Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.6% → 99.5%

Time: 17.9s

Alternatives: 20

Speedup: 6.5×

• 47.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (+
    x1
    (+
     (+
      (+
       (+
        (*
         (+
          (* (* (* 2.0 x1) t_2) (- t_2 3.0))
          (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
         t_1)
        (* t_0 t_2))
       (* (* x1 x1) x1))
      x1)
     (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
}

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(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    code = x1 + (((((((((2.0d0 * x1) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((4.0d0 * t_2) - 6.0d0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
}

Python

def code(x1, x2):
	t_0 = (3.0 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	return Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
end

MATLAB

function tmp = code(x1, x2)
	t_0 = (3.0 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (+
    x1
    (+
     (+
      (+
       (+
        (*
         (+
          (* (* (* 2.0 x1) t_2) (- t_2 3.0))
          (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
         t_1)
        (* t_0 t_2))
       (* (* x1 x1) x1))
      x1)
     (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
}

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(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    code = x1 + (((((((((2.0d0 * x1) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((4.0d0 * t_2) - 6.0d0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
}

Python

def code(x1, x2):
	t_0 = (3.0 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	return Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
end

MATLAB

function tmp = code(x1, x2)
	t_0 = (3.0 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := \mathsf{fma}\left(x1 \cdot x1, 3, x2 + x2\right) - x1\\ t_3 := x1 \cdot x1 + 1\\ t_4 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_3}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_4\right) \cdot \left(t\_4 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_4 - 6\right)\right) \cdot t\_3 + t\_1 \cdot t\_4\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_3}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \left(t\_2 \cdot \frac{4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6\right), \left(t\_2 \cdot \frac{x1 + x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\frac{t\_2}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x1\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (- (fma (* x1 x1) 3.0 (+ x2 x2)) x1))
        (t_3 (+ (* x1 x1) 1.0))
        (t_4 (/ (- (+ t_1 (* 2.0 x2)) x1) t_3)))
   (if (<=
        (+
         x1
         (+
          (+
           (+
            (+
             (*
              (+
               (* (* (* 2.0 x1) t_4) (- t_4 3.0))
               (* (* x1 x1) (- (* 4.0 t_4) 6.0)))
              t_3)
             (* t_1 t_4))
            t_0)
           x1)
          (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_3))))
        INFINITY)
     (fma
      (/ (fma (* x1 x1) 3.0 (- (* -2.0 x2) x1)) (fma x1 x1 1.0))
      3.0
      (+
       (fma
        (fma x1 x1 1.0)
        (fma
         x1
         (* x1 (- (* t_2 (/ 4.0 (fma x1 x1 1.0))) 6.0))
         (*
          (* t_2 (/ (+ x1 x1) (fma x1 x1 1.0)))
          (- (/ t_2 (fma x1 x1 1.0)) 3.0)))
        (fma
         (/ (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1) (fma x1 x1 1.0))
         (* (* x1 x1) 3.0)
         (* (fma x1 x1 1.0) x1)))
       x1))
     (* (* t_0 x1) 6.0))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = fma((x1 * x1), 3.0, (x2 + x2)) - x1;
	double t_3 = (x1 * x1) + 1.0;
	double t_4 = ((t_1 + (2.0 * x2)) - x1) / t_3;
	double tmp;
	if ((x1 + (((((((((2.0 * x1) * t_4) * (t_4 - 3.0)) + ((x1 * x1) * ((4.0 * t_4) - 6.0))) * t_3) + (t_1 * t_4)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_3)))) <= ((double) INFINITY)) {
		tmp = fma((fma((x1 * x1), 3.0, ((-2.0 * x2) - x1)) / fma(x1, x1, 1.0)), 3.0, (fma(fma(x1, x1, 1.0), fma(x1, (x1 * ((t_2 * (4.0 / fma(x1, x1, 1.0))) - 6.0)), ((t_2 * ((x1 + x1) / fma(x1, x1, 1.0))) * ((t_2 / fma(x1, x1, 1.0)) - 3.0))), fma(((fma((3.0 * x1), x1, (x2 + x2)) - x1) / fma(x1, x1, 1.0)), ((x1 * x1) * 3.0), (fma(x1, x1, 1.0) * x1))) + x1));
	} else {
		tmp = (t_0 * x1) * 6.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(fma(Float64(x1 * x1), 3.0, Float64(x2 + x2)) - x1)
	t_3 = Float64(Float64(x1 * x1) + 1.0)
	t_4 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_3)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_4) * Float64(t_4 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_4) - 6.0))) * t_3) + Float64(t_1 * t_4)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_3)))) <= Inf)
		tmp = fma(Float64(fma(Float64(x1 * x1), 3.0, Float64(Float64(-2.0 * x2) - x1)) / fma(x1, x1, 1.0)), 3.0, Float64(fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * Float64(Float64(t_2 * Float64(4.0 / fma(x1, x1, 1.0))) - 6.0)), Float64(Float64(t_2 * Float64(Float64(x1 + x1) / fma(x1, x1, 1.0))) * Float64(Float64(t_2 / fma(x1, x1, 1.0)) - 3.0))), fma(Float64(Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1) / fma(x1, x1, 1.0)), Float64(Float64(x1 * x1) * 3.0), Float64(fma(x1, x1, 1.0) * x1))) + x1));
	else
		tmp = Float64(Float64(t_0 * x1) * 6.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t$95$4 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$4), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(N[(-2.0 * x2), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(N[(t$95$2 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * N[(N[(x1 + x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x1), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{4 \cdot \left(\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \color{blue}{\mathsf{fma}\left(x1, x1 \cdot \left(\left(\mathsf{fma}\left(x1 \cdot x1, 3, x2 + x2\right) - x1\right) \cdot \frac{4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6\right), \left(\left(\mathsf{fma}\left(x1 \cdot x1, 3, x2 + x2\right) - x1\right) \cdot \frac{x1 + x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right)\right)}, \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      3. metadata-eval N/A

         \[\leadsto {x1}^{\left(3 + 1\right)} \cdot 6 \]

      4. pow-plus N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      6. pow3 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      8. lift-*.f64 99.1

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := \mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1\\ t_5 := \frac{t\_4}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_2 + t\_1 \cdot t\_3\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{4 \cdot t\_4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(t\_5 - 3\right) \cdot \left(t\_5 \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(t\_5, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x1\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4 (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1))
        (t_5 (/ t_4 (fma x1 x1 1.0))))
   (if (<=
        (+
         x1
         (+
          (+
           (+
            (+
             (*
              (+
               (* (* (* 2.0 x1) t_3) (- t_3 3.0))
               (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
              t_2)
             (* t_1 t_3))
            t_0)
           x1)
          (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))
        INFINITY)
     (+
      (fma
       (fma x1 x1 1.0)
       (fma
        (* x1 x1)
        (- (/ (* 4.0 t_4) (fma x1 x1 1.0)) 6.0)
        (* (- t_5 3.0) (* t_5 (+ x1 x1))))
       (fma t_5 (* (* x1 x1) 3.0) (* (fma x1 x1 1.0) x1)))
      (fma (/ (fma (* x1 x1) 3.0 (- (* -2.0 x2) x1)) (fma x1 x1 1.0)) 3.0 x1))
     (* (* t_0 x1) 6.0))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_4 = fma((3.0 * x1), x1, (x2 + x2)) - x1;
	double t_5 = t_4 / fma(x1, x1, 1.0);
	double tmp;
	if ((x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_2) + (t_1 * t_3)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)))) <= ((double) INFINITY)) {
		tmp = fma(fma(x1, x1, 1.0), fma((x1 * x1), (((4.0 * t_4) / fma(x1, x1, 1.0)) - 6.0), ((t_5 - 3.0) * (t_5 * (x1 + x1)))), fma(t_5, ((x1 * x1) * 3.0), (fma(x1, x1, 1.0) * x1))) + fma((fma((x1 * x1), 3.0, ((-2.0 * x2) - x1)) / fma(x1, x1, 1.0)), 3.0, x1);
	} else {
		tmp = (t_0 * x1) * 6.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1)
	t_5 = Float64(t_4 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_2) + Float64(t_1 * t_3)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2)))) <= Inf)
		tmp = Float64(fma(fma(x1, x1, 1.0), fma(Float64(x1 * x1), Float64(Float64(Float64(4.0 * t_4) / fma(x1, x1, 1.0)) - 6.0), Float64(Float64(t_5 - 3.0) * Float64(t_5 * Float64(x1 + x1)))), fma(t_5, Float64(Float64(x1 * x1) * 3.0), Float64(fma(x1, x1, 1.0) * x1))) + fma(Float64(fma(Float64(x1 * x1), 3.0, Float64(Float64(-2.0 * x2) - x1)) / fma(x1, x1, 1.0)), 3.0, x1));
	else
		tmp = Float64(Float64(t_0 * x1) * 6.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(N[(N[(4.0 * t$95$4), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision] + N[(N[(t$95$5 - 3.0), $MachinePrecision] * N[(t$95$5 * N[(x1 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 * N[(N[(x1 * x1), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(N[(-2.0 * x2), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x1), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{4 \cdot \left(\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)} \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      3. metadata-eval N/A

