Rosa's TurbineBenchmark

Percentage Accurate: 84.1% → 99.8%

Time: 7.3s

Alternatives: 14

Speedup: 1.3×

• 47.3% 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: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(v + v\right) - 3\\ t_1 := 3 + \frac{2}{\left|r\right| \cdot \left|r\right|}\\ t_2 := w \cdot \left|r\right|\\ \mathbf{if}\;\left|r\right| \leq 10^{+144}:\\ \;\;\;\;\left(t\_1 - \left(\left(\frac{-0.125}{1 - v} \cdot t\_0\right) \cdot w\right) \cdot \left(t\_2 \cdot \left|r\right|\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \left(\left(t\_2 \cdot w\right) \cdot \left|r\right|\right) \cdot \frac{t\_0 \cdot 0.125}{v - 1}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (- (+ v v) 3.0))
       (t_1 (+ 3.0 (/ 2.0 (* (fabs r) (fabs r)))))
       (t_2 (* w (fabs r))))
  (if (<= (fabs r) 1e+144)
    (-
     (- t_1 (* (* (* (/ -0.125 (- 1.0 v)) t_0) w) (* t_2 (fabs r))))
     4.5)
    (-
     (- t_1 (* (* (* t_2 w) (fabs r)) (/ (* t_0 0.125) (- v 1.0))))
     4.5))))

C

double code(double v, double w, double r) {
	double t_0 = (v + v) - 3.0;
	double t_1 = 3.0 + (2.0 / (fabs(r) * fabs(r)));
	double t_2 = w * fabs(r);
	double tmp;
	if (fabs(r) <= 1e+144) {
		tmp = (t_1 - ((((-0.125 / (1.0 - v)) * t_0) * w) * (t_2 * fabs(r)))) - 4.5;
	} else {
		tmp = (t_1 - (((t_2 * w) * fabs(r)) * ((t_0 * 0.125) / (v - 1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (v + v) - 3.0d0
    t_1 = 3.0d0 + (2.0d0 / (abs(r) * abs(r)))
    t_2 = w * abs(r)
    if (abs(r) <= 1d+144) then
        tmp = (t_1 - (((((-0.125d0) / (1.0d0 - v)) * t_0) * w) * (t_2 * abs(r)))) - 4.5d0
    else
        tmp = (t_1 - (((t_2 * w) * abs(r)) * ((t_0 * 0.125d0) / (v - 1.0d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (v + v) - 3.0;
	double t_1 = 3.0 + (2.0 / (Math.abs(r) * Math.abs(r)));
	double t_2 = w * Math.abs(r);
	double tmp;
	if (Math.abs(r) <= 1e+144) {
		tmp = (t_1 - ((((-0.125 / (1.0 - v)) * t_0) * w) * (t_2 * Math.abs(r)))) - 4.5;
	} else {
		tmp = (t_1 - (((t_2 * w) * Math.abs(r)) * ((t_0 * 0.125) / (v - 1.0)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (v + v) - 3.0
	t_1 = 3.0 + (2.0 / (math.fabs(r) * math.fabs(r)))
	t_2 = w * math.fabs(r)
	tmp = 0
	if math.fabs(r) <= 1e+144:
		tmp = (t_1 - ((((-0.125 / (1.0 - v)) * t_0) * w) * (t_2 * math.fabs(r)))) - 4.5
	else:
		tmp = (t_1 - (((t_2 * w) * math.fabs(r)) * ((t_0 * 0.125) / (v - 1.0)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(v + v) - 3.0)
	t_1 = Float64(3.0 + Float64(2.0 / Float64(abs(r) * abs(r))))
	t_2 = Float64(w * abs(r))
	tmp = 0.0
	if (abs(r) <= 1e+144)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(Float64(-0.125 / Float64(1.0 - v)) * t_0) * w) * Float64(t_2 * abs(r)))) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(t_2 * w) * abs(r)) * Float64(Float64(t_0 * 0.125) / Float64(v - 1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (v + v) - 3.0;
	t_1 = 3.0 + (2.0 / (abs(r) * abs(r)));
	t_2 = w * abs(r);
	tmp = 0.0;
	if (abs(r) <= 1e+144)
		tmp = (t_1 - ((((-0.125 / (1.0 - v)) * t_0) * w) * (t_2 * abs(r)))) - 4.5;
	else
		tmp = (t_1 - (((t_2 * w) * abs(r)) * ((t_0 * 0.125) / (v - 1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(v + v), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(w * N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 1e+144], N[(N[(t$95$1 - N[(N[(N[(N[(-0.125 / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * w), $MachinePrecision] * N[(t$95$2 * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(t$95$2 * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * 0.125), $MachinePrecision] / N[(v - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if r < 1e144

   1. Initial program 84.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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 94.5%

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

   3. Applied rewrites 94.5%

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

   5. Applied rewrites 97.3%

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

   if 1e144 < r

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.4%

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

   5. Applied rewrites 96.4%

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

Alternative 2: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := w \cdot \left|r\right|\\ t_1 := 3 + \frac{2}{\left|r\right| \cdot \left|r\right|}\\ \mathbf{if}\;\left|r\right| \leq 1.45 \cdot 10^{-66}:\\ \;\;\;\;\left(t\_1 - \frac{\left(0.375 \cdot t\_0\right) \cdot t\_0}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \left(\left(t\_0 \cdot w\right) \cdot \left|r\right|\right) \cdot \frac{\left(\left(v + v\right) - 3\right) \cdot 0.125}{v - 1}\right) - 4.5\\ \end{array} \]

