v, 0.375), Float64(Float64(Float64(Float64(r_m / Float64(1.0 - v)) * w) * r_m) * w), Float64(1.5 - t_0)));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(1.0 / Float64(Float64(Float64(1.0 - v) / r_m) * Float64(1.0 / Float64(Float64(Float64(w * fma(-2.0, v, 3.0)) * 0.125) * Float64(w * r_m)))))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 4.3e+176], (-N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(N[(N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * r$95$m), $MachinePrecision] * w), $MachinePrecision] + N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(1.0 / N[(N[(N[(1.0 - v), $MachinePrecision] / r$95$m), $MachinePrecision] * N[(1.0 / N[(N[(N[(w * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.30000000000000026e176

   1. Initial program 85.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 95.0

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

   3. Applied rewrites 95.0%

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

   5. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 96.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. +-commutative N/A

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

   9. Applied rewrites 98.0%

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

   if 4.30000000000000026e176 < r

   1. Initial program 85.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.2

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-to-mult N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 91.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{r\_m}{1 - v}\\ t_1 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 2 \cdot 10^{+187}:\\ \;\;\;\;-\mathsf{fma}\left(\mathsf{fma}\left(-0.25, v, 0.375\right), \left(\left(t\_0 \cdot w\right) \cdot r\_m\right) \cdot w, 1.5 - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - -3\right) - \mathsf{fma}\left(\left(\left(w \cdot r\_m\right) \cdot w\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right), t\_0, 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ r_m (- 1.0 v))) (t_1 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 2e+187)
     (- (fma (fma -0.25 v 0.375) (* (* (* t_0 w) r_m) w) (- 1.5 t_1)))
     (- (- t_1 -3.0) (fma (* (* (* w r_m) w) (fma -0.25 v 0.375)) t_0 4.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = r_m / (1.0 - v);
	double t_1 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 2e+187) {
		tmp = -fma(fma(-0.25, v, 0.375), (((t_0 * w) * r_m) * w), (1.5 - t_1));
	} else {
		tmp = (t_1 - -3.0) - fma((((w * r_m) * w) * fma(-0.25, v, 0.375)), t_0, 4.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(r_m / Float64(1.0 - v))
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 2e+187)
		tmp = Float64(-fma(fma(-0.25, v, 0.375), Float64(Float64(Float64(t_0 * w) * r_m) * w), Float64(1.5 - t_1)));
	else
		tmp = Float64(Float64(t_1 - -3.0) - fma(Float64(Float64(Float64(w * r_m) * w) * fma(-0.25, v, 0.375)), t_0, 4.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 2e+187], (-N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(N[(t$95$0 * w), $MachinePrecision] * r$95$m), $MachinePrecision] * w), $MachinePrecision] + N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), N[(N[(t$95$1 - -3.0), $MachinePrecision] - N[(N[(N[(N[(w * r$95$m), $MachinePrecision] * w), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision] * t$95$0 + 4.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.99999999999999981e187

   1. Initial program 85.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 95.0

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

   3. Applied rewrites 95.0%

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

   5. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 96.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. +-commutative N/A

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

   9. Applied rewrites 98.0%

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

   if 1.99999999999999981e187 < r

   1. Initial program 85.2%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   6. Applied rewrites 86.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 92.4

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

   8. Applied rewrites 92.4%

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

Alternative 3: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ -\mathsf{fma}\left(\mathsf{fma}\left(-0.25, v, 0.375\right), \left(\left(\frac{r\_m}{1 - v} \cdot w\right) \cdot r\_m\right) \cdot w, 1.5 - \frac{2}{r\_m \cdot r\_m}\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (-
  (fma
   (fma -0.25 v 0.375)
   (* (* (* (/ r_m (- 1.0 v)) w) r_m) w)
   (- 1.5 (/ 2.0 (* r_m r_m))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return -fma(fma(-0.25, v, 0.375), ((((r_m / (1.0 - v)) * w) * r_m) * w), (1.5 - (2.0 / (r_m * r_m))));
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(-fma(fma(-0.25, v, 0.375), Float64(Float64(Float64(Float64(r_m / Float64(1.0 - v)) * w) * r_m) * w), Float64(1.5 - Float64(2.0 / Float64(r_m * r_m)))))
end