         \[\leadsto {x1}^{\left(3 + 1\right)} \cdot 6 \]

      4. pow-plus N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      6. pow3 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      8. lift-*.f64 99.1

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := \left(x1 \cdot x1\right) \cdot 3\\ t_3 := x1 \cdot x1 + 1\\ t_4 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_3}\\ t_5 := \frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_4\right) \cdot \left(t\_4 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_4 - 6\right)\right) \cdot t\_3 + t\_1 \cdot t\_4\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_3}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\left(t\_2 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(t\_5 - 3\right) \cdot \left(t\_5 \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(t\_5, t\_2, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x1\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (* (* x1 x1) 3.0))
        (t_3 (+ (* x1 x1) 1.0))
        (t_4 (/ (- (+ t_1 (* 2.0 x2)) x1) t_3))
        (t_5 (/ (- (fma (* 3.0 x1) x1 (+ x2 x2)) x1) (fma x1 x1 1.0))))
   (if (<=
        (+
         x1
         (+
          (+
           (+
            (+
             (*
              (+
               (* (* (* 2.0 x1) t_4) (- t_4 3.0))
               (* (* x1 x1) (- (* 4.0 t_4) 6.0)))
              t_3)
             (* t_1 t_4))
            t_0)
           x1)
          (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_3))))
        INFINITY)
     (fma
      (/ (fma (* x1 x1) 3.0 (- (* -2.0 x2) x1)) (fma x1 x1 1.0))
      3.0
      (+
       (fma
        (fma x1 x1 1.0)
        (fma
         (* x1 x1)
         (- (/ (* (- t_2 x1) 4.0) (fma x1 x1 1.0)) 6.0)
         (* (- t_5 3.0) (* t_5 (+ x1 x1))))
        (fma t_5 t_2 (* (fma x1 x1 1.0) x1)))
       x1))
     (* (* t_0 x1) 6.0))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) * 3.0;
	double t_3 = (x1 * x1) + 1.0;
	double t_4 = ((t_1 + (2.0 * x2)) - x1) / t_3;
	double t_5 = (fma((3.0 * x1), x1, (x2 + x2)) - x1) / fma(x1, x1, 1.0);
	double tmp;
	if ((x1 + (((((((((2.0 * x1) * t_4) * (t_4 - 3.0)) + ((x1 * x1) * ((4.0 * t_4) - 6.0))) * t_3) + (t_1 * t_4)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_3)))) <= ((double) INFINITY)) {
		tmp = fma((fma((x1 * x1), 3.0, ((-2.0 * x2) - x1)) / fma(x1, x1, 1.0)), 3.0, (fma(fma(x1, x1, 1.0), fma((x1 * x1), ((((t_2 - x1) * 4.0) / fma(x1, x1, 1.0)) - 6.0), ((t_5 - 3.0) * (t_5 * (x1 + x1)))), fma(t_5, t_2, (fma(x1, x1, 1.0) * x1))) + x1));
	} else {
		tmp = (t_0 * x1) * 6.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(x1 * x1) * 3.0)
	t_3 = Float64(Float64(x1 * x1) + 1.0)
	t_4 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_3)
	t_5 = Float64(Float64(fma(Float64(3.0 * x1), x1, Float64(x2 + x2)) - x1) / fma(x1, x1, 1.0))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_4) * Float64(t_4 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_4) - 6.0))) * t_3) + Float64(t_1 * t_4)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_3)))) <= Inf)
		tmp = fma(Float64(fma(Float64(x1 * x1), 3.0, Float64(Float64(-2.0 * x2) - x1)) / fma(x1, x1, 1.0)), 3.0, Float64(fma(fma(x1, x1, 1.0), fma(Float64(x1 * x1), Float64(Float64(Float64(Float64(t_2 - x1) * 4.0) / fma(x1, x1, 1.0)) - 6.0), Float64(Float64(t_5 - 3.0) * Float64(t_5 * Float64(x1 + x1)))), fma(t_5, t_2, Float64(fma(x1, x1, 1.0) * x1))) + x1));
	else
		tmp = Float64(Float64(t_0 * x1) * 6.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 + x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t$95$4 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$4), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(N[(-2.0 * x2), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(N[(N[(N[(t$95$2 - x1), $MachinePrecision] * 4.0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision] + N[(N[(t$95$5 - 3.0), $MachinePrecision] * N[(t$95$5 * N[(x1 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 * t$95$2 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x1), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{4 \cdot \left(\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right)} \]

   4. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\color{blue}{4 \cdot \left(3 \cdot {x1}^{2} - x1\right)}}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\left(3 \cdot {x1}^{2} - x1\right) \cdot \color{blue}{4}}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\left(3 \cdot {x1}^{2} - x1\right) \cdot \color{blue}{4}}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\left(3 \cdot {x1}^{2} - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\left({x1}^{2} \cdot 3 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\left({x1}^{2} \cdot 3 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\left(\left(x1 \cdot x1\right) \cdot 3 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

      7. lift-*.f64 98.1

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\left(\left(x1 \cdot x1\right) \cdot 3 - x1\right) \cdot 4}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

   6. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, -2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1 \cdot x1, \frac{\color{blue}{\left(\left(x1 \cdot x1\right) \cdot 3 - x1\right) \cdot 4}}{\mathsf{fma}\left(x1, x1, 1\right)} - 6, \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 + x1\right)\right)\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 + x2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(x1 \cdot x1\right) \cdot 3, \mathsf{fma}\left(x1, x1, 1\right) \cdot x1\right)\right) + x1\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      3. metadata-eval N/A

         \[\leadsto {x1}^{\left(3 + 1\right)} \cdot 6 \]

      4. pow-plus N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      6. pow3 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      8. lift-*.f64 99.1

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\\ t_1 := \left(x2 + x2\right) - 3\\ \mathbf{if}\;x1 \leq -52:\\ \;\;\;\;\left(6 - \frac{\frac{\mathsf{fma}\left(-4, t\_1, \frac{-1 + -2 \cdot \left(t\_1 \cdot 3\right)}{x1}\right) + -9}{x1} + 3}{x1}\right) \cdot t\_0\\ \mathbf{elif}\;x1 \leq 11600000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(t\_1, 4, 9\right)}{x1}}{x1}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* (* x1 x1) x1) x1)) (t_1 (- (+ x2 x2) 3.0)))
   (if (<= x1 -52.0)
     (*
      (-
       6.0
       (/
        (+
         (/ (+ (fma -4.0 t_1 (/ (+ -1.0 (* -2.0 (* t_1 3.0))) x1)) -9.0) x1)
         3.0)
        x1))
      t_0)
     (if (<= x1 11600000000000.0)
       (fma (fma (* x2 x1) 8.0 (- (* -12.0 x1) 6.0)) x2 (- x1))
       (* (- 6.0 (/ (- 3.0 (/ (fma t_1 4.0 9.0) x1)) x1)) t_0)))))