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) 1.45e-66)
    (- (- t_1 (/ (* (* 0.375 t_0) t_0) 1.0)) 4.5)
    (-
     (-
      t_1
      (*
       (* (* t_0 w) (fabs r))
       (/ (* (- (+ v v) 3.0) 0.125) (- v 1.0))))
     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) <= 1.45e-66) {
		tmp = (t_1 - (((0.375 * t_0) * t_0) / 1.0)) - 4.5;
	} else {
		tmp = (t_1 - (((t_0 * w) * fabs(r)) * ((((v + v) - 3.0) * 0.125) / (v - 1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = w * abs(r)
    t_1 = 3.0d0 + (2.0d0 / (abs(r) * abs(r)))
    if (abs(r) <= 1.45d-66) then
        tmp = (t_1 - (((0.375d0 * t_0) * t_0) / 1.0d0)) - 4.5d0
    else
        tmp = (t_1 - (((t_0 * w) * abs(r)) * ((((v + v) - 3.0d0) * 0.125d0) / (v - 1.0d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * Math.abs(r);
	double t_1 = 3.0 + (2.0 / (Math.abs(r) * Math.abs(r)));
	double tmp;
	if (Math.abs(r) <= 1.45e-66) {
		tmp = (t_1 - (((0.375 * t_0) * t_0) / 1.0)) - 4.5;
	} else {
		tmp = (t_1 - (((t_0 * w) * Math.abs(r)) * ((((v + v) - 3.0) * 0.125) / (v - 1.0)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * math.fabs(r)
	t_1 = 3.0 + (2.0 / (math.fabs(r) * math.fabs(r)))
	tmp = 0
	if math.fabs(r) <= 1.45e-66:
		tmp = (t_1 - (((0.375 * t_0) * t_0) / 1.0)) - 4.5
	else:
		tmp = (t_1 - (((t_0 * w) * math.fabs(r)) * ((((v + v) - 3.0) * 0.125) / (v - 1.0)))) - 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) <= 1.45e-66)
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(0.375 * t_0) * t_0) / 1.0)) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(t_0 * w) * abs(r)) * Float64(Float64(Float64(Float64(v + v) - 3.0) * 0.125) / Float64(v - 1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * abs(r);
	t_1 = 3.0 + (2.0 / (abs(r) * abs(r)));
	tmp = 0.0;
	if (abs(r) <= 1.45e-66)
		tmp = (t_1 - (((0.375 * t_0) * t_0) / 1.0)) - 4.5;
	else
		tmp = (t_1 - (((t_0 * w) * abs(r)) * ((((v + v) - 3.0) * 0.125) / (v - 1.0)))) - 4.5;
	end
	tmp_2 = 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], 1.45e-66], N[(N[(t$95$1 - N[(N[(N[(0.375 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(t$95$0 * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(v + v), $MachinePrecision] - 3.0), $MachinePrecision] * 0.125), $MachinePrecision] / N[(v - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.4500000000000001e-66

   1. Initial program 84.1%

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

   2. 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 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.0%

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

   6. Applied rewrites 85.0%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 93.0%

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

   if 1.4500000000000001e-66 < r

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.4%

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

   5. Applied rewrites 96.4%

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

Alternative 3: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(\left(3 + t\_0\right) - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5\\ \mathbf{if}\;v \leq -19000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;-\left(1.5 - \left(t\_0 - \frac{\left(\left(0.375 \cdot r\right) \cdot w\right) \cdot \left(w \cdot r\right)}{1 - v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (/ 2.0 (* r r)))
       (t_1 (- (- (+ 3.0 t_0) (* (* (* (* w r) w) r) 0.25)) 4.5)))
  (if (<= v -19000000000000.0)
    t_1
    (if (<= v 1.5e-12)
      (- (- 1.5 (- t_0 (/ (* (* (* 0.375 r) w) (* w r)) (- 1.0 v)))))
      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 * r) * w) * r) * 0.25)) - 4.5;
	double tmp;
	if (v <= -19000000000000.0) {
		tmp = t_1;
	} else if (v <= 1.5e-12) {
		tmp = -(1.5 - (t_0 - ((((0.375 * r) * w) * (w * r)) / (1.0 - v))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = ((3.0d0 + t_0) - ((((w * r) * w) * r) * 0.25d0)) - 4.5d0
    if (v <= (-19000000000000.0d0)) then
        tmp = t_1
    else if (v <= 1.5d-12) then
        tmp = -(1.5d0 - (t_0 - ((((0.375d0 * r) * w) * (w * r)) / (1.0d0 - v))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = ((3.0 + t_0) - ((((w * r) * w) * r) * 0.25)) - 4.5;
	double tmp;
	if (v <= -19000000000000.0) {
		tmp = t_1;
	} else if (v <= 1.5e-12) {
		tmp = -(1.5 - (t_0 - ((((0.375 * r) * w) * (w * r)) / (1.0 - v))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = ((3.0 + t_0) - ((((w * r) * w) * r) * 0.25)) - 4.5
	tmp = 0
	if v <= -19000000000000.0:
		tmp = t_1
	elif v <= 1.5e-12:
		tmp = -(1.5 - (t_0 - ((((0.375 * r) * w) * (w * r)) / (1.0 - v))))
	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(Float64(Float64(w * r) * w) * r) * 0.25)) - 4.5)
	tmp = 0.0
	if (v <= -19000000000000.0)
		tmp = t_1;
	elseif (v <= 1.5e-12)
		tmp = Float64(-Float64(1.5 - Float64(t_0 - Float64(Float64(Float64(Float64(0.375 * r) * w) * Float64(w * r)) / Float64(1.0 - v)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = ((3.0 + t_0) - ((((w * r) * w) * r) * 0.25)) - 4.5;
	tmp = 0.0;
	if (v <= -19000000000000.0)
		tmp = t_1;
	elseif (v <= 1.5e-12)
		tmp = -(1.5 - (t_0 - ((((0.375 * r) * w) * (w * r)) / (1.0 - v))));
	else
		tmp = t_1;
	end
	tmp_2 = 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[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -19000000000000.0], t$95$1, If[LessEqual[v, 1.5e-12], (-N[(1.5 - N[(t$95$0 - N[(N[(N[(N[(0.375 * r), $MachinePrecision] * w), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.9e13 or 1.5000000000000001e-12 < v

