Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.8%

Time: 5.7s

Alternatives: 11

Speedup: 1.1×

• 47.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (* (fma -2.0 v 3.0) (/ (* (* (* w 0.125) 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) * ((((w * 0.125) * 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(fma(-2.0, v, 3.0) * Float64(Float64(Float64(Float64(w * 0.125) * 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[(-2.0 * v + 3.0), $MachinePrecision] * N[(N[(N[(N[(w * 0.125), $MachinePrecision] * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 87.7%

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. unswap-sqr N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.8%

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.8%

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 87.7%

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. unswap-sqr N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.8%

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

5. Applied rewrites 99.8%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (/ 2.0 (* r r))
  (fma (* (* 0.125 w) r) (* (* (/ r (- 1.0 v)) w) (fma v -2.0 3.0)) 1.5)))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) - fma(((0.125 * w) * r), (((r / (1.0 - v)) * w) * fma(v, -2.0, 3.0)), 1.5);
}

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(0.125 * w) * r), Float64(Float64(Float64(r / Float64(1.0 - v)) * w) * fma(v, -2.0, 3.0)), 1.5))
end

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 87.7%

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. unswap-sqr N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 96.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

9. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (/ 2.0 (* r r))
  (fma (* (* 0.125 w) r) (* w (* (fma v -2.0 3.0) (/ r (- 1.0 v)))) 1.5)))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) - fma(((0.125 * w) * r), (w * (fma(v, -2.0, 3.0) * (r / (1.0 - v)))), 1.5);
}

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) - fma(Float64(Float64(0.125 * w) * r), Float64(w * Float64(fma(v, -2.0, 3.0) * Float64(r / Float64(1.0 - v)))), 1.5))
end

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 87.7%

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. unswap-sqr N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 96.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.3%

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

9. Applied rewrites 99.3%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - \left(\left(\frac{r}{1 - v} \cdot w\right) \cdot \left(w \cdot r\right)\right) \cdot \left(-0.25 \cdot v\right)\right) - 4.5\\ \mathbf{if}\;v \leq -8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1:\\ \;\;\;\;\left(t\_0 - \frac{0.375 \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1
         (- (- t_0 (* (* (* (/ r (- 1.0 v)) w) (* w r)) (* -0.25 v))) 4.5)))
   (if (<= v -8.0)
     t_1
     (if (<= v 1.0)
       (- (- t_0 (/ (* 0.375 (* (* (* w r) w) r)) (- 1.0 v))) 4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - ((((r / (1.0 - v)) * w) * (w * r)) * (-0.25 * v))) - 4.5;
	double tmp;
	if (v <= -8.0) {
		tmp = t_1;
	} else if (v <= 1.0) {
		tmp = (t_0 - ((0.375 * (((w * r) * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    t_1 = (t_0 - ((((r / (1.0d0 - v)) * w) * (w * r)) * ((-0.25d0) * v))) - 4.5d0
    if (v <= (-8.0d0)) then
        tmp = t_1
    else if (v <= 1.0d0) then
        tmp = (t_0 - ((0.375d0 * (((w * r) * w) * r)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - ((((r / (1.0 - v)) * w) * (w * r)) * (-0.25 * v))) - 4.5;
	double tmp;
	if (v <= -8.0) {
		tmp = t_1;
	} else if (v <= 1.0) {
		tmp = (t_0 - ((0.375 * (((w * r) * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = (t_0 - ((((r / (1.0 - v)) * w) * (w * r)) * (-0.25 * v))) - 4.5
	tmp = 0
	if v <= -8.0:
		tmp = t_1
	elif v <= 1.0:
		tmp = (t_0 - ((0.375 * (((w * r) * w) * r)) / (1.0 - v))) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(r / Float64(1.0 - v)) * w) * Float64(w * r)) * Float64(-0.25 * v))) - 4.5)
	tmp = 0.0
	if (v <= -8.0)
		tmp = t_1;
	elseif (v <= 1.0)
		tmp = Float64(Float64(t_0 - Float64(Float64(0.375 * Float64(Float64(Float64(w * r) * w) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = (t_0 - ((((r / (1.0 - v)) * w) * (w * r)) * (-0.25 * v))) - 4.5;
	tmp = 0.0;
	if (v <= -8.0)
		tmp = t_1;
	elseif (v <= 1.0)
		tmp = (t_0 - ((0.375 * (((w * r) * w) * r)) / (1.0 - v))) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(N[(N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -8.0], t$95$1, If[LessEqual[v, 1.0], N[(N[(t$95$0 - N[(N[(0.375 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -8 or 1 < v

