Rosa's TurbineBenchmark

Percentage Accurate: 84.3% → 99.5%

Time: 6.0s

Alternatives: 16

Speedup: 1.3×

• 46.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|w\right| \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left|w\right| \leq 10^{-21}:\\ \;\;\;\;-\left(1.5 - \left(t\_1 - \left(\left(\left(\left|w\right| \cdot \mathsf{fma}\left(-2, v, 3\right)\right) \cdot 0.125\right) \cdot t\_0\right) \cdot \frac{r}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_1\right) - \left(\left|w\right| \cdot \left(t\_0 \cdot r\right)\right) \cdot \frac{\mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (fabs w) r)) (t_1 (/ 2.0 (* r r))))
   (if (<= (fabs w) 1e-21)
     (-
      (-
       1.5
       (-
        t_1
        (* (* (* (* (fabs w) (fma -2.0 v 3.0)) 0.125) t_0) (/ r (- 1.0 v))))))
     (-
      (-
       (+ 3.0 t_1)
       (* (* (fabs w) (* t_0 r)) (/ (fma -0.25 v 0.375) (- 1.0 v))))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = fabs(w) * r;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (fabs(w) <= 1e-21) {
		tmp = -(1.5 - (t_1 - ((((fabs(w) * fma(-2.0, v, 3.0)) * 0.125) * t_0) * (r / (1.0 - v)))));
	} else {
		tmp = ((3.0 + t_1) - ((fabs(w) * (t_0 * r)) * (fma(-0.25, v, 0.375) / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(abs(w) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (abs(w) <= 1e-21)
		tmp = Float64(-Float64(1.5 - Float64(t_1 - Float64(Float64(Float64(Float64(abs(w) * fma(-2.0, v, 3.0)) * 0.125) * t_0) * Float64(r / Float64(1.0 - v))))));
	else
		tmp = Float64(Float64(Float64(3.0 + t_1) - Float64(Float64(abs(w) * Float64(t_0 * r)) * Float64(fma(-0.25, v, 0.375) / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[Abs[w], $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[w], $MachinePrecision], 1e-21], (-N[(1.5 - N[(t$95$1 - N[(N[(N[(N[(N[Abs[w], $MachinePrecision] * N[(-2.0 * v + 3.0), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[Abs[w], $MachinePrecision] * N[(t$95$0 * r), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.25 * v + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if w < 9.99999999999999908e-22

   1. Initial program 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   5. Applied rewrites 91.4%

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

   if 9.99999999999999908e-22 < w

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := w \cdot \left|r\right|\\ t_1 := 3 + \frac{2}{\left|r\right| \cdot \left|r\right|}\\ \mathbf{if}\;\left|r\right| \leq 10^{+170}:\\ \;\;\;\;\left(t\_1 - \left(w \cdot \left(t\_0 \cdot \left|r\right|\right)\right) \cdot \frac{\mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \frac{\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot t\_0\right) \cdot t\_0}{1 - v}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (fabs r))) (t_1 (+ 3.0 (/ 2.0 (* (fabs r) (fabs r))))))
   (if (<= (fabs r) 1e+170)
     (-
      (- t_1 (* (* w (* t_0 (fabs r))) (/ (fma -0.25 v 0.375) (- 1.0 v))))
      4.5)
     (- (- t_1 (/ (* (* (fma -0.25 v 0.375) t_0) t_0) (- 1.0 v))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = w * fabs(r);
	double t_1 = 3.0 + (2.0 / (fabs(r) * fabs(r)));
	double tmp;
	if (fabs(r) <= 1e+170) {
		tmp = (t_1 - ((w * (t_0 * fabs(r))) * (fma(-0.25, v, 0.375) / (1.0 - v)))) - 4.5;
	} else {
		tmp = (t_1 - (((fma(-0.25, v, 0.375) * t_0) * t_0) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(w * abs(r))
	t_1 = Float64(3.0 + Float64(2.0 / Float64(abs(r) * abs(r))))
	tmp = 0.0
	if (abs(r) <= 1e+170)
		tmp = Float64(Float64(t_1 - Float64(Float64(w * Float64(t_0 * abs(r))) * Float64(fma(-0.25, v, 0.375) / Float64(1.0 - v)))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * t_0) * t_0) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[Abs[r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 1e+170], N[(N[(t$95$1 - N[(N[(w * N[(t$95$0 * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.25 * v + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.00000000000000003e170