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6%

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

   8. Applied rewrites 90.6%

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

   if -1.9e13 < v < 1.5000000000000001e-12

   1. Initial program 84.1%

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

   2. 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 76.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 76.5%

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

   6. Applied rewrites 76.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 85.0%

      \[\leadsto \color{blue}{-\left(1.5 - \left(\frac{2}{r \cdot r} - \frac{\left(\left(0.375 \cdot r\right) \cdot w\right) \cdot \left(w \cdot r\right)}{1 - v}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5\\ \mathbf{if}\;v \leq -7.5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\left(t\_0 - \frac{\left(0.375 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
       (t_1 (- (- t_0 (* (* (* (* w r) w) r) 0.25)) 4.5)))
  (if (<= v -7.5e+16)
    t_1
    (if (<= v 1.5e-12)
      (- (- t_0 (/ (* (* 0.375 (* w r)) (* w r)) 1.0)) 4.5)
      t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	double tmp;
	if (v <= -7.5e+16) {
		tmp = t_1;
	} else if (v <= 1.5e-12) {
		tmp = (t_0 - (((0.375 * (w * r)) * (w * r)) / 1.0)) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    t_1 = (t_0 - ((((w * r) * w) * r) * 0.25d0)) - 4.5d0
    if (v <= (-7.5d+16)) then
        tmp = t_1
    else if (v <= 1.5d-12) then
        tmp = (t_0 - (((0.375d0 * (w * r)) * (w * r)) / 1.0d0)) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	double tmp;
	if (v <= -7.5e+16) {
		tmp = t_1;
	} else if (v <= 1.5e-12) {
		tmp = (t_0 - (((0.375 * (w * r)) * (w * r)) / 1.0)) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5
	tmp = 0
	if v <= -7.5e+16:
		tmp = t_1
	elif v <= 1.5e-12:
		tmp = (t_0 - (((0.375 * (w * r)) * (w * r)) / 1.0)) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(w * r) * w) * r) * 0.25)) - 4.5)
	tmp = 0.0
	if (v <= -7.5e+16)
		tmp = t_1;
	elseif (v <= 1.5e-12)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(0.375 * Float64(w * r)) * Float64(w * r)) / 1.0)) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	tmp = 0.0;
	if (v <= -7.5e+16)
		tmp = t_1;
	elseif (v <= 1.5e-12)
		tmp = (t_0 - (((0.375 * (w * r)) * (w * r)) / 1.0)) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -7.5e+16], t$95$1, If[LessEqual[v, 1.5e-12], N[(N[(t$95$0 - N[(N[(N[(0.375 * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -7.5e16 or 1.5000000000000001e-12 < v

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6%

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

   8. Applied rewrites 90.6%

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

   if -7.5e16 < v < 1.5000000000000001e-12

   1. Initial program 84.1%

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

   2. 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 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.0%

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

   6. Applied rewrites 85.0%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 93.0%

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

Alternative 5: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \frac{\left(\left(0.375 + -0.25 \cdot v\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
  (if (<=
       (-
        (-
         t_0
         (/
          (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r))
          (- 1.0 v)))
        4.5)
       (- INFINITY))
    (- (- t_0 (* (* (* (* w r) w) r) 0.25)) 4.5)
    (-
     (-
      t_0
      (/ (* (* (+ 0.375 (* -0.25 v)) (* w r)) (* w r)) (- 1.0 v)))
     4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -((double) INFINITY)) {
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else {
		tmp = (t_0 - ((((0.375 + (-0.25 * v)) * (w * r)) * (w * r)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else {
		tmp = (t_0 - ((((0.375 + (-0.25 * v)) * (w * r)) * (w * r)) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if ((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -math.inf:
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5
	else:
		tmp = (t_0 - ((((0.375 + (-0.25 * v)) * (w * r)) * (w * r)) / (1.0 - v))) - 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 (Float64(Float64(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) <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(w * r) * w) * r) * 0.25)) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(0.375 + Float64(-0.25 * v)) * Float64(w * r)) * Float64(w * r)) / Float64(1.0 - v))) - 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 (((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -Inf)
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	else
		tmp = (t_0 - ((((0.375 + (-0.25 * v)) * (w * r)) * (w * r)) / (1.0 - v))) - 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[N[(N[(t$95$0 - 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], (-Infinity)], N[(N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(N[(N[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6%