C

double code(double x1, double x2) {
	double t_0 = ((x1 * x1) * x1) * x1;
	double t_1 = (x2 + x2) - 3.0;
	double tmp;
	if (x1 <= -52.0) {
		tmp = (6.0 - ((((fma(-4.0, t_1, ((-1.0 + (-2.0 * (t_1 * 3.0))) / x1)) + -9.0) / x1) + 3.0) / x1)) * t_0;
	} else if (x1 <= 11600000000000.0) {
		tmp = fma(fma((x2 * x1), 8.0, ((-12.0 * x1) - 6.0)), x2, -x1);
	} else {
		tmp = (6.0 - ((3.0 - (fma(t_1, 4.0, 9.0) / x1)) / x1)) * t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(x1 * x1) * x1) * x1)
	t_1 = Float64(Float64(x2 + x2) - 3.0)
	tmp = 0.0
	if (x1 <= -52.0)
		tmp = Float64(Float64(6.0 - Float64(Float64(Float64(Float64(fma(-4.0, t_1, Float64(Float64(-1.0 + Float64(-2.0 * Float64(t_1 * 3.0))) / x1)) + -9.0) / x1) + 3.0) / x1)) * t_0);
	elseif (x1 <= 11600000000000.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, Float64(Float64(-12.0 * x1) - 6.0)), x2, Float64(-x1));
	else
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(t_1, 4.0, 9.0) / x1)) / x1)) * t_0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x2 + x2), $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[x1, -52.0], N[(N[(6.0 - N[(N[(N[(N[(N[(-4.0 * t$95$1 + N[(N[(-1.0 + N[(-2.0 * N[(t$95$1 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] + -9.0), $MachinePrecision] / x1), $MachinePrecision] + 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x1, 11600000000000.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(N[(6.0 - N[(N[(3.0 - N[(N[(t$95$1 * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -52

   1. Initial program 34.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{1 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(6 - \frac{\frac{\mathsf{fma}\left(-4, \left(x2 + x2\right) - 3, \frac{-1 + -2 \cdot \left(\left(\left(x2 + x2\right) - 3\right) \cdot 3\right)}{x1}\right) + -9}{x1} + 3}{x1}\right) \cdot \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right)} \]

   if -52 < x1 < 1.16e13

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(8 \cdot \left(x1 \cdot x2\right) + -12 \cdot x1\right) - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(8 \cdot \left(x1 \cdot x2\right) + \left(-12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(x1 \cdot x2\right) \cdot 8 + \left(-12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      13. lower-neg.f64 97.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, -x1\right) \]

   7. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), \color{blue}{x2}, -x1\right) \]

   if 1.16e13 < x1

   1. Initial program 46.9%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\left(x2 + x2\right) - 3, 4, 9\right)}{x1}}{x1}\right) \cdot \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\left(x2 + x2\right) - 3, 4, 9\right)}{x1}}{x1}\right) \cdot \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right)\\ \mathbf{if}\;x1 \leq -52:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 11600000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (- 6.0 (/ (- 3.0 (/ (fma (- (+ x2 x2) 3.0) 4.0 9.0) x1)) x1))
          (* (* (* x1 x1) x1) x1))))
   (if (<= x1 -52.0)
     t_0
     (if (<= x1 11600000000000.0)
       (fma (fma (* x2 x1) 8.0 (- (* -12.0 x1) 6.0)) x2 (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (6.0 - ((3.0 - (fma(((x2 + x2) - 3.0), 4.0, 9.0) / x1)) / x1)) * (((x1 * x1) * x1) * x1);
	double tmp;
	if (x1 <= -52.0) {
		tmp = t_0;
	} else if (x1 <= 11600000000000.0) {
		tmp = fma(fma((x2 * x1), 8.0, ((-12.0 * x1) - 6.0)), x2, -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(Float64(Float64(x2 + x2) - 3.0), 4.0, 9.0) / x1)) / x1)) * Float64(Float64(Float64(x1 * x1) * x1) * x1))
	tmp = 0.0
	if (x1 <= -52.0)
		tmp = t_0;
	elseif (x1 <= 11600000000000.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, Float64(Float64(-12.0 * x1) - 6.0)), x2, Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(N[(x2 + x2), $MachinePrecision] - 3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -52.0], t$95$0, If[LessEqual[x1, 11600000000000.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -52 or 1.16e13 < x1

   1. Initial program 40.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\left(x2 + x2\right) - 3, 4, 9\right)}{x1}}{x1}\right) \cdot \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right)} \]

   if -52 < x1 < 1.16e13

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(8 \cdot \left(x1 \cdot x2\right) + -12 \cdot x1\right) - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(8 \cdot \left(x1 \cdot x2\right) + \left(-12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(x1 \cdot x2\right) \cdot 8 + \left(-12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      13. lower-neg.f64 97.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, -x1\right) \]

   7. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), \color{blue}{x2}, -x1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(t\_2 + 2 \cdot x2\right) - x1}{t\_1}\\ t_4 := t\_2 \cdot t\_3\\ t_5 := t\_3 - 3\\ t_6 := \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\\ t_7 := 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot t\_5 + t\_6\right) \cdot t\_1 + t\_4\right) + t\_0\right) + x1\right) + t\_7\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(-2, x1, 4 \cdot x2\right) \cdot x1\right) \cdot t\_5 + t\_6\right) \cdot t\_1 + t\_4\right) + t\_0\right) + x1\right) + t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x1\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (* (* 3.0 x1) x1))
        (t_3 (/ (- (+ t_2 (* 2.0 x2)) x1) t_1))
        (t_4 (* t_2 t_3))
        (t_5 (- t_3 3.0))
        (t_6 (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
        (t_7 (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_1))))
   (if (<=
        (+
         x1
         (+
          (+ (+ (+ (* (+ (* (* (* 2.0 x1) t_3) t_5) t_6) t_1) t_4) t_0) x1)
          t_7))
        INFINITY)
     (+
      x1
      (+
       (+
        (+ (+ (* (+ (* (* (fma -2.0 x1 (* 4.0 x2)) x1) t_5) t_6) t_1) t_4) t_0)
        x1)
       t_7))
     (* (* t_0 x1) 6.0))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	double t_4 = t_2 * t_3;
	double t_5 = t_3 - 3.0;
	double t_6 = (x1 * x1) * ((4.0 * t_3) - 6.0);
	double t_7 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + (((((((((2.0 * x1) * t_3) * t_5) + t_6) * t_1) + t_4) + t_0) + x1) + t_7)) <= ((double) INFINITY)) {
		tmp = x1 + ((((((((fma(-2.0, x1, (4.0 * x2)) * x1) * t_5) + t_6) * t_1) + t_4) + t_0) + x1) + t_7);
	} else {
		tmp = (t_0 * x1) * 6.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	t_3 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_1)
	t_4 = Float64(t_2 * t_3)
	t_5 = Float64(t_3 - 3.0)
	t_6 = Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))
	t_7 = Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_1))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * t_5) + t_6) * t_1) + t_4) + t_0) + x1) + t_7)) <= Inf)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(fma(-2.0, x1, Float64(4.0 * x2)) * x1) * t_5) + t_6) * t_1) + t_4) + t_0) + x1) + t_7));
	else
		tmp = Float64(Float64(t_0 * x1) * 6.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 - 3.0), $MachinePrecision]}, Block[{t$95$6 = N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$5), $MachinePrecision] + t$95$6), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(-2.0 * x1 + N[(4.0 * x2), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * t$95$5), $MachinePrecision] + t$95$6), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x1), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\color{blue}{\left(x1 \cdot \left(-2 \cdot x1 + 4 \cdot x2\right)\right)} \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(-2 \cdot x1 + 4 \cdot x2\right) \cdot \color{blue}{x1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(-2 \cdot x1 + 4 \cdot x2\right) \cdot \color{blue}{x1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(-2, x1, 4 \cdot x2\right) \cdot x1\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      4. lower-*.f64 94.2

