Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.4%

Time: 5.2s

Alternatives: 11

Speedup: 1.3×

• 54.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{r\_m}{1 - v}\\ t_1 := 3 + \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\left(t\_1 - \mathsf{fma}\left(-2, v, 3\right) \cdot \left(\left(w \cdot 0.125\right) \cdot \left(t\_0 \cdot \left(w \cdot r\_m\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \left(\left(\left(w \cdot r\_m\right) \cdot w\right) \cdot t\_0\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ r_m (- 1.0 v))) (t_1 (+ 3.0 (/ 2.0 (* r_m r_m)))))
   (if (<= r_m 2e+78)
     (- (- t_1 (* (fma -2.0 v 3.0) (* (* w 0.125) (* t_0 (* w r_m))))) 4.5)
     (- (- t_1 (* (* (* (* w r_m) w) t_0) (fma -0.25 v 0.375))) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = r_m / (1.0 - v);
	double t_1 = 3.0 + (2.0 / (r_m * r_m));
	double tmp;
	if (r_m <= 2e+78) {
		tmp = (t_1 - (fma(-2.0, v, 3.0) * ((w * 0.125) * (t_0 * (w * r_m))))) - 4.5;
	} else {
		tmp = (t_1 - ((((w * r_m) * w) * t_0) * fma(-0.25, v, 0.375))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(r_m / Float64(1.0 - v))
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r_m * r_m)))
	tmp = 0.0
	if (r_m <= 2e+78)
		tmp = Float64(Float64(t_1 - Float64(fma(-2.0, v, 3.0) * Float64(Float64(w * 0.125) * Float64(t_0 * Float64(w * r_m))))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(Float64(w * r_m) * w) * t_0) * fma(-0.25, v, 0.375))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 2e+78], N[(N[(t$95$1 - N[(N[(-2.0 * v + 3.0), $MachinePrecision] * N[(N[(w * 0.125), $MachinePrecision] * N[(t$95$0 * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(N[(w * r$95$m), $MachinePrecision] * w), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.00000000000000002e78

   1. Initial program 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. lower-unsound-/.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 94.8

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

   3. Applied rewrites 94.8%

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

   5. Applied rewrites 97.7%

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

   if 2.00000000000000002e78 < r

   1. Initial program 84.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 87.3

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

   6. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := \left(\left(t\_0 - -3\right) - \left(\left(\frac{r\_m}{1 - v} \cdot w\right) \cdot \left(w \cdot r\_m\right)\right) \cdot \left(-0.25 \cdot v\right)\right) - 4.5\\ \mathbf{if}\;v \leq -400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{\left(w \cdot r\_m\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\_m\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1
         (-
          (-
           (- t_0 -3.0)
           (* (* (* (/ r_m (- 1.0 v)) w) (* w r_m)) (* -0.25 v)))
          4.5)))
   (if (<= v -400000.0)
     t_1
     (if (<= v 1.5)
       (- (- (+ 3.0 t_0) (/ (* (* w r_m) (* (* 0.375 w) r_m)) (- 1.0 v))) 4.5)
       t_1))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = ((t_0 - -3.0) - ((((r_m / (1.0 - v)) * w) * (w * r_m)) * (-0.25 * v))) - 4.5;
	double tmp;
	if (v <= -400000.0) {
		tmp = t_1;
	} else if (v <= 1.5) {
		tmp = ((3.0 + t_0) - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(v, w, r_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    t_1 = ((t_0 - (-3.0d0)) - ((((r_m / (1.0d0 - v)) * w) * (w * r_m)) * ((-0.25d0) * v))) - 4.5d0
    if (v <= (-400000.0d0)) then
        tmp = t_1
    else if (v <= 1.5d0) then
        tmp = ((3.0d0 + t_0) - (((w * r_m) * ((0.375d0 * w) * r_m)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = ((t_0 - -3.0) - ((((r_m / (1.0 - v)) * w) * (w * r_m)) * (-0.25 * v))) - 4.5;
	double tmp;
	if (v <= -400000.0) {
		tmp = t_1;
	} else if (v <= 1.5) {
		tmp = ((3.0 + t_0) - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	t_1 = ((t_0 - -3.0) - ((((r_m / (1.0 - v)) * w) * (w * r_m)) * (-0.25 * v))) - 4.5
	tmp = 0
	if v <= -400000.0:
		tmp = t_1
	elif v <= 1.5:
		tmp = ((3.0 + t_0) - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	t_1 = Float64(Float64(Float64(t_0 - -3.0) - Float64(Float64(Float64(Float64(r_m / Float64(1.0 - v)) * w) * Float64(w * r_m)) * Float64(-0.25 * v))) - 4.5)
	tmp = 0.0
	if (v <= -400000.0)
		tmp = t_1;
	elseif (v <= 1.5)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(w * r_m) * Float64(Float64(0.375 * w) * r_m)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 2.0 / (r_m * r_m);
	t_1 = ((t_0 - -3.0) - ((((r_m / (1.0 - v)) * w) * (w * r_m)) * (-0.25 * v))) - 4.5;
	tmp = 0.0;
	if (v <= -400000.0)
		tmp = t_1;
	elseif (v <= 1.5)
		tmp = ((3.0 + t_0) - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(N[(N[(N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -400000.0], t$95$1, If[LessEqual[v, 1.5], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(w * r$95$m), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r$95$m), $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 < -4e5 or 1.5 < v