   1. Initial program 84.9%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.9%

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

   4. Applied rewrites 73.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.7%

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

   6. Applied rewrites 84.7%

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

   if -8 < v < 1

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 92.7%

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 82.4%

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

Alternative 6: 90.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 8.6 \cdot 10^{-37}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{0.375 \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - \left(-0.25 \cdot v\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{-v}\right)\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 8.6e-37)
     (- (- (+ 3.0 t_0) (/ (* 0.375 (* (* (* w r) w) r)) (- 1.0 v))) 4.5)
     (- (- 1.5 (- t_0 (* (* -0.25 v) (/ (* (* (* w w) r) r) (- v)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 8.6e-37) {
		tmp = ((3.0 + t_0) - ((0.375 * (((w * r) * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = -(1.5 - (t_0 - ((-0.25 * v) * ((((w * w) * r) * r) / -v))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (v <= 8.6d-37) then
        tmp = ((3.0d0 + t_0) - ((0.375d0 * (((w * r) * w) * r)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = -(1.5d0 - (t_0 - (((-0.25d0) * v) * ((((w * w) * r) * r) / -v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 8.6e-37) {
		tmp = ((3.0 + t_0) - ((0.375 * (((w * r) * w) * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = -(1.5 - (t_0 - ((-0.25 * v) * ((((w * w) * r) * r) / -v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= 8.6e-37:
		tmp = ((3.0 + t_0) - ((0.375 * (((w * r) * w) * r)) / (1.0 - v))) - 4.5
	else:
		tmp = -(1.5 - (t_0 - ((-0.25 * v) * ((((w * w) * r) * r) / -v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 8.6e-37)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(0.375 * Float64(Float64(Float64(w * r) * w) * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(Float64(-0.25 * v) * Float64(Float64(Float64(Float64(w * w) * r) * r) / Float64(-v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= 8.6e-37)
		tmp = ((3.0 + t_0) - ((0.375 * (((w * r) * w) * r)) / (1.0 - v))) - 4.5;
	else
		tmp = -(1.5 - (t_0 - ((-0.25 * v) * ((((w * w) * r) * r) / -v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 8.6e-37], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.375 * N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-N[(1.5 - N[(t$95$0 - N[(N[(-0.25 * v), $MachinePrecision] * N[(N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision] / (-v)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if v < 8.59999999999999936e-37

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 92.7%

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 82.4%

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

   if 8.59999999999999936e-37 < v

   1. Initial program 84.9%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.9%

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in v around inf

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

      1. lower-*.f64 80.1%

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

   7. Applied rewrites 80.1%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

   9. Applied rewrites 82.6%

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

Alternative 7: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|r\right| \leq 1.12 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\left|r\right| \cdot \left|r\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{\frac{2}{\left|r\right|}}{\left|r\right|}\right) - \frac{0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot \left|r\right|\right) \cdot \left|r\right|\right)}{1}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= (fabs r) 1.12e-119)
   (/ 2.0 (* (fabs r) (fabs r)))
   (-
    (-
     (+ 3.0 (/ (/ 2.0 (fabs r)) (fabs r)))
     (/ (* 0.375 (* (* (* w w) (fabs r)) (fabs r))) 1.0))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (fabs(r) <= 1.12e-119) {
		tmp = 2.0 / (fabs(r) * fabs(r));
	} else {
		tmp = ((3.0 + ((2.0 / fabs(r)) / fabs(r))) - ((0.375 * (((w * w) * fabs(r)) * fabs(r))) / 1.0)) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (abs(r) <= 1.12d-119) then
        tmp = 2.0d0 / (abs(r) * abs(r))
    else
        tmp = ((3.0d0 + ((2.0d0 / abs(r)) / abs(r))) - ((0.375d0 * (((w * w) * abs(r)) * abs(r))) / 1.0d0)) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (Math.abs(r) <= 1.12e-119) {
		tmp = 2.0 / (Math.abs(r) * Math.abs(r));
	} else {
		tmp = ((3.0 + ((2.0 / Math.abs(r)) / Math.abs(r))) - ((0.375 * (((w * w) * Math.abs(r)) * Math.abs(r))) / 1.0)) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if math.fabs(r) <= 1.12e-119:
		tmp = 2.0 / (math.fabs(r) * math.fabs(r))
	else:
		tmp = ((3.0 + ((2.0 / math.fabs(r)) / math.fabs(r))) - ((0.375 * (((w * w) * math.fabs(r)) * math.fabs(r))) / 1.0)) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (abs(r) <= 1.12e-119)
		tmp = Float64(2.0 / Float64(abs(r) * abs(r)));
	else
		tmp = Float64(Float64(Float64(3.0 + Float64(Float64(2.0 / abs(r)) / abs(r))) - Float64(Float64(0.375 * Float64(Float64(Float64(w * w) * abs(r)) * abs(r))) / 1.0)) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (abs(r) <= 1.12e-119)
		tmp = 2.0 / (abs(r) * abs(r));
	else
		tmp = ((3.0 + ((2.0 / abs(r)) / abs(r))) - ((0.375 * (((w * w) * abs(r)) * abs(r))) / 1.0)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[Abs[r], $MachinePrecision], 1.12e-119], N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + N[(N[(2.0 / N[Abs[r], $MachinePrecision]), $MachinePrecision] / N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 * N[(N[(N[(w * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.11999999999999998e-119