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   if 1.00000000000000003e170 < r

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 94.8%

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 94.8%

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

   6. Applied rewrites 94.8%

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

Alternative 3: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left|w\right| \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - \frac{\mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v} \cdot \left(\left(\left(\left|w\right| \cdot \left|w\right|\right) \cdot r\right) \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(\left|w\right| \cdot \left(\left(\left|w\right| \cdot r\right) \cdot r\right)\right) \cdot 0.375\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (fabs w) 2.2e+138)
     (-
      (-
       1.5
       (-
        t_0
        (*
         (/ (fma -0.25 v 0.375) (- 1.0 v))
         (* (* (* (fabs w) (fabs w)) r) r)))))
     (- (- (+ 3.0 t_0) (* (* (fabs w) (* (* (fabs w) r) r)) 0.375)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (fabs(w) <= 2.2e+138) {
		tmp = -(1.5 - (t_0 - ((fma(-0.25, v, 0.375) / (1.0 - v)) * (((fabs(w) * fabs(w)) * r) * r))));
	} else {
		tmp = ((3.0 + t_0) - ((fabs(w) * ((fabs(w) * r) * r)) * 0.375)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (abs(w) <= 2.2e+138)
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(Float64(fma(-0.25, v, 0.375) / Float64(1.0 - v)) * Float64(Float64(Float64(abs(w) * abs(w)) * r) * r)))));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(abs(w) * Float64(Float64(abs(w) * r) * r)) * 0.375)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[w], $MachinePrecision], 2.2e+138], (-N[(1.5 - N[(t$95$0 - N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[w], $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[Abs[w], $MachinePrecision] * N[(N[(N[Abs[w], $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.2000000000000001e138

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

   8. Applied rewrites 87.2%

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

   if 2.2000000000000001e138 < w

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 91.8%

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

Alternative 4: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left|w\right| \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v} \cdot \left(\left|w\right| \cdot \left|w\right|\right)\right) \cdot r, r, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(\left|w\right| \cdot \left(\left(\left|w\right| \cdot r\right) \cdot r\right)\right) \cdot 0.375\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (fabs w) 2.2e+138)
     (-
      t_0
      (fma
       (* (* (/ (fma -0.25 v 0.375) (- 1.0 v)) (* (fabs w) (fabs w))) r)
       r
       1.5))
     (- (- (+ 3.0 t_0) (* (* (fabs w) (* (* (fabs w) r) r)) 0.375)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (fabs(w) <= 2.2e+138) {
		tmp = t_0 - fma((((fma(-0.25, v, 0.375) / (1.0 - v)) * (fabs(w) * fabs(w))) * r), r, 1.5);
	} else {
		tmp = ((3.0 + t_0) - ((fabs(w) * ((fabs(w) * r) * r)) * 0.375)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (abs(w) <= 2.2e+138)
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(fma(-0.25, v, 0.375) / Float64(1.0 - v)) * Float64(abs(w) * abs(w))) * r), r, 1.5));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(abs(w) * Float64(Float64(abs(w) * r) * r)) * 0.375)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[w], $MachinePrecision], 2.2e+138], N[(t$95$0 - N[(N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[w], $MachinePrecision] * N[Abs[w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] * r + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[Abs[w], $MachinePrecision] * N[(N[(N[Abs[w], $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 2.2000000000000001e138

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   10. Applied rewrites 87.2%