   1. Initial program 84.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 76.8

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

   6. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval 85.2

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

   8. Applied rewrites 85.2%

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

   if -4e5 < v < 1.5

   1. Initial program 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. lower-unsound-/.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 94.8

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

   3. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-down N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.9

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

   8. Applied rewrites 84.9%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{r\_m}{1 - v}\\ t_1 := 3 + \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\left(t\_1 - w \cdot \left(t\_0 \cdot \left(\left(-0.25 \cdot v\right) \cdot \left(w \cdot r\_m\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \left(\left(\left(w \cdot r\_m\right) \cdot w\right) \cdot t\_0\right) \cdot \mathsf{fma}\left(-0.25, v, 0.375\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ r_m (- 1.0 v))) (t_1 (+ 3.0 (/ 2.0 (* r_m r_m)))))
   (if (<= r_m 3e-109)
     (- (- t_1 (* w (* t_0 (* (* -0.25 v) (* w r_m))))) 4.5)
     (- (- t_1 (* (* (* (* w r_m) w) t_0) (fma -0.25 v 0.375))) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = r_m / (1.0 - v);
	double t_1 = 3.0 + (2.0 / (r_m * r_m));
	double tmp;
	if (r_m <= 3e-109) {
		tmp = (t_1 - (w * (t_0 * ((-0.25 * v) * (w * r_m))))) - 4.5;
	} else {
		tmp = (t_1 - ((((w * r_m) * w) * t_0) * fma(-0.25, v, 0.375))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(r_m / Float64(1.0 - v))
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r_m * r_m)))
	tmp = 0.0
	if (r_m <= 3e-109)
		tmp = Float64(Float64(t_1 - Float64(w * Float64(t_0 * Float64(Float64(-0.25 * v) * Float64(w * r_m))))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(Float64(w * r_m) * w) * t_0) * fma(-0.25, v, 0.375))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 3e-109], N[(N[(t$95$1 - N[(w * N[(t$95$0 * N[(N[(-0.25 * v), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(N[(w * r$95$m), $MachinePrecision] * w), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.25 * v + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.00000000000000021e-109