   1. Initial program 84.9%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 44.2%

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

   4. Applied rewrites 44.2%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 44.2%

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

   6. Applied rewrites 44.2%

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

   if 1.11999999999999998e-119 < r

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 84.8%

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 76.4%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 83.2%

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

Alternative 8: 67.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(w \cdot w\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := 3 + t\_1\\ \mathbf{if}\;\left(t\_2 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_0 \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(t\_2 - \left(\left(-0.25 \cdot v\right) \cdot t\_0\right) \cdot r\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* w w) r)) (t_1 (/ 2.0 (* r r))) (t_2 (+ 3.0 t_1)))
   (if (<=
        (- (- t_2 (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_0 r)) (- 1.0 v))) 4.5)
        (- INFINITY))
     (- (- t_2 (* (* (* -0.25 v) t_0) r)) 4.5)
     (- t_1 1.5))))

C

double code(double v, double w, double r) {
	double t_0 = (w * w) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = 3.0 + t_1;
	double tmp;
	if (((t_2 - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) - 4.5) <= -((double) INFINITY)) {
		tmp = (t_2 - (((-0.25 * v) * t_0) * r)) - 4.5;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = (w * w) * r;
	double t_1 = 2.0 / (r * r);
	double t_2 = 3.0 + t_1;
	double tmp;
	if (((t_2 - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) - 4.5) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_2 - (((-0.25 * v) * t_0) * r)) - 4.5;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (w * w) * r
	t_1 = 2.0 / (r * r)
	t_2 = 3.0 + t_1
	tmp = 0
	if ((t_2 - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) - 4.5) <= -math.inf:
		tmp = (t_2 - (((-0.25 * v) * t_0) * r)) - 4.5
	else:
		tmp = t_1 - 1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(w * w) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(3.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(t_2 - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_0 * r)) / Float64(1.0 - v))) - 4.5) <= Float64(-Inf))
		tmp = Float64(Float64(t_2 - Float64(Float64(Float64(-0.25 * v) * t_0) * r)) - 4.5);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (w * w) * r;
	t_1 = 2.0 / (r * r);
	t_2 = 3.0 + t_1;
	tmp = 0.0;
	if (((t_2 - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r)) / (1.0 - v))) - 4.5) <= -Inf)
		tmp = (t_2 - (((-0.25 * v) * t_0) * r)) - 4.5;
	else
		tmp = t_1 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-Infinity)], N[(N[(t$95$2 - N[(N[(N[(-0.25 * v), $MachinePrecision] * t$95$0), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$1 - 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)) < -inf.0

   1. Initial program 84.9%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 73.9%

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in v around inf

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

      1. lower-*.f64 80.1%

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

   7. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 78.5%

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 78.5%

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

   9. Applied rewrites 78.5%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 62.2%

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

   if -inf.0 < (-.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 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 87.7%

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

   3. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. unswap-sqr N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.8%

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 96.6%

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

   8. Taylor expanded in w around 0

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

   10. Applied rewrites 57.1%

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

Alternative 9: 57.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 87.7%

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. unswap-sqr N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 96.6%

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

8. Taylor expanded in w around 0

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

10. Applied rewrites 57.1%

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

Alternative 10: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|r\right| \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left|r\right| \cdot \left|r\right|}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= (fabs r) 1.2e-7) (/ 2.0 (* (fabs r) (fabs r))) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (fabs(r) <= 1.2e-7) {
		tmp = 2.0 / (fabs(r) * fabs(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 (abs(r) <= 1.2d-7) then
        tmp = 2.0d0 / (abs(r) * abs(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 (Math.abs(r) <= 1.2e-7) {
		tmp = 2.0 / (Math.abs(r) * Math.abs(r));
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if math.fabs(r) <= 1.2e-7:
		tmp = 2.0 / (math.fabs(r) * math.fabs(r))
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (abs(r) <= 1.2e-7)
		tmp = Float64(2.0 / Float64(abs(r) * abs(r)));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (abs(r) <= 1.2e-7)
		tmp = 2.0 / (abs(r) * abs(r));
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[Abs[r], $MachinePrecision], 1.2e-7], N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.19999999999999989e-7

   1. Initial program 84.9%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 44.2%

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

   4. Applied rewrites 44.2%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 44.2%

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

   6. Applied rewrites 44.2%

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

   if 1.19999999999999989e-7 < r

   1. Initial program 84.9%

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

   2. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 57.1%

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

   4. Applied rewrites 57.1%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 13.9%

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

Alternative 11: 13.9% accurate, 41.6× speedup

Math

\[-1.5 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 57.1%

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in r around inf

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

7. Applied rewrites 13.9%

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

Reproduce

herbie shell --seed 2025187 
(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))