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

   if 2.2000000000000001e138 < w

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 91.8%

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

Alternative 5: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{if}\;v \leq -8.6 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\left(t\_0 - -3\right) - \mathsf{fma}\left(\frac{r}{1 - v} \cdot \left(0.375 \cdot w\right), w \cdot r, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (- (- (+ 3.0 t_0) (* (* w (* (* w r) r)) 0.25)) 4.5)))
   (if (<= v -8.6e+103)
     t_1
     (if (<= v 8e-7)
       (- (- t_0 -3.0) (fma (* (/ r (- 1.0 v)) (* 0.375 w)) (* w r) 4.5))
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - ((w * ((w * r) * r)) * 0.25)) - 4.5;
	double tmp;
	if (v <= -8.6e+103) {
		tmp = t_1;
	} else if (v <= 8e-7) {
		tmp = (t_0 - -3.0) - fma(((r / (1.0 - v)) * (0.375 * w)), (w * r), 4.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(w * Float64(Float64(w * r) * r)) * 0.25)) - 4.5)
	tmp = 0.0
	if (v <= -8.6e+103)
		tmp = t_1;
	elseif (v <= 8e-7)
		tmp = Float64(Float64(t_0 - -3.0) - fma(Float64(Float64(r / Float64(1.0 - v)) * Float64(0.375 * w)), Float64(w * r), 4.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(w * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -8.6e+103], t$95$1, If[LessEqual[v, 8e-7], N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(N[(N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(0.375 * w), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -8.59999999999999938e103 or 7.9999999999999996e-7 < v

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 91.7%

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

   if -8.59999999999999938e103 < v < 7.9999999999999996e-7

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 77.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   6. Applied rewrites 82.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 84.2%

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

   8. Applied rewrites 84.2%

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

Alternative 6: 95.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (* (* w (* (* w r) r)) (/ (fma -0.25 v 0.375) (- 1.0 v))))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((w * ((w * r) * r)) * (fma(-0.25, v, 0.375) / (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(w * Float64(Float64(w * r) * r)) * Float64(fma(-0.25, v, 0.375) / Float64(1.0 - v)))) - 4.5)
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(w * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.25 * v + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.3%

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

2. Taylor expanded in v around 0

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 84.3%

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

4. Applied rewrites 84.3%

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

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 87.2%

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 87.2%

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

6. Applied rewrites 87.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 97.0%

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

8. Applied rewrites 97.0%

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

Alternative 7: 95.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\left(t\_0 - \frac{0.375 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) \cdot 0.25\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= v 8e-7)
     (- (- t_0 (/ (* 0.375 (* (* w r) (* w r))) (- 1.0 v))) 4.5)
     (- (- t_0 (* (* w (* (* w r) r)) 0.25)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= 8e-7) {
		tmp = (t_0 - ((0.375 * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - ((w * ((w * r) * r)) * 0.25)) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    if (v <= 8d-7) then
        tmp = (t_0 - ((0.375d0 * ((w * r) * (w * r))) / (1.0d0 - v))) - 4.5d0
    else
        tmp = (t_0 - ((w * ((w * r) * r)) * 0.25d0)) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= 8e-7) {
		tmp = (t_0 - ((0.375 * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - ((w * ((w * r) * r)) * 0.25)) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if v <= 8e-7:
		tmp = (t_0 - ((0.375 * ((w * r) * (w * r))) / (1.0 - v))) - 4.5
	else:
		tmp = (t_0 - ((w * ((w * r) * r)) * 0.25)) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (v <= 8e-7)
		tmp = Float64(Float64(t_0 - Float64(Float64(0.375 * Float64(Float64(w * r) * Float64(w * r))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(w * Float64(Float64(w * r) * r)) * 0.25)) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (v <= 8e-7)
		tmp = (t_0 - ((0.375 * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	else
		tmp = (t_0 - ((w * ((w * r) * r)) * 0.25)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 8e-7], N[(N[(t$95$0 - N[(N[(0.375 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(w * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 7.9999999999999996e-7