   1. Initial program 84.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 76.8

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

   6. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 80.2

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

   8. Applied rewrites 80.2%

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

   if 3.00000000000000021e-109 < r

   1. Initial program 84.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 87.3

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

   6. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

Alternative 4: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{r\_m}{1 - v}\\ t_1 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 3.1 \cdot 10^{-109}:\\ \;\;\;\;\left(\left(3 + t\_1\right) - w \cdot \left(t\_0 \cdot \left(\left(-0.25 \cdot v\right) \cdot \left(w \cdot r\_m\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \mathsf{fma}\left(\left(\left(w \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot 0.125\right) \cdot \left(w \cdot r\_m\right), t\_0, 1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ r_m (- 1.0 v))) (t_1 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 3.1e-109)
     (- (- (+ 3.0 t_1) (* w (* t_0 (* (* -0.25 v) (* w r_m))))) 4.5)
     (- t_1 (fma (* (* (* w (fma -2.0 v 3.0)) 0.125) (* w r_m)) t_0 1.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = r_m / (1.0 - v);
	double t_1 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 3.1e-109) {
		tmp = ((3.0 + t_1) - (w * (t_0 * ((-0.25 * v) * (w * r_m))))) - 4.5;
	} else {
		tmp = t_1 - fma((((w * fma(-2.0, v, 3.0)) * 0.125) * (w * r_m)), t_0, 1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(r_m / Float64(1.0 - v))
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 3.1e-109)
		tmp = Float64(Float64(Float64(3.0 + t_1) - Float64(w * Float64(t_0 * Float64(Float64(-0.25 * v) * Float64(w * r_m))))) - 4.5);
	else
		tmp = Float64(t_1 - fma(Float64(Float64(Float64(w * fma(-2.0, v, 3.0)) * 0.125) * Float64(w * r_m)), t_0, 1.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 3.1e-109], N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(w * N[(t$95$0 * N[(N[(-0.25 * v), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$1 - N[(N[(N[(N[(w * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.1e-109

   1. Initial program 84.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 76.8

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

   6. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 80.2

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

   8. Applied rewrites 80.2%

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

   if 3.1e-109 < r

   1. Initial program 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. lower-unsound-/.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 94.8

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

   3. Applied rewrites 94.8%

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

   5. Applied rewrites 91.4%

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

Alternative 5: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := -\left(1.5 - \left(t\_0 - \left(\left(-0.25 \cdot v\right) \cdot \left(\left(\frac{r\_m}{1 - v} \cdot w\right) \cdot w\right)\right) \cdot r\_m\right)\right)\\ \mathbf{if}\;v \leq -7.5 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \frac{\left(w \cdot r\_m\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\_m\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1
         (-
          (-
           1.5
           (- t_0 (* (* (* -0.25 v) (* (* (/ r_m (- 1.0 v)) w) w)) r_m))))))
   (if (<= v -7.5e+77)
     t_1
     (if (<= v 1.5)
       (- (- (+ 3.0 t_0) (/ (* (* w r_m) (* (* 0.375 w) r_m)) (- 1.0 v))) 4.5)
       t_1))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = -(1.5 - (t_0 - (((-0.25 * v) * (((r_m / (1.0 - v)) * w) * w)) * r_m)));
	double tmp;
	if (v <= -7.5e+77) {
		tmp = t_1;
	} else if (v <= 1.5) {
		tmp = ((3.0 + t_0) - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(v, w, r_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    t_1 = -(1.5d0 - (t_0 - ((((-0.25d0) * v) * (((r_m / (1.0d0 - v)) * w) * w)) * r_m)))
    if (v <= (-7.5d+77)) then
        tmp = t_1
    else if (v <= 1.5d0) then
        tmp = ((3.0d0 + t_0) - (((w * r_m) * ((0.375d0 * w) * r_m)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = -(1.5 - (t_0 - (((-0.25 * v) * (((r_m / (1.0 - v)) * w) * w)) * r_m)));
	double tmp;
	if (v <= -7.5e+77) {
		tmp = t_1;
	} else if (v <= 1.5) {
		tmp = ((3.0 + t_0) - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	t_1 = -(1.5 - (t_0 - (((-0.25 * v) * (((r_m / (1.0 - v)) * w) * w)) * r_m)))
	tmp = 0
	if v <= -7.5e+77:
		tmp = t_1
	elif v <= 1.5:
		tmp = ((3.0 + t_0) - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	t_1 = Float64(-Float64(1.5 - Float64(t_0 - Float64(Float64(Float64(-0.25 * v) * Float64(Float64(Float64(r_m / Float64(1.0 - v)) * w) * w)) * r_m))))
	tmp = 0.0
	if (v <= -7.5e+77)
		tmp = t_1;
	elseif (v <= 1.5)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(w * r_m) * Float64(Float64(0.375 * w) * r_m)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 2.0 / (r_m * r_m);
	t_1 = -(1.5 - (t_0 - (((-0.25 * v) * (((r_m / (1.0 - v)) * w) * w)) * r_m)));
	tmp = 0.0;
	if (v <= -7.5e+77)
		tmp = t_1;
	elseif (v <= 1.5)
		tmp = ((3.0 + t_0) - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(1.5 - N[(t$95$0 - N[(N[(N[(-0.25 * v), $MachinePrecision] * N[(N[(N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[v, -7.5e+77], t$95$1, If[LessEqual[v, 1.5], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(w * r$95$m), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -7.49999999999999955e77 or 1.5 < v