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(-2, x1, 4 \cdot x2\right) \cdot x1\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   4. Applied rewrites 94.2%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\color{blue}{\left(\mathsf{fma}\left(-2, x1, 4 \cdot x2\right) \cdot x1\right)} \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      3. metadata-eval N/A

         \[\leadsto {x1}^{\left(3 + 1\right)} \cdot 6 \]

      4. pow-plus N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      6. pow3 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      8. lift-*.f64 99.1

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -52:\\ \;\;\;\;x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 3900:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -52.0)
   (+ x1 (* (- (* 6.0 x1) 3.0) (* (* x1 x1) x1)))
   (if (<= x1 3900.0)
     (fma (fma (* x2 x1) 8.0 (- (* -12.0 x1) 6.0)) x2 (- x1))
     (+ x1 (* (- 6.0 (/ 3.0 x1)) (pow x1 4.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -52.0) {
		tmp = x1 + (((6.0 * x1) - 3.0) * ((x1 * x1) * x1));
	} else if (x1 <= 3900.0) {
		tmp = fma(fma((x2 * x1), 8.0, ((-12.0 * x1) - 6.0)), x2, -x1);
	} else {
		tmp = x1 + ((6.0 - (3.0 / x1)) * pow(x1, 4.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -52.0)
		tmp = Float64(x1 + Float64(Float64(Float64(6.0 * x1) - 3.0) * Float64(Float64(x1 * x1) * x1)));
	elseif (x1 <= 3900.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, Float64(Float64(-12.0 * x1) - 6.0)), x2, Float64(-x1));
	else
		tmp = Float64(x1 + Float64(Float64(6.0 - Float64(3.0 / x1)) * (x1 ^ 4.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -52.0], N[(x1 + N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3900.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(x1 + N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -52

   1. Initial program 34.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 28.7

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   4. Applied rewrites 28.7%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{2 \cdot x2} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 25.6

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 25.6%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{\left(x2 + x2\right)} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   8. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(2 \cdot x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   9. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 6.9

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   10. Applied rewrites 6.9%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(x2 + x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   12. Step-by-step derivation

       1. count-2-rev N/A

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

       2. lower-+.f64 6.9

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   13. Applied rewrites 6.9%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   14. Taylor expanded in x2 around inf

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(x2 \cdot \left(\left(4 \cdot \frac{3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \frac{x1}{1 + {x1}^{2}}}{x2} + 8 \cdot \frac{1}{1 + {x1}^{2}}\right) - 6 \cdot \frac{1}{x2}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   16. Applied rewrites 24.2%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(\frac{8}{\mathsf{fma}\left(x1, x1, 1\right)} + \frac{\frac{\left(x1 \cdot x1\right) \cdot 3 - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 4 - 6}{x2}\right) \cdot x2\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   17. Taylor expanded in x1 around inf

       \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   18. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot {\color{blue}{x1}}^{4} \]

       4. associate-*r/ N/A

          \[\leadsto x1 + \left(6 - \frac{3 \cdot 1}{x1}\right) \cdot {x1}^{4} \]

       5. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       6. lower-/.f64 N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       7. sqr-pow N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

   19. Applied rewrites 88.3%

       \[\leadsto x1 + \color{blue}{\left(6 - \frac{3}{x1}\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]

   20. Taylor expanded in x1 around 0

       \[\leadsto x1 + {x1}^{3} \cdot \color{blue}{\left(6 \cdot x1 - 3\right)} \]
   21. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{\color{blue}{3}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{\color{blue}{3}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{3} \]

       4. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{3} \]

       5. pow3 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

       6. lift-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

       7. lift-*.f64 88.4

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

   22. Applied rewrites 88.4%

       \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot x1\right)} \]

   if -52 < x1 < 3900

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(8 \cdot \left(x1 \cdot x2\right) + -12 \cdot x1\right) - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(8 \cdot \left(x1 \cdot x2\right) + \left(-12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(x1 \cdot x2\right) \cdot 8 + \left(-12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      13. lower-neg.f64 98.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, -x1\right) \]

   7. Applied rewrites 98.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), \color{blue}{x2}, -x1\right) \]

   if 3900 < x1

   1. Initial program 48.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 42.5

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   4. Applied rewrites 42.5%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{2 \cdot x2} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 42.8

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 42.8%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{\left(x2 + x2\right)} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   8. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(2 \cdot x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   9. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 11.9

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   10. Applied rewrites 11.9%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(x2 + x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   12. Step-by-step derivation

       1. count-2-rev N/A

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

       2. lower-+.f64 11.9

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   13. Applied rewrites 11.9%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   14. Taylor expanded in x2 around inf

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(x2 \cdot \left(\left(4 \cdot \frac{3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \frac{x1}{1 + {x1}^{2}}}{x2} + 8 \cdot \frac{1}{1 + {x1}^{2}}\right) - 6 \cdot \frac{1}{x2}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   16. Applied rewrites 38.2%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(\frac{8}{\mathsf{fma}\left(x1, x1, 1\right)} + \frac{\frac{\left(x1 \cdot x1\right) \cdot 3 - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 4 - 6}{x2}\right) \cdot x2\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   17. Taylor expanded in x1 around inf

       \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   18. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot {\color{blue}{x1}}^{4} \]

       4. associate-*r/ N/A

          \[\leadsto x1 + \left(6 - \frac{3 \cdot 1}{x1}\right) \cdot {x1}^{4} \]

       5. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       6. lower-/.f64 N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       7. sqr-pow N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

   19. Applied rewrites 90.2%

       \[\leadsto x1 + \color{blue}{\left(6 - \frac{3}{x1}\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]
   20. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \color{blue}{\left(x1 \cdot x1\right)}\right) \]

       2. pow2 N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {\left(x1 \cdot x1\right)}^{\color{blue}{2}} \]

       3. lift-*.f64 N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {\left(x1 \cdot x1\right)}^{2} \]

       4. pow-prod-down N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot \color{blue}{{x1}^{2}}\right) \]

       5. pow-prod-up N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{\color{blue}{\left(2 + 2\right)}} \]

       6. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       7. lower-pow.f64 90.3

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{\color{blue}{4}} \]

   21. Applied rewrites 90.3%

       \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{\color{blue}{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right)\\ \mathbf{if}\;x1 \leq -52:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3900:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ x1 (* (- (* 6.0 x1) 3.0) (* (* x1 x1) x1)))))
   (if (<= x1 -52.0)
     t_0
     (if (<= x1 3900.0)
       (fma (fma (* x2 x1) 8.0 (- (* -12.0 x1) 6.0)) x2 (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (((6.0 * x1) - 3.0) * ((x1 * x1) * x1));
	double tmp;
	if (x1 <= -52.0) {
		tmp = t_0;
	} else if (x1 <= 3900.0) {
		tmp = fma(fma((x2 * x1), 8.0, ((-12.0 * x1) - 6.0)), x2, -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(Float64(Float64(6.0 * x1) - 3.0) * Float64(Float64(x1 * x1) * x1)))
	tmp = 0.0
	if (x1 <= -52.0)
		tmp = t_0;
	elseif (x1 <= 3900.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, Float64(Float64(-12.0 * x1) - 6.0)), x2, Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -52.0], t$95$0, If[LessEqual[x1, 3900.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -52 or 3900 < x1

   1. Initial program 41.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 35.6

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   4. Applied rewrites 35.6%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{2 \cdot x2} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 34.1

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 34.1%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{\left(x2 + x2\right)} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   8. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(2 \cdot x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   9. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 9.3

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   10. Applied rewrites 9.3%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(x2 + x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   12. Step-by-step derivation

       1. count-2-rev N/A

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

       2. lower-+.f64 9.3

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   13. Applied rewrites 9.3%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   14. Taylor expanded in x2 around inf

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(x2 \cdot \left(\left(4 \cdot \frac{3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \frac{x1}{1 + {x1}^{2}}}{x2} + 8 \cdot \frac{1}{1 + {x1}^{2}}\right) - 6 \cdot \frac{1}{x2}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   16. Applied rewrites 31.1%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(\frac{8}{\mathsf{fma}\left(x1, x1, 1\right)} + \frac{\frac{\left(x1 \cdot x1\right) \cdot 3 - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 4 - 6}{x2}\right) \cdot x2\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   17. Taylor expanded in x1 around inf