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

   8. Applied rewrites 90.6%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

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

   2. 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 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.0%

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

   6. Applied rewrites 85.0%

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

   7. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5 \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 94.5%

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

   9. Applied rewrites 94.5%

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

Alternative 6: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \leq -\infty:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \frac{\left(0.375 + -0.25 \cdot v\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
  (if (<=
       (-
        (-
         t_0
         (/
          (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r))
          (- 1.0 v)))
        4.5)
       (- INFINITY))
    (- (- t_0 (* (* (* (* w r) w) r) 0.25)) 4.5)
    (-
     (-
      t_0
      (/ (* (+ 0.375 (* -0.25 v)) (* (* w r) (* w r))) (- 1.0 v)))
     4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -((double) INFINITY)) {
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else {
		tmp = (t_0 - (((0.375 + (-0.25 * v)) * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else {
		tmp = (t_0 - (((0.375 + (-0.25 * v)) * ((w * r) * (w * r))) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if ((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -math.inf:
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5
	else:
		tmp = (t_0 - (((0.375 + (-0.25 * v)) * ((w * r) * (w * r))) / (1.0 - v))) - 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 (Float64(Float64(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) <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(w * r) * w) * r) * 0.25)) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(0.375 + Float64(-0.25 * v)) * Float64(Float64(w * r) * Float64(w * r))) / Float64(1.0 - v))) - 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 (((t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5) <= -Inf)
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	else
		tmp = (t_0 - (((0.375 + (-0.25 * v)) * ((w * r) * (w * r))) / (1.0 - v))) - 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[N[(N[(t$95$0 - 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], (-Infinity)], N[(N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(N[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6%

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

   8. Applied rewrites 90.6%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64))

   1. Initial program 84.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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 94.5%

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

   3. Applied rewrites 94.5%

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

   4. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 94.5%

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

   6. Applied rewrites 94.5%

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

Alternative 7: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 - \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot r\right) \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;\left(3 - \frac{\left(\left(0.375 + -0.25 \cdot v\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot 0.25\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
       (t_1
        (-
         (-
          t_0
          (/
           (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r))
           (- 1.0 v)))
         4.5)))
  (if (<= t_1 (- INFINITY))
    (- (- t_0 (* (* (* (* w r) w) r) 0.25)) 4.5)
    (if (<= t_1 -1.0)
      (-
       (-
        3.0
        (/ (* (* (+ 0.375 (* -0.25 v)) (* 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 t_1 = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else if (t_1 <= -1.0) {
		tmp = (3.0 - ((((0.375 + (-0.25 * v)) * (w * r)) * (w * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - ((w * (w * (r * r))) * 0.25)) - 4.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	} else if (t_1 <= -1.0) {
		tmp = (3.0 - ((((0.375 + (-0.25 * v)) * (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))
	t_1 = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5
	elif t_1 <= -1.0:
		tmp = (3.0 - ((((0.375 + (-0.25 * v)) * (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)))
	t_1 = Float64(Float64(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)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(w * r) * w) * r) * 0.25)) - 4.5);
	elseif (t_1 <= -1.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(0.375 + Float64(-0.25 * v)) * Float64(w * r)) * Float64(w * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(w * Float64(w * Float64(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));
	t_1 = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (t_0 - ((((w * r) * w) * r) * 0.25)) - 4.5;
	elseif (t_1 <= -1.0)
		tmp = (3.0 - ((((0.375 + (-0.25 * v)) * (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]}, Block[{t$95$1 = N[(N[(t$95$0 - 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$0 - N[(N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -1.0], N[(N[(3.0 - N[(N[(N[(N[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - N[(N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6%

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

   8. Applied rewrites 90.6%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1

   1. Initial program 84.1%

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

   2. 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 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.0%

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

   6. Applied rewrites 85.0%

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

   7. Taylor expanded in v around 0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)} \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v}\right) - 4.5 \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 94.5%

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

   9. Applied rewrites 94.5%

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

   10. Taylor expanded in r around inf

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

   12. Applied rewrites 51.6%

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

   if -1 < (-.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 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 87.5%