   1. Initial program 84.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 94.8%

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

   3. Applied rewrites 94.8%

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

   4. Taylor expanded in v around 0

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

   6. Applied rewrites 86.2%

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

   if 7.9999999999999996e-7 < v

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 91.7%

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

Alternative 8: 95.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := w \cdot \left(\left(w \cdot \left|r\right|\right) \cdot \left|r\right|\right)\\ \mathbf{if}\;\left|r\right| \leq 360000:\\ \;\;\;\;\left(\left(3 + \frac{2}{\left|r\right| \cdot \left|r\right|}\right) - t\_0 \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - t\_0 \cdot \frac{\mathsf{fma}\left(-0.25, v, 0.375\right)}{1 - v}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* (* w (fabs r)) (fabs r)))))
   (if (<= (fabs r) 360000.0)
     (- (- (+ 3.0 (/ 2.0 (* (fabs r) (fabs r)))) (* t_0 0.25)) 4.5)
     (- (- 3.0 (* t_0 (/ (fma -0.25 v 0.375) (- 1.0 v)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = w * ((w * fabs(r)) * fabs(r));
	double tmp;
	if (fabs(r) <= 360000.0) {
		tmp = ((3.0 + (2.0 / (fabs(r) * fabs(r)))) - (t_0 * 0.25)) - 4.5;
	} else {
		tmp = (3.0 - (t_0 * (fma(-0.25, v, 0.375) / (1.0 - v)))) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(Float64(w * abs(r)) * abs(r)))
	tmp = 0.0
	if (abs(r) <= 360000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(abs(r) * abs(r)))) - Float64(t_0 * 0.25)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(t_0 * Float64(fma(-0.25, v, 0.375) / Float64(1.0 - v)))) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(N[(w * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 360000.0], N[(N[(N[(3.0 + N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(t$95$0 * N[(N[(-0.25 * v + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.6e5

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 91.7%

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

   if 3.6e5 < r

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 52.8%

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

Alternative 9: 94.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := w \cdot \left(\left(w \cdot r\right) \cdot r\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\left(t\_1 - t\_0 \cdot 0.375\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - t\_0 \cdot 0.25\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* (* w r) r))) (t_1 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= v 5e-14)
     (- (- t_1 (* t_0 0.375)) 4.5)
     (- (- t_1 (* t_0 0.25)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = w * ((w * r) * r);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= 5e-14) {
		tmp = (t_1 - (t_0 * 0.375)) - 4.5;
	} else {
		tmp = (t_1 - (t_0 * 0.25)) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = w * ((w * r) * r)
    t_1 = 3.0d0 + (2.0d0 / (r * r))
    if (v <= 5d-14) then
        tmp = (t_1 - (t_0 * 0.375d0)) - 4.5d0
    else
        tmp = (t_1 - (t_0 * 0.25d0)) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * ((w * r) * r);
	double t_1 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (v <= 5e-14) {
		tmp = (t_1 - (t_0 * 0.375)) - 4.5;
	} else {
		tmp = (t_1 - (t_0 * 0.25)) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * ((w * r) * r)
	t_1 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if v <= 5e-14:
		tmp = (t_1 - (t_0 * 0.375)) - 4.5
	else:
		tmp = (t_1 - (t_0 * 0.25)) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(Float64(w * r) * r))
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (v <= 5e-14)
		tmp = Float64(Float64(t_1 - Float64(t_0 * 0.375)) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(t_0 * 0.25)) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * ((w * r) * r);
	t_1 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (v <= 5e-14)
		tmp = (t_1 - (t_0 * 0.375)) - 4.5;
	else
		tmp = (t_1 - (t_0 * 0.25)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 5e-14], N[(N[(t$95$1 - N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < 5.0000000000000002e-14

