Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.6%

Time: 5.2s

Alternatives: 13

Speedup: 1.6×

• 52.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{if}\;v \leq -2 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\left(t\_0 - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1 (- (- t_0 (* 0.25 (* (* w r) (* w r)))) 4.5)))
   (if (<= v -2e+71)
     t_1
     (if (<= v 2.5e+14)
       (-
        (-
         t_0
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* r w) (* r w))) (- 1.0 v)))
        4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	double tmp;
	if (v <= -2e+71) {
		tmp = t_1;
	} else if (v <= 2.5e+14) {
		tmp = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    t_1 = (t_0 - (0.25d0 * ((w * r) * (w * r)))) - 4.5d0
    if (v <= (-2d+71)) then
        tmp = t_1
    else if (v <= 2.5d+14) then
        tmp = (t_0 - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * ((r * w) * (r * w))) / (1.0d0 - v))) - 4.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	double tmp;
	if (v <= -2e+71) {
		tmp = t_1;
	} else if (v <= 2.5e+14) {
		tmp = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	t_1 = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5
	tmp = 0
	if v <= -2e+71:
		tmp = t_1
	elif v <= 2.5e+14:
		tmp = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(t_0 - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5)
	tmp = 0.0
	if (v <= -2e+71)
		tmp = t_1;
	elseif (v <= 2.5e+14)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	t_1 = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	tmp = 0.0;
	if (v <= -2e+71)
		tmp = t_1;
	elseif (v <= 2.5e+14)
		tmp = (t_0 - (((0.125 * (3.0 - (2.0 * v))) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -2e+71], t$95$1, If[LessEqual[v, 2.5e+14], N[(N[(t$95$0 - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.0000000000000001e71 or 2.5e14 < v

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 88.8

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

   3. Applied rewrites 88.8%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if -2.0000000000000001e71 < v < 2.5e14

   1. Initial program 88.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.5

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

   3. Applied rewrites 99.5%

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

Alternative 2: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(w \cdot r\right)}^{2}\\ \mathbf{if}\;r \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;{r}^{-2} \cdot 2 - \mathsf{fma}\left(t\_0, 0.375, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(\mathsf{fma}\left(v, -2, 3\right) \cdot 0.125, \frac{t\_0}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (pow (* w r) 2.0)))
   (if (<= r 3.6e-12)
     (- (* (pow r -2.0) 2.0) (fma t_0 0.375 1.5))
     (- 3.0 (fma (* (fma v -2.0 3.0) 0.125) (/ t_0 (- 1.0 v)) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = pow((w * r), 2.0);
	double tmp;
	if (r <= 3.6e-12) {
		tmp = (pow(r, -2.0) * 2.0) - fma(t_0, 0.375, 1.5);
	} else {
		tmp = 3.0 - fma((fma(v, -2.0, 3.0) * 0.125), (t_0 / (1.0 - v)), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(w * r) ^ 2.0
	tmp = 0.0
	if (r <= 3.6e-12)
		tmp = Float64(Float64((r ^ -2.0) * 2.0) - fma(t_0, 0.375, 1.5));
	else
		tmp = Float64(3.0 - fma(Float64(fma(v, -2.0, 3.0) * 0.125), Float64(t_0 / Float64(1.0 - v)), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[Power[N[(w * r), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[r, 3.6e-12], N[(N[(N[Power[r, -2.0], $MachinePrecision] * 2.0), $MachinePrecision] - N[(t$95$0 * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(t$95$0 / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.6e-12

   1. Initial program 83.3%

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

   2. Taylor expanded in v around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 94.8

          \[\leadsto {r}^{-2} \cdot 2 - \mathsf{fma}\left({\left(w \cdot r\right)}^{2}, 0.375, 1.5\right) \]

   4. Applied rewrites 94.8%

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

   if 3.6e-12 < r

   1. Initial program 89.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in r around inf

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

      1. metadata-eval 91.5

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

      2. pow2 91.5

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

      3. associate-*r/ 91.5

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

      4. +-commutative 91.5

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

      5. pow-flip 91.5

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

      6. metadata-eval 91.5

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 91.5

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

   6. Applied rewrites 91.5%

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

   8. Applied rewrites 97.7%

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

Alternative 3: 97.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 92.2

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

   3. Applied rewrites 92.2%

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

   4. Taylor expanded in r around inf

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

      1. metadata-eval 89.7

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

      2. pow2 89.7

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

      3. associate-*r/ 89.7

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

      4. +-commutative 89.7

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

      5. pow-flip 89.7

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

      6. metadata-eval 89.7

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 89.7

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

   6. Applied rewrites 89.7%

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

   8. Applied rewrites 96.3%

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

   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 85.2%

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

   2. Taylor expanded in r around 0

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

      1. metadata-eval N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 4: 98.0% accurate, 0.4× speedup