       \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   18. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot {\color{blue}{x1}}^{4} \]

       4. associate-*r/ N/A

          \[\leadsto x1 + \left(6 - \frac{3 \cdot 1}{x1}\right) \cdot {x1}^{4} \]

       5. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       6. lower-/.f64 N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       7. sqr-pow N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

   19. Applied rewrites 89.3%

       \[\leadsto x1 + \color{blue}{\left(6 - \frac{3}{x1}\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]

   20. Taylor expanded in x1 around 0

       \[\leadsto x1 + {x1}^{3} \cdot \color{blue}{\left(6 \cdot x1 - 3\right)} \]
   21. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{\color{blue}{3}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{\color{blue}{3}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{3} \]

       4. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{3} \]

       5. pow3 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

       6. lift-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

       7. lift-*.f64 89.3

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

   22. Applied rewrites 89.3%

       \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot x1\right)} \]

   if -52 < x1 < 3900

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(8 \cdot \left(x1 \cdot x2\right) + -12 \cdot x1\right) - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(8 \cdot \left(x1 \cdot x2\right) + \left(-12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(x1 \cdot x2\right) \cdot 8 + \left(-12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x2, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, \mathsf{neg}\left(x1\right)\right) \]

      13. lower-neg.f64 98.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), x2, -x1\right) \]

   7. Applied rewrites 98.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, -12 \cdot x1 - 6\right), \color{blue}{x2}, -x1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_2 + t\_1 \cdot t\_3\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+250}:\\ \;\;\;\;\left(\left(x2 \cdot x1\right) \cdot \frac{x2}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot 8\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(6 \cdot x1 - 3\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
               t_2)
              (* t_1 t_3))
             t_0)
            x1)
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -5e+250)
     (* (* (* x2 x1) (/ x2 (fma x1 x1 1.0))) 8.0)
     (if (<= t_4 4e+116)
       (fma (- (* -12.0 x1) 6.0) x2 (- x1))
       (+ x1 (* (- (* 6.0 x1) 3.0) t_0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_2) + (t_1 * t_3)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -5e+250) {
		tmp = ((x2 * x1) * (x2 / fma(x1, x1, 1.0))) * 8.0;
	} else if (t_4 <= 4e+116) {
		tmp = fma(((-12.0 * x1) - 6.0), x2, -x1);
	} else {
		tmp = x1 + (((6.0 * x1) - 3.0) * t_0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_2) + Float64(t_1 * t_3)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -5e+250)
		tmp = Float64(Float64(Float64(x2 * x1) * Float64(x2 / fma(x1, x1, 1.0))) * 8.0);
	elseif (t_4 <= 4e+116)
		tmp = fma(Float64(Float64(-12.0 * x1) - 6.0), x2, Float64(-x1));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(6.0 * x1) - 3.0) * t_0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+250], N[(N[(N[(x2 * x1), $MachinePrecision] * N[(x2 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision], If[LessEqual[t$95$4, 4e+116], N[(N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(x1 + N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000002e250

   1. Initial program 99.9%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around inf

      \[\leadsto \color{blue}{8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}} \cdot \color{blue}{8} \]

      2. *-commutative N/A

         \[\leadsto \frac{{x2}^{2} \cdot x1}{1 + {x1}^{2}} \cdot 8 \]

      3. associate-/l* N/A

         \[\leadsto \left({x2}^{2} \cdot \frac{x1}{1 + {x1}^{2}}\right) \cdot 8 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x2}^{2} \cdot \frac{x1}{1 + {x1}^{2}}\right) \cdot \color{blue}{8} \]

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(x1 \cdot \left(x2 \cdot \frac{x2}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right) \cdot 8} \]

   6. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\left(\left(x2 \cdot x1\right) \cdot \frac{x2}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot 8} \]

   if -5.0000000000000002e250 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.00000000000000006e116

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{-12 \cdot x1} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(-12 \cdot x1 - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-12 \cdot x1 - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      7. lower-neg.f64 84.4

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right) \]

   7. Applied rewrites 84.4%

      \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, \color{blue}{x2}, -x1\right) \]

   if 4.00000000000000006e116 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 45.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 41.8

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   4. Applied rewrites 41.8%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{2 \cdot x2} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 40.5

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 40.5%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{\left(x2 + x2\right)} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   8. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(2 \cdot x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   9. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 21.6

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   10. Applied rewrites 21.6%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(x2 + x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   12. Step-by-step derivation

       1. count-2-rev N/A

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

       2. lower-+.f64 21.6

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   13. Applied rewrites 21.6%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   14. Taylor expanded in x2 around inf

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(x2 \cdot \left(\left(4 \cdot \frac{3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \frac{x1}{1 + {x1}^{2}}}{x2} + 8 \cdot \frac{1}{1 + {x1}^{2}}\right) - 6 \cdot \frac{1}{x2}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   16. Applied rewrites 38.4%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(\frac{8}{\mathsf{fma}\left(x1, x1, 1\right)} + \frac{\frac{\left(x1 \cdot x1\right) \cdot 3 - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 4 - 6}{x2}\right) \cdot x2\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   17. Taylor expanded in x1 around inf

       \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   18. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot {\color{blue}{x1}}^{4} \]

       4. associate-*r/ N/A

          \[\leadsto x1 + \left(6 - \frac{3 \cdot 1}{x1}\right) \cdot {x1}^{4} \]

       5. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       6. lower-/.f64 N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       7. sqr-pow N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

   19. Applied rewrites 80.1%

       \[\leadsto x1 + \color{blue}{\left(6 - \frac{3}{x1}\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]

   20. Taylor expanded in x1 around 0

       \[\leadsto x1 + {x1}^{3} \cdot \color{blue}{\left(6 \cdot x1 - 3\right)} \]
   21. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{\color{blue}{3}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{\color{blue}{3}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{3} \]

       4. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{3} \]

       5. pow3 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

       6. lift-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

       7. lift-*.f64 80.1

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

   22. Applied rewrites 80.1%

       \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot x1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_2 + t\_1 \cdot t\_3\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(6 \cdot x1 - 3\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
               t_2)
              (* t_1 t_3))
             t_0)
            x1)
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2000000000.0)
     (fma (* (* x2 x2) 8.0) x1 (* -6.0 x2))
     (if (<= t_4 4e+116)
       (fma (- (* -12.0 x1) 6.0) x2 (- x1))
       (+ x1 (* (- (* 6.0 x1) 3.0) t_0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_2) + (t_1 * t_3)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2000000000.0) {
		tmp = fma(((x2 * x2) * 8.0), x1, (-6.0 * x2));
	} else if (t_4 <= 4e+116) {
		tmp = fma(((-12.0 * x1) - 6.0), x2, -x1);
	} else {
		tmp = x1 + (((6.0 * x1) - 3.0) * t_0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_2) + Float64(t_1 * t_3)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2000000000.0)
		tmp = fma(Float64(Float64(x2 * x2) * 8.0), x1, Float64(-6.0 * x2));
	elseif (t_4 <= 4e+116)
		tmp = fma(Float64(Float64(-12.0 * x1) - 6.0), x2, Float64(-x1));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(6.0 * x1) - 3.0) * t_0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2000000000.0], N[(N[(N[(x2 * x2), $MachinePrecision] * 8.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+116], N[(N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(x1 + N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e9

   1. Initial program 99.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(8 \cdot {x2}^{2}, x1, -6 \cdot x2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({x2}^{2} \cdot 8, x1, -6 \cdot x2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({x2}^{2} \cdot 8, x1, -6 \cdot x2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

      4. lower-*.f64 72.6

         \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

   7. Applied rewrites 72.6%

      \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

   if -2e9 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.00000000000000006e116

   1. Initial program 99.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{-12 \cdot x1} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(-12 \cdot x1 - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-12 \cdot x1 - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      7. lower-neg.f64 89.3

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right) \]