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 91.8%

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

   if 5.0000000000000002e-14 < v

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 91.7%

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

Alternative 10: 93.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 4.4 \cdot 10^{+27}:\\ \;\;\;\;\left(t\_0 - -3\right) - \mathsf{fma}\left(\left(0.375 \cdot w\right) \cdot \left(w \cdot r\right), r, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right) \cdot 0.25\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 4.4e+27)
     (- (- t_0 -3.0) (fma (* (* 0.375 w) (* w r)) r 4.5))
     (- (- (+ 3.0 t_0) (* (* w (* (* w r) r)) 0.25)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 4.4e+27) {
		tmp = (t_0 - -3.0) - fma(((0.375 * w) * (w * r)), r, 4.5);
	} else {
		tmp = ((3.0 + t_0) - ((w * ((w * r) * r)) * 0.25)) - 4.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 4.4e+27)
		tmp = Float64(Float64(t_0 - -3.0) - fma(Float64(Float64(0.375 * w) * Float64(w * r)), r, 4.5));
	else
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(w * Float64(Float64(w * r) * r)) * 0.25)) - 4.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 4.4e+27], N[(N[(t$95$0 - -3.0), $MachinePrecision] - N[(N[(N[(0.375 * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * r + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(w * N[(N[(w * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 4.3999999999999997e27

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 77.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   6. Applied rewrites 82.6%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 90.9%

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

   if 4.3999999999999997e27 < v

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 97.0%

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in v around inf

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

   11. Applied rewrites 91.7%

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

Alternative 11: 92.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 77.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   6. Applied rewrites 82.6%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 90.9%

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

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

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

   8. Applied rewrites 87.2%

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

   9. Taylor expanded in w around 0

      \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-pow.64 57.9%

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

   11. Applied rewrites 57.9%

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

Alternative 12: 90.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{\left|r\right| \cdot \left|r\right|}\\ \mathbf{if}\;\left|r\right| \leq 9.4 \cdot 10^{-116}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - 0.375 \cdot \left(\left(\left(w \cdot w\right) \cdot \left|r\right|\right) \cdot \left|r\right|\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* (fabs r) (fabs r)))))
   (if (<= (fabs r) 9.4e-116)
     t_0
     (- (- 1.5 (- t_0 (* 0.375 (* (* (* w w) (fabs r)) (fabs r)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (fabs(r) * fabs(r));
	double tmp;
	if (fabs(r) <= 9.4e-116) {
		tmp = t_0;
	} else {
		tmp = -(1.5 - (t_0 - (0.375 * (((w * w) * fabs(r)) * fabs(r)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (abs(r) * abs(r))
    if (abs(r) <= 9.4d-116) then
        tmp = t_0
    else
        tmp = -(1.5d0 - (t_0 - (0.375d0 * (((w * w) * abs(r)) * abs(r)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (Math.abs(r) * Math.abs(r));
	double tmp;
	if (Math.abs(r) <= 9.4e-116) {
		tmp = t_0;
	} else {
		tmp = -(1.5 - (t_0 - (0.375 * (((w * w) * Math.abs(r)) * Math.abs(r)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (math.fabs(r) * math.fabs(r))
	tmp = 0
	if math.fabs(r) <= 9.4e-116:
		tmp = t_0
	else:
		tmp = -(1.5 - (t_0 - (0.375 * (((w * w) * math.fabs(r)) * math.fabs(r)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(abs(r) * abs(r)))
	tmp = 0.0
	if (abs(r) <= 9.4e-116)
		tmp = t_0;
	else
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(0.375 * Float64(Float64(Float64(w * w) * abs(r)) * abs(r))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (abs(r) * abs(r));
	tmp = 0.0;
	if (abs(r) <= 9.4e-116)
		tmp = t_0;
	else
		tmp = -(1.5 - (t_0 - (0.375 * (((w * w) * abs(r)) * abs(r)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 9.4e-116], t$95$0, (-N[(1.5 - N[(t$95$0 - N[(0.375 * N[(N[(N[(w * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if r < 9.39999999999999989e-116