Wolfram

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

Derivation

1. Initial program 85.2%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 95.0

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

3. Applied rewrites 95.0%

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

5. Applied rewrites 91.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

7. Applied rewrites 96.9%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--r- N/A

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

   4. +-commutative N/A

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

9. Applied rewrites 98.0%

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

Alternative 4: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \frac{0.375 \cdot \left(\left(w \cdot r\_m\right) \cdot \left(w \cdot r\_m\right)\right)}{1}\right) - 4.5 \end{array} \]

FPCore

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

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return ((3.0 + (2.0 / (r_m * r_m))) - ((0.375 * ((w * r_m) * (w * r_m))) / 1.0)) - 4.5;
}

Fortran

r_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = ((3.0d0 + (2.0d0 / (r_m * r_m))) - ((0.375d0 * ((w * r_m) * (w * r_m))) / 1.0d0)) - 4.5d0
end function

Java

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

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return ((3.0 + (2.0 / (r_m * r_m))) - ((0.375 * ((w * r_m) * (w * r_m))) / 1.0)) - 4.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(0.375 * Float64(Float64(w * r_m) * Float64(w * r_m))) / 1.0)) - 4.5)
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((0.375 * ((w * r_m) * (w * r_m))) / 1.0)) - 4.5;
end

Wolfram

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

Derivation

1. Initial program 85.2%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 95.0

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

3. Applied rewrites 95.0%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 85.0%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 93.5%

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

Alternative 5: 90.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 10^{-141}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(\left(\left(w \cdot w\right) \cdot r\_m\right) \cdot r\_m\right) \cdot 0.25\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 1e-141)
     t_0
     (- (- (+ 3.0 t_0) (* (* (* (* w w) r_m) r_m) 0.25)) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 1e-141) {
		tmp = t_0;
	} else {
		tmp = ((3.0 + t_0) - ((((w * w) * r_m) * r_m) * 0.25)) - 4.5;
	}
	return tmp;
}

Fortran

r_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    if (r_m <= 1d-141) then
        tmp = t_0
    else
        tmp = ((3.0d0 + t_0) - ((((w * w) * r_m) * r_m) * 0.25d0)) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 1e-141) {
		tmp = t_0;
	} else {
		tmp = ((3.0 + t_0) - ((((w * w) * r_m) * r_m) * 0.25)) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	tmp = 0
	if r_m <= 1e-141:
		tmp = t_0
	else:
		tmp = ((3.0 + t_0) - ((((w * w) * r_m) * r_m) * 0.25)) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 1e-141)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(w * w) * r_m) * r_m) * 0.25)) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 2.0 / (r_m * r_m);
	tmp = 0.0;
	if (r_m <= 1e-141)
		tmp = t_0;
	else
		tmp = ((3.0 + t_0) - ((((w * w) * r_m) * r_m) * 0.25)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 1e-141], t$95$0, N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1e-141

   1. Initial program 85.2%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 44.1

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

   4. Applied rewrites 44.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 44.1

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

   6. Applied rewrites 44.1%

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

   if 1e-141 < r

   1. Initial program 85.2%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 74.4

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

   4. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 77.4

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

   6. Applied rewrites 77.4%

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

   7. Taylor expanded in v around inf

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

   9. Applied rewrites 83.9%

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

Alternative 6: 57.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

r_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

2. Taylor expanded in v around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 85.2

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

4. Applied rewrites 85.2%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lower--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. add-flip N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

6. Applied rewrites 86.0%

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

7. Taylor expanded in w around 0

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

9. Applied rewrites 57.5%

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

Alternative 7: 44.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

r_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 44.1

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

4. Applied rewrites 44.1%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 44.1

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

6. Applied rewrites 44.1%

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

Reproduce

herbie shell --seed 2025151 
(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))