   7. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, \color{blue}{x2}, -x1\right) \]

   if 4.00000000000000006e116 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 45.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 41.8

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   4. Applied rewrites 41.8%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{2 \cdot x2} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 40.5

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + \color{blue}{x2}\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 40.5%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\color{blue}{\left(x2 + x2\right)} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   8. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(2 \cdot x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   9. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-+.f64 21.6

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + \color{blue}{x2}\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   10. Applied rewrites 21.6%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \color{blue}{\left(x2 + x2\right)} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(2 \cdot x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   12. Step-by-step derivation

       1. count-2-rev N/A

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

       2. lower-+.f64 21.6

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + \color{blue}{x2}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   13. Applied rewrites 21.6%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \left(x2 + x2\right) - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(x2 + x2\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   14. Taylor expanded in x2 around inf

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(x2 \cdot \left(\left(4 \cdot \frac{3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \frac{x1}{1 + {x1}^{2}}}{x2} + 8 \cdot \frac{1}{1 + {x1}^{2}}\right) - 6 \cdot \frac{1}{x2}\right)\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   16. Applied rewrites 38.4%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \left(x2 + x2\right)\right) \cdot \left(\left(x2 + x2\right) - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(\frac{8}{\mathsf{fma}\left(x1, x1, 1\right)} + \frac{\frac{\left(x1 \cdot x1\right) \cdot 3 - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 4 - 6}{x2}\right) \cdot x2\right)}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \left(x2 + x2\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   17. Taylor expanded in x1 around inf

       \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - 3 \cdot \frac{1}{x1}\right)} \]
   18. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \color{blue}{{x1}^{4}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 - 3 \cdot \frac{1}{x1}\right) \cdot {\color{blue}{x1}}^{4} \]

       4. associate-*r/ N/A

          \[\leadsto x1 + \left(6 - \frac{3 \cdot 1}{x1}\right) \cdot {x1}^{4} \]

       5. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       6. lower-/.f64 N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot {x1}^{4} \]

       7. sqr-pow N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto x1 + \left(6 - \frac{3}{x1}\right) \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

   19. Applied rewrites 80.1%

       \[\leadsto x1 + \color{blue}{\left(6 - \frac{3}{x1}\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]

   20. Taylor expanded in x1 around 0

       \[\leadsto x1 + {x1}^{3} \cdot \color{blue}{\left(6 \cdot x1 - 3\right)} \]
   21. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{\color{blue}{3}} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{\color{blue}{3}} \]

       3. lower--.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{3} \]

       4. lower-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot {x1}^{3} \]

       5. pow3 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

       6. lift-*.f64 N/A

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

       7. lift-*.f64 80.1

          \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot x1\right) \]

   22. Applied rewrites 80.1%

       \[\leadsto x1 + \left(6 \cdot x1 - 3\right) \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot x1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_2 + t\_1 \cdot t\_3\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(t\_0 \cdot x1\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
               t_2)
              (* t_1 t_3))
             t_0)
            x1)
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2000000000.0)
     (fma (* (* x2 x2) 8.0) x1 (* -6.0 x2))
     (if (<= t_4 4e+116)
       (fma (- (* -12.0 x1) 6.0) x2 (- x1))
       (+ x1 (* (* t_0 x1) 6.0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_2) + (t_1 * t_3)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2000000000.0) {
		tmp = fma(((x2 * x2) * 8.0), x1, (-6.0 * x2));
	} else if (t_4 <= 4e+116) {
		tmp = fma(((-12.0 * x1) - 6.0), x2, -x1);
	} else {
		tmp = x1 + ((t_0 * x1) * 6.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_2) + Float64(t_1 * t_3)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2000000000.0)
		tmp = fma(Float64(Float64(x2 * x2) * 8.0), x1, Float64(-6.0 * x2));
	elseif (t_4 <= 4e+116)
		tmp = fma(Float64(Float64(-12.0 * x1) - 6.0), x2, Float64(-x1));
	else
		tmp = Float64(x1 + Float64(Float64(t_0 * x1) * 6.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2000000000.0], N[(N[(N[(x2 * x2), $MachinePrecision] * 8.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+116], N[(N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(x1 + N[(N[(t$95$0 * x1), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e9

   1. Initial program 99.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(8 \cdot {x2}^{2}, x1, -6 \cdot x2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({x2}^{2} \cdot 8, x1, -6 \cdot x2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({x2}^{2} \cdot 8, x1, -6 \cdot x2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

      4. lower-*.f64 72.6

         \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

   7. Applied rewrites 72.6%

      \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

   if -2e9 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.00000000000000006e116

   1. Initial program 99.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{-12 \cdot x1} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(-12 \cdot x1 - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-12 \cdot x1 - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      7. lower-neg.f64 89.3

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right) \]

   7. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, \color{blue}{x2}, -x1\right) \]

   if 4.00000000000000006e116 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 45.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{6} \]

      3. metadata-eval N/A

         \[\leadsto x1 + {x1}^{\left(3 + 1\right)} \cdot 6 \]

      4. pow-plus N/A

         \[\leadsto x1 + \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      5. lower-*.f64 N/A

         \[\leadsto x1 + \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      6. pow3 N/A

         \[\leadsto x1 + \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      7. lift-*.f64 N/A

         \[\leadsto x1 + \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      8. lift-*.f64 80.1

         \[\leadsto x1 + \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

   4. Applied rewrites 80.1%

      \[\leadsto x1 + \color{blue}{\left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_2 + t\_1 \cdot t\_3\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x1\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
               t_2)
              (* t_1 t_3))
             t_0)
            x1)
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2000000000.0)
     (fma (* (* x2 x2) 8.0) x1 (* -6.0 x2))
     (if (<= t_4 4e+116)
       (fma (- (* -12.0 x1) 6.0) x2 (- x1))
       (* (* t_0 x1) 6.0)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_2) + (t_1 * t_3)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2000000000.0) {
		tmp = fma(((x2 * x2) * 8.0), x1, (-6.0 * x2));
	} else if (t_4 <= 4e+116) {
		tmp = fma(((-12.0 * x1) - 6.0), x2, -x1);
	} else {
		tmp = (t_0 * x1) * 6.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_2) + Float64(t_1 * t_3)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2000000000.0)
		tmp = fma(Float64(Float64(x2 * x2) * 8.0), x1, Float64(-6.0 * x2));
	elseif (t_4 <= 4e+116)
		tmp = fma(Float64(Float64(-12.0 * x1) - 6.0), x2, Float64(-x1));
	else
		tmp = Float64(Float64(t_0 * x1) * 6.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2000000000.0], N[(N[(N[(x2 * x2), $MachinePrecision] * 8.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+116], N[(N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(N[(t$95$0 * x1), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e9

   1. Initial program 99.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(8 \cdot {x2}^{2}, x1, -6 \cdot x2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({x2}^{2} \cdot 8, x1, -6 \cdot x2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({x2}^{2} \cdot 8, x1, -6 \cdot x2\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

      4. lower-*.f64 72.6

         \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

   7. Applied rewrites 72.6%

      \[\leadsto \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right) \]

   if -2e9 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.00000000000000006e116

   1. Initial program 99.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{-12 \cdot x1} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(-12 \cdot x1 - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-12 \cdot x1 - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      7. lower-neg.f64 89.3

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right) \]

   7. Applied rewrites 89.3%

      \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, \color{blue}{x2}, -x1\right) \]

   if 4.00000000000000006e116 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 45.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      3. metadata-eval N/A

         \[\leadsto {x1}^{\left(3 + 1\right)} \cdot 6 \]

      4. pow-plus N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      6. pow3 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      8. lift-*.f64 80.0