   1. Initial program 84.3%

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

   2. 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.64 44.7%

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

   4. Applied rewrites 44.7%

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

      1. lift-pow.64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 44.7%

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

   6. Applied rewrites 44.7%

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

   if 9.39999999999999989e-116 < r

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

   8. Applied rewrites 87.2%

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

   9. Taylor expanded in v around 0

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

   11. Applied rewrites 82.9%

       \[\leadsto -\left(1.5 - \left(\frac{2}{r \cdot r} - \color{blue}{0.375} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 90.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{\left|r\right| \cdot \left|r\right|}\\ \mathbf{if}\;\left|r\right| \leq 9.4 \cdot 10^{-116}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - 0.25 \cdot \left(\left(\left(w \cdot w\right) \cdot \left|r\right|\right) \cdot \left|r\right|\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* (fabs r) (fabs r)))))
   (if (<= (fabs r) 9.4e-116)
     t_0
     (- (- 1.5 (- t_0 (* 0.25 (* (* (* w w) (fabs r)) (fabs r)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (fabs(r) * fabs(r));
	double tmp;
	if (fabs(r) <= 9.4e-116) {
		tmp = t_0;
	} else {
		tmp = -(1.5 - (t_0 - (0.25 * (((w * w) * fabs(r)) * fabs(r)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (abs(r) * abs(r))
    if (abs(r) <= 9.4d-116) then
        tmp = t_0
    else
        tmp = -(1.5d0 - (t_0 - (0.25d0 * (((w * w) * abs(r)) * abs(r)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (Math.abs(r) * Math.abs(r));
	double tmp;
	if (Math.abs(r) <= 9.4e-116) {
		tmp = t_0;
	} else {
		tmp = -(1.5 - (t_0 - (0.25 * (((w * w) * Math.abs(r)) * Math.abs(r)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (math.fabs(r) * math.fabs(r))
	tmp = 0
	if math.fabs(r) <= 9.4e-116:
		tmp = t_0
	else:
		tmp = -(1.5 - (t_0 - (0.25 * (((w * w) * math.fabs(r)) * math.fabs(r)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(abs(r) * abs(r)))
	tmp = 0.0
	if (abs(r) <= 9.4e-116)
		tmp = t_0;
	else
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(0.25 * Float64(Float64(Float64(w * w) * abs(r)) * abs(r))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (abs(r) * abs(r));
	tmp = 0.0;
	if (abs(r) <= 9.4e-116)
		tmp = t_0;
	else
		tmp = -(1.5 - (t_0 - (0.25 * (((w * w) * abs(r)) * abs(r)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 9.4e-116], t$95$0, (-N[(1.5 - N[(t$95$0 - N[(0.25 * N[(N[(N[(w * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if r < 9.39999999999999989e-116

   1. Initial program 84.3%

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

   2. 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.64 44.7%

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

   4. Applied rewrites 44.7%

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

      1. lift-pow.64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 44.7%

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

   6. Applied rewrites 44.7%

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

   if 9.39999999999999989e-116 < r

   1. Initial program 84.3%

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

   2. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 84.3%

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

   4. Applied rewrites 84.3%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.2%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.2%

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

   6. Applied rewrites 87.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

   8. Applied rewrites 87.2%

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

   9. Taylor expanded in v around inf

      \[\leadsto -\left(1.5 - \left(\frac{2}{r \cdot r} - \color{blue}{\frac{1}{4}} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 82.9%

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

Alternative 14: 57.9% accurate, 1.7× speedup

Math

\[-\left(1.5 - \frac{2}{{r}^{2}}\right) \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

2. Taylor expanded in v around 0

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 84.3%

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

4. Applied rewrites 84.3%

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

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 87.2%

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 87.2%

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

6. Applied rewrites 87.2%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. associate--r+ N/A

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

   8. lower--.f64 N/A

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

8. Applied rewrites 87.2%

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

9. Taylor expanded in w around 0

   \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

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

    2. lower-pow.64 57.9%

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

11. Applied rewrites 57.9%

    \[\leadsto -\left(1.5 - \color{blue}{\frac{2}{{r}^{2}}}\right) \]
12. Add Preprocessing

Alternative 15: 57.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 77.5%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lower--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. add-flip N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

6. Applied rewrites 82.6%

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

7. Taylor expanded in w around 0

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

9. Applied rewrites 57.9%

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

Alternative 16: 44.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

2. 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.64 44.7%

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

4. Applied rewrites 44.7%

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

   1. lift-pow.64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 44.7%

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

6. Applied rewrites 44.7%

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

Reproduce

herbie shell --seed 2025183 
(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))