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

   8. Applied rewrites 87.5%

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

Alternative 8: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(w \cdot r\right) \cdot w\right) \cdot r\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t\_1 - t\_0 \cdot 0.25\right) - 4.5\\ \mathbf{if}\;v \leq -1.2 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;v \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\left(t\_1 - t\_0 \cdot 0.375\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* (* (* w r) w) r))
       (t_1 (+ 3.0 (/ 2.0 (* r r))))
       (t_2 (- (- t_1 (* t_0 0.25)) 4.5)))
  (if (<= v -1.2e+16)
    t_2
    (if (<= v 1.5e-12) (- (- t_1 (* t_0 0.375)) 4.5) t_2))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * r) * w) * r;
	double t_1 = 3.0 + (2.0 / (r * r));
	double t_2 = (t_1 - (t_0 * 0.25)) - 4.5;
	double tmp;
	if (v <= -1.2e+16) {
		tmp = t_2;
	} else if (v <= 1.5e-12) {
		tmp = (t_1 - (t_0 * 0.375)) - 4.5;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = ((w * r) * w) * r
    t_1 = 3.0d0 + (2.0d0 / (r * r))
    t_2 = (t_1 - (t_0 * 0.25d0)) - 4.5d0
    if (v <= (-1.2d+16)) then
        tmp = t_2
    else if (v <= 1.5d-12) then
        tmp = (t_1 - (t_0 * 0.375d0)) - 4.5d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = ((w * r) * w) * r;
	double t_1 = 3.0 + (2.0 / (r * r));
	double t_2 = (t_1 - (t_0 * 0.25)) - 4.5;
	double tmp;
	if (v <= -1.2e+16) {
		tmp = t_2;
	} else if (v <= 1.5e-12) {
		tmp = (t_1 - (t_0 * 0.375)) - 4.5;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((w * r) * w) * r
	t_1 = 3.0 + (2.0 / (r * r))
	t_2 = (t_1 - (t_0 * 0.25)) - 4.5
	tmp = 0
	if v <= -1.2e+16:
		tmp = t_2
	elif v <= 1.5e-12:
		tmp = (t_1 - (t_0 * 0.375)) - 4.5
	else:
		tmp = t_2
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * r) * w) * r)
	t_1 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_2 = Float64(Float64(t_1 - Float64(t_0 * 0.25)) - 4.5)
	tmp = 0.0
	if (v <= -1.2e+16)
		tmp = t_2;
	elseif (v <= 1.5e-12)
		tmp = Float64(Float64(t_1 - Float64(t_0 * 0.375)) - 4.5);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((w * r) * w) * r;
	t_1 = 3.0 + (2.0 / (r * r));
	t_2 = (t_1 - (t_0 * 0.25)) - 4.5;
	tmp = 0.0;
	if (v <= -1.2e+16)
		tmp = t_2;
	elseif (v <= 1.5e-12)
		tmp = (t_1 - (t_0 * 0.375)) - 4.5;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * r), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -1.2e+16], t$95$2, If[LessEqual[v, 1.5e-12], N[(N[(t$95$1 - N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.2e16 or 1.5000000000000001e-12 < v

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6%

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

   8. Applied rewrites 90.6%

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

   if -1.2e16 < v < 1.5000000000000001e-12

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.4%

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 90.2%

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

Alternative 9: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|r\right| \cdot \left|r\right|\\ t_1 := 3 + \frac{2}{t\_0}\\ \mathbf{if}\;\left|r\right| \leq 10^{+88}:\\ \;\;\;\;\left(t\_1 - \left(w \cdot \left(w \cdot t\_0\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - \left(\left(\left(w \cdot \left|r\right|\right) \cdot w\right) \cdot \left|r\right|\right) \cdot 0.25\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* (fabs r) (fabs r))) (t_1 (+ 3.0 (/ 2.0 t_0))))
  (if (<= (fabs r) 1e+88)
    (- (- t_1 (* (* w (* w t_0)) 0.25)) 4.5)
    (- (- t_1 (* (* (* (* w (fabs r)) w) (fabs r)) 0.25)) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = fabs(r) * fabs(r);
	double t_1 = 3.0 + (2.0 / t_0);
	double tmp;
	if (fabs(r) <= 1e+88) {
		tmp = (t_1 - ((w * (w * t_0)) * 0.25)) - 4.5;
	} else {
		tmp = (t_1 - ((((w * fabs(r)) * w) * fabs(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) :: t_1
    real(8) :: tmp
    t_0 = abs(r) * abs(r)
    t_1 = 3.0d0 + (2.0d0 / t_0)
    if (abs(r) <= 1d+88) then
        tmp = (t_1 - ((w * (w * t_0)) * 0.25d0)) - 4.5d0
    else
        tmp = (t_1 - ((((w * abs(r)) * w) * abs(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 = Math.abs(r) * Math.abs(r);
	double t_1 = 3.0 + (2.0 / t_0);
	double tmp;
	if (Math.abs(r) <= 1e+88) {
		tmp = (t_1 - ((w * (w * t_0)) * 0.25)) - 4.5;
	} else {
		tmp = (t_1 - ((((w * Math.abs(r)) * w) * Math.abs(r)) * 0.25)) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = math.fabs(r) * math.fabs(r)
	t_1 = 3.0 + (2.0 / t_0)
	tmp = 0
	if math.fabs(r) <= 1e+88:
		tmp = (t_1 - ((w * (w * t_0)) * 0.25)) - 4.5
	else:
		tmp = (t_1 - ((((w * math.fabs(r)) * w) * math.fabs(r)) * 0.25)) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(abs(r) * abs(r))
	t_1 = Float64(3.0 + Float64(2.0 / t_0))
	tmp = 0.0
	if (abs(r) <= 1e+88)
		tmp = Float64(Float64(t_1 - Float64(Float64(w * Float64(w * t_0)) * 0.25)) - 4.5);
	else
		tmp = Float64(Float64(t_1 - Float64(Float64(Float64(Float64(w * abs(r)) * w) * abs(r)) * 0.25)) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = abs(r) * abs(r);
	t_1 = 3.0 + (2.0 / t_0);
	tmp = 0.0;
	if (abs(r) <= 1e+88)
		tmp = (t_1 - ((w * (w * t_0)) * 0.25)) - 4.5;
	else
		tmp = (t_1 - ((((w * abs(r)) * w) * abs(r)) * 0.25)) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 1e+88], N[(N[(t$95$1 - N[(N[(w * N[(w * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(N[(w * N[Abs[r], $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 9.9999999999999996e87