Math

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

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 (* 0.25 (* (* w r) (* w r)))) 4.5)
     (if (<= t_1 -1.0)
       (-
        (-
         3.0
         (/ (* (* (* (* (fma v -2.0 3.0) 0.125) w) r) (* w r)) (- 1.0 v)))
        4.5)
       (/ (/ 2.0 r) r)))))

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 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_1 <= -1.0) {
		tmp = (3.0 - (((((fma(v, -2.0, 3.0) * 0.125) * w) * r) * (w * r)) / (1.0 - v))) - 4.5;
	} else {
		tmp = (2.0 / r) / r;
	}
	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(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5);
	elseif (t_1 <= -1.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(Float64(fma(v, -2.0, 3.0) * 0.125) * w) * r) * Float64(w * r)) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(2.0 / r) / r);
	end
	return 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[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -1.0], N[(N[(3.0 - N[(N[(N[(N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * w), $MachinePrecision] * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 / r), $MachinePrecision] / r), $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 82.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 88.6

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 97.9

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

   8. Applied rewrites 97.9%

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

   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 87.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 97.5

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in r around inf

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

      1. metadata-eval 96.4

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

      2. pow2 96.4

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

      3. associate-*r/ 96.4

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

      4. +-commutative 96.4

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

      5. pow-flip 96.4

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

      6. metadata-eval 96.4

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

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

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 96.5

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

   10. Applied rewrites 96.5%

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

   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 85.2%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in r around 0

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

   9. Applied rewrites 99.1%

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

   11. Applied rewrites 99.1%

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

Alternative 5: 98.0% accurate, 0.4× speedup

Math

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

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 (* 0.25 (* (* w r) (* w r)))) 4.5)
     (if (<= t_1 -1.0)
       (-
        (- 3.0 (/ (* (fma -0.25 v 0.375) (* (* r w) (* r w))) (- 1.0 v)))
        4.5)
       (/ (/ 2.0 r) r)))))

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 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	} else if (t_1 <= -1.0) {
		tmp = (3.0 - ((fma(-0.25, v, 0.375) * ((r * w) * (r * w))) / (1.0 - v))) - 4.5;
	} else {
		tmp = (2.0 / r) / r;
	}
	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(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5);
	elseif (t_1 <= -1.0)
		tmp = Float64(Float64(3.0 - Float64(Float64(fma(-0.25, v, 0.375) * Float64(Float64(r * w) * Float64(r * w))) / Float64(1.0 - v))) - 4.5);
	else
		tmp = Float64(Float64(2.0 / r) / r);
	end
	return 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[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[t$95$1, -1.0], N[(N[(3.0 - N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 / r), $MachinePrecision] / r), $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 82.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 88.6

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 97.9

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

   8. Applied rewrites 97.9%

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

   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 87.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 97.5

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in r around inf

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

      1. metadata-eval 96.4

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

      2. pow2 96.4

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

      3. associate-*r/ 96.4

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

      4. +-commutative 96.4

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

      5. pow-flip 96.4

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

      6. metadata-eval 96.4

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.4

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

   9. Applied rewrites 96.4%

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

   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 85.2%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in r around 0

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

   9. Applied rewrites 99.1%

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

   11. Applied rewrites 99.1%

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

Alternative 6: 91.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := If[LessEqual[N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], -1.0], N[(N[(3.0 - N[(N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(2.0 / r), $MachinePrecision] / r), $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

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 92.2

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

   3. Applied rewrites 92.2%

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

   4. Taylor expanded in r around inf

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

      1. metadata-eval 89.7

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

      2. pow2 89.7

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

      3. associate-*r/ 89.7

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

      4. +-commutative 89.7

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

      5. pow-flip 89.7

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

      6. metadata-eval 89.7

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 89.7

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

   6. Applied rewrites 89.7%

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

   7. Taylor expanded in v around 0

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

   9. Applied rewrites 71.7%

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

   10. Taylor expanded in v around 0

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

   12. Applied rewrites 85.5%

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

   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 85.2%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in r around 0

