Rosa's TurbineBenchmark

Percentage Accurate: 84.4% → 99.7%

Time: 4.6s

Alternatives: 10

Speedup: 1.0×

• 52.3% 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: 84.4% 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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((fma(-2.0, v, 3.0) * 0.125) * (((w * r) * (w * 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(fma(-2.0, v, 3.0) * 0.125) * Float64(Float64(Float64(w * r) * Float64(w * r)) / Float64(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[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 2: 98.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r r)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
       4.5)
      -1.5)
   (-
    (- 3.0 (* (* (fma -2.0 v 3.0) 0.125) (/ (* (* w r) (* w r)) (- 1.0 v))))
    4.5)
   (- (* (/ 1.0 (* r r)) 2.0) 1.5)))

C

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

Julia

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

Wolfram

code[v_, w_, r_] := If[LessEqual[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], -1.5], N[(N[(3.0 - N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $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)) < -1.5

   1. Initial program 83.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 96.8%

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

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 85.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1.5) {
		tmp = (3.0 - ((0.375 * ((w * (w * r)) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = ((1.0 / (r * 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) :: tmp
    if ((((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-1.5d0)) then
        tmp = (3.0d0 - ((0.375d0 * ((w * (w * r)) * r)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = ((1.0d0 / (r * r)) * 2.0d0) - 1.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := If[LessEqual[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], -1.5], N[(N[(3.0 - N[(N[(0.375 * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $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)) < -1.5

   1. Initial program 83.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in r around inf

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

   10. Applied rewrites 70.2%

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

   if -1.5 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 85.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -5 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot r} \cdot 2 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r r)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
       4.5)
      -5e+191)
   (/ (/ (fma (* r r) -1.5 2.0) r) r)
   (- (* (/ 1.0 (* r r)) 2.0) 1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -5e+191) {
		tmp = (fma((r * r), -1.5, 2.0) / r) / r;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (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) <= -5e+191)
		tmp = Float64(Float64(fma(Float64(r * r), -1.5, 2.0) / r) / r);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[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], -5e+191], N[(N[(N[(N[(r * r), $MachinePrecision] * -1.5 + 2.0), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $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)) < -5.0000000000000002e191

   1. Initial program 83.4%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 5.2

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

   5. Applied rewrites 5.2%

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

      1. associate--l+ 5.2

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

      2. associate-/r* 5.2

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r} \]

      3. associate--l+ 5.2

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 46.3%

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

   if -5.0000000000000002e191 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 85.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 87.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 86.9

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

   9. Applied rewrites 86.9%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 70.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -1 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot r} \cdot 2 - 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<=
      (-
       (-
        (+ 3.0 (/ 2.0 (* r r)))
        (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
       4.5)
      -1e+53)
   (/ (/ (* (* r r) -1.5) r) r)
   (- (* (/ 1.0 (* r r)) 2.0) 1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+53) {
		tmp = (((r * r) * -1.5) / r) / r;
	} else {
		tmp = ((1.0 / (r * 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) :: tmp
    if ((((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0) <= (-1d+53)) then
        tmp = (((r * r) * (-1.5d0)) / r) / r
    else
        tmp = ((1.0d0 / (r * r)) * 2.0d0) - 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+53) {
		tmp = (((r * r) * -1.5) / r) / r;
	} else {
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+53:
		tmp = (((r * r) * -1.5) / r) / r
	else:
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (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) <= -1e+53)
		tmp = Float64(Float64(Float64(Float64(r * r) * -1.5) / r) / r);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -1e+53)
		tmp = (((r * r) * -1.5) / r) / r;
	else
		tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[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], -1e+53], N[(N[(N[(N[(r * r), $MachinePrecision] * -1.5), $MachinePrecision] / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $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)) < -9.9999999999999999e52

   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. Add Preprocessing

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

      1. associate--l+ 5.0

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

      2. associate-/r* 5.0

         \[\leadsto \frac{\mathsf{fma}\left(-1.5, r \cdot r, 2\right)}{r \cdot r} \]

      3. associate--l+ 5.0

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   7. Applied rewrites 42.2%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 39.1

         \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r} \]

   10. Applied rewrites 39.1%

       \[\leadsto \frac{\frac{\left(r \cdot r\right) \cdot -1.5}{r}}{r} \]

   if -9.9999999999999999e52 < (-.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 84.2%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 92.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 92.5