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 87.5%

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

   8. Applied rewrites 87.5%

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

   if 9.9999999999999996e87 < r

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.6%

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

   8. Applied rewrites 90.6%

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

Alternative 10: 91.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|r\right| \cdot \left|r\right|\\ t_1 := \frac{2}{t\_0}\\ \mathbf{if}\;\left|r\right| \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(3 + t\_1\right) - \left(w \cdot \left(w \cdot t\_0\right)\right) \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-\left(1.5 - \left(t\_1 - \left(0.25 \cdot \left|r\right|\right) \cdot \left(\left(w \cdot w\right) \cdot \left|r\right|\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* (fabs r) (fabs r))) (t_1 (/ 2.0 t_0)))
  (if (<= (fabs r) 5e+69)
    (- (- (+ 3.0 t_1) (* (* w (* w t_0)) 0.25)) 4.5)
    (- (- 1.5 (- t_1 (* (* 0.25 (fabs r)) (* (* w w) (fabs r)))))))))

C

double code(double v, double w, double r) {
	double t_0 = fabs(r) * fabs(r);
	double t_1 = 2.0 / t_0;
	double tmp;
	if (fabs(r) <= 5e+69) {
		tmp = ((3.0 + t_1) - ((w * (w * t_0)) * 0.25)) - 4.5;
	} else {
		tmp = -(1.5 - (t_1 - ((0.25 * fabs(r)) * ((w * w) * 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) :: t_1
    real(8) :: tmp
    t_0 = abs(r) * abs(r)
    t_1 = 2.0d0 / t_0
    if (abs(r) <= 5d+69) then
        tmp = ((3.0d0 + t_1) - ((w * (w * t_0)) * 0.25d0)) - 4.5d0
    else
        tmp = -(1.5d0 - (t_1 - ((0.25d0 * abs(r)) * ((w * w) * abs(r)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = Math.abs(r) * Math.abs(r);
	double t_1 = 2.0 / t_0;
	double tmp;
	if (Math.abs(r) <= 5e+69) {
		tmp = ((3.0 + t_1) - ((w * (w * t_0)) * 0.25)) - 4.5;
	} else {
		tmp = -(1.5 - (t_1 - ((0.25 * Math.abs(r)) * ((w * w) * Math.abs(r)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = math.fabs(r) * math.fabs(r)
	t_1 = 2.0 / t_0
	tmp = 0
	if math.fabs(r) <= 5e+69:
		tmp = ((3.0 + t_1) - ((w * (w * t_0)) * 0.25)) - 4.5
	else:
		tmp = -(1.5 - (t_1 - ((0.25 * math.fabs(r)) * ((w * w) * math.fabs(r)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(abs(r) * abs(r))
	t_1 = Float64(2.0 / t_0)
	tmp = 0.0
	if (abs(r) <= 5e+69)
		tmp = Float64(Float64(Float64(3.0 + t_1) - Float64(Float64(w * Float64(w * t_0)) * 0.25)) - 4.5);
	else
		tmp = Float64(-Float64(1.5 - Float64(t_1 - Float64(Float64(0.25 * abs(r)) * Float64(Float64(w * w) * abs(r))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = abs(r) * abs(r);
	t_1 = 2.0 / t_0;
	tmp = 0.0;
	if (abs(r) <= 5e+69)
		tmp = ((3.0 + t_1) - ((w * (w * t_0)) * 0.25)) - 4.5;
	else
		tmp = -(1.5 - (t_1 - ((0.25 * abs(r)) * ((w * w) * abs(r)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[r], $MachinePrecision], 5e+69], N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(w * N[(w * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], (-N[(1.5 - N[(t$95$1 - N[(N[(0.25 * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if r < 5.0000000000000004e69