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_2 + t\_1 \cdot t\_3\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+250}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x1\right) \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
               t_2)
              (* t_1 t_3))
             t_0)
            x1)
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -5e+250)
     (* (* (* x2 x2) x1) 8.0)
     (if (<= t_4 4e+116)
       (fma (- (* -12.0 x1) 6.0) x2 (- x1))
       (* (* t_0 x1) 6.0)))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_2) + (t_1 * t_3)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -5e+250) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else if (t_4 <= 4e+116) {
		tmp = fma(((-12.0 * x1) - 6.0), x2, -x1);
	} else {
		tmp = (t_0 * x1) * 6.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_2) + Float64(t_1 * t_3)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -5e+250)
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	elseif (t_4 <= 4e+116)
		tmp = fma(Float64(Float64(-12.0 * x1) - 6.0), x2, Float64(-x1));
	else
		tmp = Float64(Float64(t_0 * x1) * 6.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+250], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], If[LessEqual[t$95$4, 4e+116], N[(N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(N[(t$95$0 * x1), $MachinePrecision] * 6.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000002e250

   1. Initial program 99.9%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto 8 \cdot \color{blue}{\left(x1 \cdot {x2}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x1 \cdot {x2}^{2}\right) \cdot 8 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot {x2}^{2}\right) \cdot 8 \]

      3. *-commutative N/A

         \[\leadsto \left({x2}^{2} \cdot x1\right) \cdot 8 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x2}^{2} \cdot x1\right) \cdot 8 \]

      5. unpow2 N/A

         \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 \]

      6. lower-*.f64 77.3

         \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 \]

   7. Applied rewrites 77.3%

      \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot \color{blue}{8} \]

   if -5.0000000000000002e250 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.00000000000000006e116

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{-12 \cdot x1} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(-12 \cdot x1 - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-12 \cdot x1 - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      7. lower-neg.f64 84.4

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right) \]

   7. Applied rewrites 84.4%

      \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, \color{blue}{x2}, -x1\right) \]

   if 4.00000000000000006e116 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 45.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{6} \]

      3. metadata-eval N/A

         \[\leadsto {x1}^{\left(3 + 1\right)} \cdot 6 \]

      4. pow-plus N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      5. lower-*.f64 N/A

         \[\leadsto \left({x1}^{3} \cdot x1\right) \cdot 6 \]

      6. pow3 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

      8. lift-*.f64 80.0

         \[\leadsto \left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6 \]

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(\left(\left(x1 \cdot x1\right) \cdot x1\right) \cdot x1\right) \cdot 6} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ t_4 := \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+250}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot x1 - 1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1)))))
        (t_4 (* (* (* x2 x2) x1) 8.0)))
   (if (<= t_3 -5e+250)
     t_4
     (if (<= t_3 2e+230)
       (fma (- (* -12.0 x1) 6.0) x2 (- x1))
       (if (<= t_3 INFINITY) t_4 (* (- (* 9.0 x1) 1.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double t_4 = ((x2 * x2) * x1) * 8.0;
	double tmp;
	if (t_3 <= -5e+250) {
		tmp = t_4;
	} else if (t_3 <= 2e+230) {
		tmp = fma(((-12.0 * x1) - 6.0), x2, -x1);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = ((9.0 * x1) - 1.0) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	t_4 = Float64(Float64(Float64(x2 * x2) * x1) * 8.0)
	tmp = 0.0
	if (t_3 <= -5e+250)
		tmp = t_4;
	elseif (t_3 <= 2e+230)
		tmp = fma(Float64(Float64(-12.0 * x1) - 6.0), x2, Float64(-x1));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(Float64(9.0 * x1) - 1.0) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+250], t$95$4, If[LessEqual[t$95$3, 2e+230], N[(N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(N[(9.0 * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000002e250 or 2.0000000000000002e230 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto 8 \cdot \color{blue}{\left(x1 \cdot {x2}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x1 \cdot {x2}^{2}\right) \cdot 8 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot {x2}^{2}\right) \cdot 8 \]

      3. *-commutative N/A

         \[\leadsto \left({x2}^{2} \cdot x1\right) \cdot 8 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x2}^{2} \cdot x1\right) \cdot 8 \]

      5. unpow2 N/A

         \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 \]

      6. lower-*.f64 49.3

         \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 \]

   7. Applied rewrites 49.3%

      \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot \color{blue}{8} \]

   if -5.0000000000000002e250 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.0000000000000002e230

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot x1 + \color{blue}{x2 \cdot \left(-12 \cdot x1 - 6\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x1\right)\right) + x2 \cdot \left(\color{blue}{-12 \cdot x1} - 6\right) \]

      2. +-commutative N/A

         \[\leadsto x2 \cdot \left(-12 \cdot x1 - 6\right) + \left(\mathsf{neg}\left(x1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(-12 \cdot x1 - 6\right) \cdot x2 + \left(\mathsf{neg}\left(x1\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, \mathsf{neg}\left(x1\right)\right) \]

      7. lower-neg.f64 77.0

         \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, x2, -x1\right) \]

   7. Applied rewrites 77.0%

      \[\leadsto \mathsf{fma}\left(-12 \cdot x1 - 6, \color{blue}{x2}, -x1\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot x2 - \left(\left(x2 + x2\right) - 3\right), 2, \mathsf{fma}\left(3 + \left(x2 + x2\right), 3, x2 \cdot 14\right) - 6\right), x1, \left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1\right), x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      4. lower-*.f64 3.6

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

   7. Applied rewrites 3.6%

      \[\leadsto \left(9 \cdot x1 - 1\right) \cdot \color{blue}{x1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ t_4 := \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+250}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+230}:\\ \;\;\;\;\mathsf{fma}\left(-1, x1, -6 \cdot x2\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot x1 - 1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1)))))
        (t_4 (* (* (* x2 x2) x1) 8.0)))
   (if (<= t_3 -5e+250)
     t_4
     (if (<= t_3 2e+230)
       (fma -1.0 x1 (* -6.0 x2))
       (if (<= t_3 INFINITY) t_4 (* (- (* 9.0 x1) 1.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double t_4 = ((x2 * x2) * x1) * 8.0;
	double tmp;
	if (t_3 <= -5e+250) {
		tmp = t_4;
	} else if (t_3 <= 2e+230) {
		tmp = fma(-1.0, x1, (-6.0 * x2));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = ((9.0 * x1) - 1.0) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	t_4 = Float64(Float64(Float64(x2 * x2) * x1) * 8.0)
	tmp = 0.0
	if (t_3 <= -5e+250)
		tmp = t_4;
	elseif (t_3 <= 2e+230)
		tmp = fma(-1.0, x1, Float64(-6.0 * x2));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(Float64(9.0 * x1) - 1.0) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+250], t$95$4, If[LessEqual[t$95$3, 2e+230], N[(-1.0 * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(N[(9.0 * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000002e250 or 2.0000000000000002e230 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto 8 \cdot \color{blue}{\left(x1 \cdot {x2}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x1 \cdot {x2}^{2}\right) \cdot 8 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot {x2}^{2}\right) \cdot 8 \]

      3. *-commutative N/A

         \[\leadsto \left({x2}^{2} \cdot x1\right) \cdot 8 \]

      4. lower-*.f64 N/A

         \[\leadsto \left({x2}^{2} \cdot x1\right) \cdot 8 \]

      5. unpow2 N/A

         \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 \]

      6. lower-*.f64 49.3

         \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8 \]

   7. Applied rewrites 49.3%

      \[\leadsto \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot \color{blue}{8} \]

   if -5.0000000000000002e250 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.0000000000000002e230

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 77.0%

      \[\leadsto \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot x2 - \left(\left(x2 + x2\right) - 3\right), 2, \mathsf{fma}\left(3 + \left(x2 + x2\right), 3, x2 \cdot 14\right) - 6\right), x1, \left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1\right), x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      4. lower-*.f64 3.6