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

   9. Applied rewrites 92.5%

      \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 7.2 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;r \leq 75000000:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(\left(\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125\right) \cdot \left(w \cdot r\right)\right) \cdot w\right) \cdot \frac{r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \frac{\left(\left(w \cdot r\right) \cdot w\right) \cdot r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 7.2e-104)
     t_0
     (if (<= r 75000000.0)
       (-
        (-
         (+ 3.0 t_0)
         (* (* (* (* (fma v -2.0 3.0) 0.125) (* w r)) w) (/ r (- 1.0 v))))
        4.5)
       (-
        (-
         3.0
         (* (* (fma -2.0 v 3.0) 0.125) (/ (* (* (* w r) w) r) (- 1.0 v))))
        4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 7.2e-104) {
		tmp = t_0;
	} else if (r <= 75000000.0) {
		tmp = ((3.0 + t_0) - ((((fma(v, -2.0, 3.0) * 0.125) * (w * r)) * w) * (r / (1.0 - v)))) - 4.5;
	} else {
		tmp = (3.0 - ((fma(-2.0, v, 3.0) * 0.125) * ((((w * r) * w) * r) / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 7.2e-104)
		tmp = t_0;
	elseif (r <= 75000000.0)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * Float64(w * r)) * w) * Float64(r / Float64(1.0 - v)))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * Float64(Float64(Float64(Float64(w * r) * w) * r) / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 7.2e-104], t$95$0, If[LessEqual[r, 75000000.0], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.1999999999999996e-104

   1. Initial program 82.3%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 56.9%

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

   if 7.1999999999999996e-104 < r < 7.5e7

   1. Initial program 89.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 96.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

   10. Applied rewrites 89.4%

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

   if 7.5e7 < r

   1. Initial program 88.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 99.8%

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

Alternative 7: 70.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.05 \cdot 10^{-121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;r \leq 7800000:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \frac{\left(\left(w \cdot r\right) \cdot w\right) \cdot r}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1.05e-121)
     t_0
     (if (<= r 7800000.0)
       (- (- (+ 3.0 t_0) (/ (* 0.375 (* (* (* w w) r) r)) 1.0)) 4.5)
       (-
        (-
         3.0
         (* (* (fma -2.0 v 3.0) 0.125) (/ (* (* (* w r) w) r) (- 1.0 v))))
        4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.05e-121) {
		tmp = t_0;
	} else if (r <= 7800000.0) {
		tmp = ((3.0 + t_0) - ((0.375 * (((w * w) * r) * r)) / 1.0)) - 4.5;
	} else {
		tmp = (3.0 - ((fma(-2.0, v, 3.0) * 0.125) * ((((w * r) * w) * r) / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1.05e-121)
		tmp = t_0;
	elseif (r <= 7800000.0)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(0.375 * Float64(Float64(Float64(w * w) * r) * r)) / 1.0)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * Float64(Float64(Float64(Float64(w * r) * w) * r) / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.05e-121], t$95$0, If[LessEqual[r, 7800000.0], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.375 * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.0499999999999999e-121

   1. Initial program 82.3%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 56.4%

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

   if 1.0499999999999999e-121 < r < 7.8e6

   1. Initial program 88.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 87.7%

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

   if 7.8e6 < r

   1. Initial program 88.2%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 99.7%

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

Alternative 8: 57.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{r \cdot r} \cdot 2 - 1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (- (* (/ 1.0 (* r r)) 2.0) 1.5))

C

double code(double v, double w, double r) {
	return ((1.0 / (r * 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 = ((1.0d0 / (r * r)) * 2.0d0) - 1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((1.0 / (r * r)) * 2.0) - 1.5;
}

Python

def code(v, w, r):
	return ((1.0 / (r * r)) * 2.0) - 1.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(1.0 / Float64(r * r)) * 2.0) - 1.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((1.0 / (r * r)) * 2.0) - 1.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(1.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - 1.5), $MachinePrecision]

Derivation

1. Initial program 84.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in w around 0

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

7. Applied rewrites 57.1%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. lower-/.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 57.0

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

9. Applied rewrites 57.0%

   \[\leadsto \frac{1}{r \cdot r} \cdot 2 - 1.5 \]
10. Add Preprocessing

Alternative 9: 50.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.16:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.16) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.16) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -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) :: tmp
    if (r <= 1.16d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.16) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.16:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.16)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.16)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.16], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.15999999999999992

   1. Initial program 83.1%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 57.5%

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

   if 1.15999999999999992 < r

   1. Initial program 88.3%

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

   3. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 19.9

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 28.0%

      \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 14.5% accurate, 73.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

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 84.4%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 53.0

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

5. Applied rewrites 53.0%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 14.5%

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

Reproduce

herbie shell --seed 2025089 
(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))