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 87.5%

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

   8. Applied rewrites 87.5%

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

   if 5.0000000000000004e69 < r

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 82.8%

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

Alternative 11: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(w \cdot w\right) \cdot r\\ t_2 := t\_1 \cdot r\\ t_3 := \left(\left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot t\_2}{1 - v}\right) - 4.5\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(3 - t\_2 \cdot 0.25\right) - 4.5\\ \mathbf{elif}\;t\_3 \leq -1.5000000000126734:\\ \;\;\;\;\left(3 - \frac{t\_1 \cdot \left(r \cdot 0.375\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - -3\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (/ 2.0 (* r r)))
       (t_1 (* (* w w) r))
       (t_2 (* t_1 r))
       (t_3
        (-
         (-
          (+ 3.0 t_0)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) t_2) (- 1.0 v)))
         4.5)))
  (if (<= t_3 (- INFINITY))
    (- (- 3.0 (* t_2 0.25)) 4.5)
    (if (<= t_3 -1.5000000000126734)
      (- (- 3.0 (/ (* t_1 (* r 0.375)) (- 1.0 v))) 4.5)
      (- (- t_0 -3.0) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double t_2 = t_1 * r;
	double t_3 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * t_2) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (3.0 - (t_2 * 0.25)) - 4.5;
	} else if (t_3 <= -1.5000000000126734) {
		tmp = (3.0 - ((t_1 * (r * 0.375)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - -3.0) - 4.5;
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (w * w) * r;
	double t_2 = t_1 * r;
	double t_3 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * t_2) / (1.0 - v))) - 4.5;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (3.0 - (t_2 * 0.25)) - 4.5;
	} else if (t_3 <= -1.5000000000126734) {
		tmp = (3.0 - ((t_1 * (r * 0.375)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (t_0 - -3.0) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (w * w) * r
	t_2 = t_1 * r
	t_3 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * t_2) / (1.0 - v))) - 4.5
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (3.0 - (t_2 * 0.25)) - 4.5
	elif t_3 <= -1.5000000000126734:
		tmp = (3.0 - ((t_1 * (r * 0.375)) / (1.0 - v))) - 4.5
	else:
		tmp = (t_0 - -3.0) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(w * w) * r)
	t_2 = Float64(t_1 * r)
	t_3 = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * t_2) / Float64(1.0 - v))) - 4.5)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(3.0 - Float64(t_2 * 0.25)) - 4.5);
	elseif (t_3 <= -1.5000000000126734)
		tmp = Float64(Float64(3.0 - Float64(Float64(t_1 * Float64(r * 0.375)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(t_0 - -3.0) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (w * w) * r;
	t_2 = t_1 * r;
	t_3 = ((3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * t_2) / (1.0 - v))) - 4.5;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (3.0 - (t_2 * 0.25)) - 4.5;
	elseif (t_3 <= -1.5000000000126734)
		tmp = (3.0 - ((t_1 * (r * 0.375)) / (1.0 - v))) - 4.5;
	else
		tmp = (t_0 - -3.0) - 4.5;
	end
	tmp_2 = 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[(w * w), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * r), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(3.0 - N[(t$95$2 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$3, -1.5000000000126734], N[(N[(3.0 - N[(N[(t$95$1 * N[(r * 0.375), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 - -3.0), $MachinePrecision] - 4.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -inf.0

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 47.0%

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

   if -inf.0 < (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5000000000126734

   1. Initial program 84.1%

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

   2. 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 76.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 76.5%

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

   6. Applied rewrites 76.5%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 40.8%

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

   if -1.5000000000126734 < (-.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 84.1%

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

   3. Applied rewrites 69.4%

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

   4. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r}} - 4.5 \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{\color{blue}{r}} - \frac{9}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      5. lower-/.f64 57.6%

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - 4.5 \]

   6. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r}} - 4.5 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{\color{blue}{r}} - \frac{9}{2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      4. add-to-fraction-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{9}{2} \]

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 57.7%

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

   8. Applied rewrites 57.7%

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

Alternative 12: 90.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|r\right| \leq 6.8 \cdot 10^{-106}:\\ \;\;\;\;\left(\frac{\frac{2}{\left|r\right|}}{\left|r\right|} - -3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-\left(1.5 - \left(\frac{2}{\left|r\right| \cdot \left|r\right|} - \left(0.25 \cdot \left|r\right|\right) \cdot \left(\left(w \cdot w\right) \cdot \left|r\right|\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (if (<= (fabs r) 6.8e-106)
  (- (- (/ (/ 2.0 (fabs r)) (fabs r)) -3.0) 4.5)
  (-
   (-
    1.5
    (-
     (/ 2.0 (* (fabs r) (fabs r)))
     (* (* 0.25 (fabs r)) (* (* w w) (fabs r))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (fabs(r) <= 6.8e-106) {
		tmp = (((2.0 / fabs(r)) / fabs(r)) - -3.0) - 4.5;
	} else {
		tmp = -(1.5 - ((2.0 / (fabs(r) * fabs(r))) - ((0.25 * fabs(r)) * ((w * w) * 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) :: tmp
    if (abs(r) <= 6.8d-106) then
        tmp = (((2.0d0 / abs(r)) / abs(r)) - (-3.0d0)) - 4.5d0
    else
        tmp = -(1.5d0 - ((2.0d0 / (abs(r) * abs(r))) - ((0.25d0 * abs(r)) * ((w * w) * abs(r)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (Math.abs(r) <= 6.8e-106) {
		tmp = (((2.0 / Math.abs(r)) / Math.abs(r)) - -3.0) - 4.5;
	} else {
		tmp = -(1.5 - ((2.0 / (Math.abs(r) * Math.abs(r))) - ((0.25 * Math.abs(r)) * ((w * w) * Math.abs(r)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if math.fabs(r) <= 6.8e-106:
		tmp = (((2.0 / math.fabs(r)) / math.fabs(r)) - -3.0) - 4.5
	else:
		tmp = -(1.5 - ((2.0 / (math.fabs(r) * math.fabs(r))) - ((0.25 * math.fabs(r)) * ((w * w) * math.fabs(r)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (abs(r) <= 6.8e-106)
		tmp = Float64(Float64(Float64(Float64(2.0 / abs(r)) / abs(r)) - -3.0) - 4.5);
	else
		tmp = Float64(-Float64(1.5 - Float64(Float64(2.0 / Float64(abs(r) * abs(r))) - Float64(Float64(0.25 * abs(r)) * Float64(Float64(w * w) * abs(r))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (abs(r) <= 6.8e-106)
		tmp = (((2.0 / abs(r)) / abs(r)) - -3.0) - 4.5;
	else
		tmp = -(1.5 - ((2.0 / (abs(r) * abs(r))) - ((0.25 * abs(r)) * ((w * w) * abs(r)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[N[Abs[r], $MachinePrecision], 6.8e-106], N[(N[(N[(N[(2.0 / N[Abs[r], $MachinePrecision]), $MachinePrecision] / N[Abs[r], $MachinePrecision]), $MachinePrecision] - -3.0), $MachinePrecision] - 4.5), $MachinePrecision], (-N[(1.5 - N[(N[(2.0 / N[(N[Abs[r], $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.25 * N[Abs[r], $MachinePrecision]), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * N[Abs[r], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if r < 6.7999999999999996e-106