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

   7. Applied rewrites 3.6%

      \[\leadsto \left(9 \cdot x1 - 1\right) \cdot \color{blue}{x1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq 4 \cdot 10^{+224}:\\ \;\;\;\;\mathsf{fma}\left(-1, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot x1 - 1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         x1
         (+
          (+
           (+
            (+
             (*
              (+
               (* (* (* 2.0 x1) t_2) (- t_2 3.0))
               (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
              t_1)
             (* t_0 t_2))
            (* (* x1 x1) x1))
           x1)
          (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))
        4e+224)
     (fma -1.0 x1 (* -6.0 x2))
     (* (- (* 9.0 x1) 1.0) x1))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 4e+224) {
		tmp = fma(-1.0, x1, (-6.0 * x2));
	} else {
		tmp = ((9.0 * x1) - 1.0) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= 4e+224)
		tmp = fma(-1.0, x1, Float64(-6.0 * x2));
	else
		tmp = Float64(Float64(Float64(9.0 * x1) - 1.0) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+224], N[(-1.0 * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(9.0 * x1), $MachinePrecision] - 1.0), $MachinePrecision] * x1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 3.99999999999999988e224

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 67.9%

      \[\leadsto \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \]

   if 3.99999999999999988e224 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 37.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot x2 - \left(\left(x2 + x2\right) - 3\right), 2, \mathsf{fma}\left(3 + \left(x2 + x2\right), 3, x2 \cdot 14\right) - 6\right), x1, \left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1\right), x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

      4. lower-*.f64 55.2

         \[\leadsto \left(9 \cdot x1 - 1\right) \cdot x1 \]

   7. Applied rewrites 55.2%

      \[\leadsto \left(9 \cdot x1 - 1\right) \cdot \color{blue}{x1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 38.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (fma -1.0 x1 (* -6.0 x2)))

C

double code(double x1, double x2) {
	return fma(-1.0, x1, (-6.0 * x2));
}

Julia

function code(x1, x2)
	return fma(-1.0, x1, Float64(-6.0 * x2))
end

Wolfram

code[x1_, x2_] := N[(-1.0 * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

2. Taylor expanded in x1 around 0

   \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

   2. *-commutative N/A

      \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

4. Applied rewrites 54.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

5. Taylor expanded in x2 around 0

   \[\leadsto \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \]
6. Step-by-step derivation

7. Applied rewrites 38.2%

   \[\leadsto \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \]
8. Add Preprocessing

Alternative 18: 30.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \leq -1.5 \cdot 10^{-169}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \leq 9 \cdot 10^{-273}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;x1 + -6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x2 -1.5e-169)
   (* -6.0 x2)
   (if (<= x2 9e-273) (- x1) (+ x1 (* -6.0 x2)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x2 <= -1.5e-169) {
		tmp = -6.0 * x2;
	} else if (x2 <= 9e-273) {
		tmp = -x1;
	} else {
		tmp = x1 + (-6.0 * x2);
	}
	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(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if (x2 <= (-1.5d-169)) then
        tmp = (-6.0d0) * x2
    else if (x2 <= 9d-273) then
        tmp = -x1
    else
        tmp = x1 + ((-6.0d0) * x2)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x2 <= -1.5e-169) {
		tmp = -6.0 * x2;
	} else if (x2 <= 9e-273) {
		tmp = -x1;
	} else {
		tmp = x1 + (-6.0 * x2);
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x2 <= -1.5e-169:
		tmp = -6.0 * x2
	elif x2 <= 9e-273:
		tmp = -x1
	else:
		tmp = x1 + (-6.0 * x2)
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x2 <= -1.5e-169)
		tmp = Float64(-6.0 * x2);
	elseif (x2 <= 9e-273)
		tmp = Float64(-x1);
	else
		tmp = Float64(x1 + Float64(-6.0 * x2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x2 <= -1.5e-169)
		tmp = -6.0 * x2;
	elseif (x2 <= 9e-273)
		tmp = -x1;
	else
		tmp = x1 + (-6.0 * x2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x2, -1.5e-169], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x2, 9e-273], (-x1), N[(x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x2 < -1.5e-169

   1. Initial program 70.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2} \]
   3. Step-by-step derivation

      1. lower-*.f64 30.3

         \[\leadsto -6 \cdot \color{blue}{x2} \]

   4. Applied rewrites 30.3%

      \[\leadsto \color{blue}{-6 \cdot x2} \]

   if -1.5e-169 < x2 < 8.99999999999999921e-273

   1. Initial program 70.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot \color{blue}{x1} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x1\right) \]

      2. lower-neg.f64 41.3

         \[\leadsto -x1 \]

   7. Applied rewrites 41.3%

      \[\leadsto -x1 \]

   if 8.99999999999999921e-273 < x2

   1. Initial program 71.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   3. Step-by-step derivation

      1. lower-*.f64 27.3

         \[\leadsto x1 + -6 \cdot \color{blue}{x2} \]

   4. Applied rewrites 27.3%

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 30.1% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \leq -1.5 \cdot 10^{-169}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \leq 9 \cdot 10^{-273}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x2 -1.5e-169) (* -6.0 x2) (if (<= x2 9e-273) (- x1) (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if (x2 <= -1.5e-169) {
		tmp = -6.0 * x2;
	} else if (x2 <= 9e-273) {
		tmp = -x1;
	} else {
		tmp = -6.0 * x2;
	}
	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(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if (x2 <= (-1.5d-169)) then
        tmp = (-6.0d0) * x2
    else if (x2 <= 9d-273) then
        tmp = -x1
    else
        tmp = (-6.0d0) * x2
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x2 <= -1.5e-169) {
		tmp = -6.0 * x2;
	} else if (x2 <= 9e-273) {
		tmp = -x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x2 <= -1.5e-169:
		tmp = -6.0 * x2
	elif x2 <= 9e-273:
		tmp = -x1
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x2 <= -1.5e-169)
		tmp = Float64(-6.0 * x2);
	elseif (x2 <= 9e-273)
		tmp = Float64(-x1);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x2 <= -1.5e-169)
		tmp = -6.0 * x2;
	elseif (x2 <= 9e-273)
		tmp = -x1;
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x2, -1.5e-169], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x2, 9e-273], (-x1), N[(-6.0 * x2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x2 < -1.5e-169 or 8.99999999999999921e-273 < x2

   1. Initial program 70.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2} \]
   3. Step-by-step derivation

      1. lower-*.f64 28.3

         \[\leadsto -6 \cdot \color{blue}{x2} \]

   4. Applied rewrites 28.3%

      \[\leadsto \color{blue}{-6 \cdot x2} \]

   if -1.5e-169 < x2 < 8.99999999999999921e-273

   1. Initial program 70.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

      2. *-commutative N/A

         \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

   5. Taylor expanded in x2 around 0

      \[\leadsto -1 \cdot \color{blue}{x1} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x1\right) \]

      2. lower-neg.f64 41.3

         \[\leadsto -x1 \]

   7. Applied rewrites 41.3%

      \[\leadsto -x1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 13.8% accurate, 93.1× speedup

Math

\[\begin{array}{l} \\ -x1 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (- x1))

C

double code(double x1, double x2) {
	return -x1;
}

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(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = -x1
end function

Java

public static double code(double x1, double x2) {
	return -x1;
}

Python

def code(x1, x2):
	return -x1

Julia

function code(x1, x2)
	return Float64(-x1)
end

MATLAB

function tmp = code(x1, x2)
	tmp = -x1;
end

Wolfram

code[x1_, x2_] := (-x1)

Derivation

1. Initial program 70.6%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

2. Taylor expanded in x1 around 0

   \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) + \color{blue}{-6 \cdot x2} \]

   2. *-commutative N/A

      \[\leadsto \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right) \cdot x1 + \color{blue}{-6} \cdot x2 \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1, \color{blue}{x1}, -6 \cdot x2\right) \]

4. Applied rewrites 54.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(x2 + x2\right) - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)} \]

5. Taylor expanded in x2 around 0

   \[\leadsto -1 \cdot \color{blue}{x1} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(x1\right) \]

   2. lower-neg.f64 13.8

      \[\leadsto -x1 \]

7. Applied rewrites 13.8%

   \[\leadsto -x1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x1 x2)
  :name "Rosa's FloatVsDoubleBenchmark"
  :precision binary64
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2.0 x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) (- (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)) 3.0)) (* (* x1 x1) (- (* 4.0 (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) 6.0))) (+ (* x1 x1) 1.0)) (* (* (* 3.0 x1) x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))) (* (* x1 x1) x1)) x1) (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))))