   1. Initial program 84.5%

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

   2. Taylor expanded in v around inf

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

      1. lower-*.f64 74.1

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

   4. Applied rewrites 74.1%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 76.8

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

   6. Applied rewrites 85.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

   8. Applied rewrites 81.8%

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

   if -7.49999999999999955e77 < v < 1.5

   1. Initial program 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. lower-unsound-/.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 94.8

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

   3. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-down N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.9

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

   8. Applied rewrites 84.9%

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

Alternative 6: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := 3 + \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;v \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;\left(t\_0 - \frac{\left(w \cdot r\_m\right) \cdot \left(\left(0.375 \cdot w\right) \cdot r\_m\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(w \cdot w\right) \cdot r\_m\right) \cdot r\_m\right) \cdot 0.375\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r_m r_m)))))
   (if (<= v 2.8e+40)
     (- (- t_0 (/ (* (* w r_m) (* (* 0.375 w) r_m)) (- 1.0 v))) 4.5)
     (- (- t_0 (* (* (* (* w w) r_m) r_m) 0.375)) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 3.0 + (2.0 / (r_m * r_m));
	double tmp;
	if (v <= 2.8e+40) {
		tmp = (t_0 - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(v, w, r_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r_m * r_m))
    if (v <= 2.8d+40) then
        tmp = (t_0 - (((w * r_m) * ((0.375d0 * w) * r_m)) / (1.0d0 - v))) - 4.5d0
    else
        tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375d0)) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 3.0 + (2.0 / (r_m * r_m));
	double tmp;
	if (v <= 2.8e+40) {
		tmp = (t_0 - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 3.0 + (2.0 / (r_m * r_m))
	tmp = 0
	if v <= 2.8e+40:
		tmp = (t_0 - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5
	else:
		tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r_m * r_m)))
	tmp = 0.0
	if (v <= 2.8e+40)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(w * r_m) * Float64(Float64(0.375 * w) * r_m)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(w * w) * r_m) * r_m) * 0.375)) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 3.0 + (2.0 / (r_m * r_m));
	tmp = 0.0;
	if (v <= 2.8e+40)
		tmp = (t_0 - (((w * r_m) * ((0.375 * w) * r_m)) / (1.0 - v))) - 4.5;
	else
		tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 2.8e+40], N[(N[(t$95$0 - N[(N[(N[(w * r$95$m), $MachinePrecision] * N[(N[(0.375 * w), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 2.8000000000000001e40

   1. Initial program 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. lower-unsound-/.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 94.8

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

   3. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-down N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 84.9

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

   8. Applied rewrites 84.9%

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

   if 2.8000000000000001e40 < v

   1. Initial program 84.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 87.3

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 74.3%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 83.0%