   1. Initial program 84.1%

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

   3. Applied rewrites 69.4%

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

   4. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r}} - 4.5 \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{\color{blue}{r}} - \frac{9}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      5. lower-/.f64 57.6%

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - 4.5 \]

   6. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r}} - 4.5 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{\color{blue}{r}} - \frac{9}{2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      4. add-to-fraction-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{9}{2} \]

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 57.7%

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

   8. Applied rewrites 57.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \left(\frac{2}{r \cdot r} - -3\right) - \frac{9}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{2}{r \cdot r} - -3\right) - \frac{9}{2} \]

      3. associate-/r* N/A

         \[\leadsto \left(\frac{\frac{2}{r}}{r} - -3\right) - \frac{9}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{2}{r}}{r} - -3\right) - \frac{9}{2} \]

      5. lower-/.f64 57.7%

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

   10. Applied rewrites 57.7%

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

   if 6.7999999999999996e-106 < r

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate--l+ N/A

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

      7. associate--r+ N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

   8. Applied rewrites 82.8%

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

Alternative 13: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(w \cdot w\right) \cdot r\right) \cdot r\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(\left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot t\_0}{1 - v}\right) - 4.5 \leq -1.5000000000126734:\\ \;\;\;\;\left(3 - t\_0 \cdot 0.25\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 - -3\right) - 4.5\\ \end{array} \]

FPCore

(FPCore (v w r)
  :precision binary64
  (let* ((t_0 (* (* (* w w) r) r)) (t_1 (/ 2.0 (* r r))))
  (if (<=
       (-
        (-
         (+ 3.0 t_1)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) t_0) (- 1.0 v)))
        4.5)
       -1.5000000000126734)
    (- (- 3.0 (* t_0 0.25)) 4.5)
    (- (- t_1 -3.0) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = ((w * w) * r) * r;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5) <= -1.5000000000126734) {
		tmp = (3.0 - (t_0 * 0.25)) - 4.5;
	} else {
		tmp = (t_1 - -3.0) - 4.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((w * w) * r) * r
    t_1 = 2.0d0 / (r * r)
    if ((((3.0d0 + t_1) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * t_0) / (1.0d0 - v))) - 4.5d0) <= (-1.5000000000126734d0)) then
        tmp = (3.0d0 - (t_0 * 0.25d0)) - 4.5d0
    else
        tmp = (t_1 - (-3.0d0)) - 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 = 2.0 / (r * r);
	double tmp;
	if ((((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5) <= -1.5000000000126734) {
		tmp = (3.0 - (t_0 * 0.25)) - 4.5;
	} else {
		tmp = (t_1 - -3.0) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((w * w) * r) * r
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5) <= -1.5000000000126734:
		tmp = (3.0 - (t_0 * 0.25)) - 4.5
	else:
		tmp = (t_1 - -3.0) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(w * w) * r) * r)
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * t_0) / Float64(1.0 - v))) - 4.5) <= -1.5000000000126734)
		tmp = Float64(Float64(3.0 - Float64(t_0 * 0.25)) - 4.5);
	else
		tmp = Float64(Float64(t_1 - -3.0) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((w * w) * r) * r;
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * t_0) / (1.0 - v))) - 4.5) <= -1.5000000000126734)
		tmp = (3.0 - (t_0 * 0.25)) - 4.5;
	else
		tmp = (t_1 - -3.0) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.5000000000126734], N[(N[(3.0 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$1 - -3.0), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) #s(literal 9/2 binary64)) < -1.5000000000126734

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. frac-2neg-rev N/A

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

      18. lower-/.f64 N/A

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

   3. Applied rewrites 87.2%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 82.8%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 47.0%

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

   if -1.5000000000126734 < (-.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 84.1%

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

   3. Applied rewrites 69.4%

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

   4. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r}} - 4.5 \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{\color{blue}{r}} - \frac{9}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      5. lower-/.f64 57.6%

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - 4.5 \]

   6. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r}} - 4.5 \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{\color{blue}{r}} - \frac{9}{2} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

      4. add-to-fraction-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{9}{2} \]

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 57.7%

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

   8. Applied rewrites 57.7%

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

Alternative 14: 57.7% accurate, 3.2× 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 84.1%

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

3. Applied rewrites 69.4%

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

4. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r}} - 4.5 \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{\color{blue}{r}} - \frac{9}{2} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

   5. lower-/.f64 57.6%

      \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - 4.5 \]

6. Applied rewrites 57.6%

   \[\leadsto \color{blue}{\frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r}} - 4.5 \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{\color{blue}{r}} - \frac{9}{2} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{3 \cdot r + 2 \cdot \frac{1}{r}}{r} - \frac{9}{2} \]

   4. add-to-fraction-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip-rev N/A

      \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{9}{2} \]

   8. associate-/r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. +-commutative N/A

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

   12. add-flip N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval 57.7%

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

8. Applied rewrites 57.7%

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

Reproduce

herbie shell --seed 2025258 
(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))