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

   9. Applied rewrites 99.1%

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

   11. Applied rewrites 99.1%

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

Alternative 7: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t\_0 - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5\\ \mathbf{if}\;v \leq -1 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\left(t\_0 - \frac{\left(\left(\mathsf{fma}\left(-2, v, 3\right) \cdot 0.125\right) \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r))))
        (t_1 (- (- t_0 (* 0.25 (* (* w r) (* w r)))) 4.5)))
   (if (<= v -1e+70)
     t_1
     (if (<= v 2e+39)
       (-
        (-
         t_0
         (/ (* (* (* (fma -2.0 v 3.0) 0.125) (* w (* w r))) r) (- 1.0 v)))
        4.5)
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - (0.25 * ((w * r) * (w * r)))) - 4.5;
	double tmp;
	if (v <= -1e+70) {
		tmp = t_1;
	} else if (v <= 2e+39) {
		tmp = (t_0 - ((((fma(-2.0, v, 3.0) * 0.125) * (w * (w * r))) * r) / (1.0 - v))) - 4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(t_0 - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5)
	tmp = 0.0
	if (v <= -1e+70)
		tmp = t_1;
	elseif (v <= 2e+39)
		tmp = Float64(Float64(t_0 - Float64(Float64(Float64(Float64(fma(-2.0, v, 3.0) * 0.125) * Float64(w * Float64(w * r))) * r) / Float64(1.0 - v))) - 4.5);
	else
		tmp = t_1;
	end
	return 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[(0.25 * N[(N[(w * r), $MachinePrecision] * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, If[LessEqual[v, -1e+70], t$95$1, If[LessEqual[v, 2e+39], N[(N[(t$95$0 - N[(N[(N[(N[(N[(-2.0 * v + 3.0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.00000000000000007e70 or 1.99999999999999988e39 < v

   1. Initial program 80.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 88.3

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in v around inf

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if -1.00000000000000007e70 < v < 1.99999999999999988e39

   1. Initial program 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. +-commutative N/A

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

      14. lower-fma.f64 88.3

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

   3. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

Alternative 8: 93.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) - 4.5 \end{array} \]

FPCore

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

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (0.25 * ((w * r) * (w * r)))) - 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))) - (0.25d0 * ((w * r) * (w * r)))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (0.25 * ((w * r) * (w * r)))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (0.25 * ((w * r) * (w * r)))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(0.25 * Float64(Float64(w * r) * Float64(w * r)))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (0.25 * ((w * r) * (w * r)))) - 4.5;
end

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift-*.f64 N/A

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

   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. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. associate-*l* N/A

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

   6. pow2 N/A

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

   7. *-commutative N/A

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

   8. pow-prod-down N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 94.8

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

3. Applied rewrites 94.8%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 93.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 93.2

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

8. Applied rewrites 93.2%

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

Alternative 9: 69.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-1.5 \cdot r, r, 2\right)}{r}}{r} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift-*.f64 N/A

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

   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. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. associate-*l* N/A

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

   6. pow2 N/A

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

   7. *-commutative N/A

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

   8. pow-prod-down N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 94.8

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

3. Applied rewrites 94.8%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 93.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 93.2

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

8. Applied rewrites 93.2%

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

9. Taylor expanded in r around 0

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

11. Applied rewrites 69.1%

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

Alternative 10: 69.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(r \cdot r, -1.5, 2\right)}{r}}{r} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 53.0

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

4. Applied rewrites 53.0%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 69.1

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

6. Applied rewrites 69.1%

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

Alternative 11: 49.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 3.6e-12) (/ (/ 2.0 r) r) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.6e-12) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 3.6d-12) then
        tmp = (2.0d0 / r) / r
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.6e-12) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3.6e-12:
		tmp = (2.0 / r) / r
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3.6e-12)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3.6e-12)
		tmp = (2.0 / r) / r;
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3.6e-12], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 3.6e-12

   1. Initial program 83.3%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 64.7

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

   4. Applied rewrites 64.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 64.7

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

   6. Applied rewrites 64.7%

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

   7. Taylor expanded in r around 0

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

   9. Applied rewrites 57.9%

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

   11. Applied rewrites 57.8%

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

   if 3.6e-12 < r

   1. Initial program 89.4%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 19.8

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

   4. Applied rewrites 19.8%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 25.4%

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

Alternative 12: 49.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 3.6e-12) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.6e-12) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(v, w, r)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 3.6d-12) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.6e-12) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3.6e-12:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3.6e-12)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3.6e-12)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3.6e-12], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 3.6e-12

   1. Initial program 83.3%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in r around 0

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

   7. Applied rewrites 57.9%

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

   if 3.6e-12 < r

   1. Initial program 89.4%

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

   2. Taylor expanded in r around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 19.8

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

   4. Applied rewrites 19.8%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 25.4%

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

Alternative 13: 14.4% accurate, 73.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 53.0

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

4. Applied rewrites 53.0%

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

5. Taylor expanded in r around inf

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

7. Applied rewrites 14.4%

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

Reproduce

herbie shell --seed 2025093 
(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))