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

Alternative 7: 90.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := 3 + \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;v \leq 2.8 \cdot 10^{+40}:\\ \;\;\;\;\left(t\_0 - \frac{\left(w \cdot r\_m\right) \cdot \left(0.375 \cdot \left(r\_m \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(w \cdot w\right) \cdot r\_m\right) \cdot r\_m\right) \cdot 0.375\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r_m r_m)))))
   (if (<= v 2.8e+40)
     (- (- t_0 (/ (* (* w r_m) (* 0.375 (* r_m w))) (- 1.0 v))) 4.5)
     (- (- t_0 (* (* (* (* w w) r_m) r_m) 0.375)) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 3.0 + (2.0 / (r_m * r_m));
	double tmp;
	if (v <= 2.8e+40) {
		tmp = (t_0 - (((w * r_m) * (0.375 * (r_m * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(v, w, r_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r_m * r_m))
    if (v <= 2.8d+40) then
        tmp = (t_0 - (((w * r_m) * (0.375d0 * (r_m * w))) / (1.0d0 - v))) - 4.5d0
    else
        tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375d0)) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 3.0 + (2.0 / (r_m * r_m));
	double tmp;
	if (v <= 2.8e+40) {
		tmp = (t_0 - (((w * r_m) * (0.375 * (r_m * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 3.0 + (2.0 / (r_m * r_m))
	tmp = 0
	if v <= 2.8e+40:
		tmp = (t_0 - (((w * r_m) * (0.375 * (r_m * w))) / (1.0 - v))) - 4.5
	else:
		tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r_m * r_m)))
	tmp = 0.0
	if (v <= 2.8e+40)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(w * r_m) * Float64(0.375 * Float64(r_m * w))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(w * w) * r_m) * r_m) * 0.375)) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 3.0 + (2.0 / (r_m * r_m));
	tmp = 0.0;
	if (v <= 2.8e+40)
		tmp = (t_0 - (((w * r_m) * (0.375 * (r_m * w))) / (1.0 - v))) - 4.5;
	else
		tmp = (t_0 - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 2.8e+40], N[(N[(t$95$0 - N[(N[(N[(w * r$95$m), $MachinePrecision] * N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 2.8000000000000001e40

   1. Initial program 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. lower-unsound-/.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 94.8

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

   3. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-down N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 84.9

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

   8. Applied rewrites 84.9%

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

   if 2.8000000000000001e40 < v

   1. Initial program 84.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 87.3

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 74.3%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 83.0%

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

Alternative 8: 90.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 2.1 \cdot 10^{-111}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(\left(\left(w \cdot w\right) \cdot r\_m\right) \cdot r\_m\right) \cdot 0.375\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 2.1e-111)
     t_0
     (- (- (+ 3.0 t_0) (* (* (* (* w w) r_m) r_m) 0.375)) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 2.1e-111) {
		tmp = t_0;
	} else {
		tmp = ((3.0 + t_0) - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(v, w, r_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    if (r_m <= 2.1d-111) then
        tmp = t_0
    else
        tmp = ((3.0d0 + t_0) - ((((w * w) * r_m) * r_m) * 0.375d0)) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 2.1e-111) {
		tmp = t_0;
	} else {
		tmp = ((3.0 + t_0) - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	tmp = 0
	if r_m <= 2.1e-111:
		tmp = t_0
	else:
		tmp = ((3.0 + t_0) - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 2.1e-111)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(Float64(w * w) * r_m) * r_m) * 0.375)) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 2.0 / (r_m * r_m);
	tmp = 0.0;
	if (r_m <= 2.1e-111)
		tmp = t_0;
	else
		tmp = ((3.0 + t_0) - ((((w * w) * r_m) * r_m) * 0.375)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 2.1e-111], t$95$0, N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.0999999999999999e-111

   1. Initial program 84.5%

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

   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 45.3

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

   4. Applied rewrites 45.3%

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   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 45.3

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

   6. Applied rewrites 45.3%

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

   if 2.0999999999999999e-111 < r

   1. Initial program 84.5%

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

   2. Taylor expanded in v around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 84.5

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

   4. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 87.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 87.3

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

   6. Applied rewrites 87.3%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 74.3%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 83.0%

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

Alternative 9: 58.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ 2 \cdot \frac{1}{{r\_m}^{2}} - 1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

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

real(8) function code(v, w, r_m)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = (2.0d0 * (1.0d0 / (r_m ** 2.0d0))) - 1.5d0
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return (2.0 * (1.0 / Math.pow(r_m, 2.0))) - 1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (2.0 * (1.0 / math.pow(r_m, 2.0))) - 1.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

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 58.1

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

4. Applied rewrites 58.1%

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

Alternative 10: 58.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. mult-flip N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

3. Applied rewrites 91.4%

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

4. Taylor expanded in w around 0

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

6. Applied rewrites 58.1%

   \[\leadsto \mathsf{fma}\left(\frac{2}{r}, \frac{1}{r}, \color{blue}{-1.5}\right) \]
7. Add Preprocessing

Alternative 11: 45.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

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 45.3

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

4. Applied rewrites 45.3%

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
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 45.3

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

6. Applied rewrites 45.3%

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

Reproduce

herbie shell --seed 2025157 
(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))