Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.1% → 99.4%

Time: 9.6s

Alternatives: 23

Speedup: 5.1×

• 47.2% 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: 70.1% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := 1 + x1 \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \left(3 \cdot x1\right) \cdot x1\\ t_4 := \frac{\left(t\_3 + 2 \cdot x2\right) - x1}{t\_2}\\ t_5 := t\_3 \cdot t\_4\\ t_6 := \left(\left(2 \cdot x1\right) \cdot t\_4\right) \cdot \left(t\_4 - 3\right)\\ t_7 := 3 \cdot \frac{\left(t\_3 - 2 \cdot x2\right) - x1}{t\_2}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(t\_6 + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_4 - 6\right)\right) \cdot t\_2 + t\_5\right) + t\_0\right) + x1\right) + t\_7\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(t\_6 + \left(x1 \cdot x1\right) \cdot \left(x2 \cdot \left(\mathsf{fma}\left(4, \frac{3 \cdot \frac{x1 \cdot x1}{t\_1} - \frac{x1}{t\_1}}{x2}, 8 \cdot {t\_1}^{-1}\right) - 6 \cdot {x2}^{-1}\right)\right)\right) \cdot t\_2 + t\_5\right) + t\_0\right) + x1\right) + t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (+ 1.0 (* x1 x1)))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (* (* 3.0 x1) x1))
        (t_4 (/ (- (+ t_3 (* 2.0 x2)) x1) t_2))
        (t_5 (* t_3 t_4))
        (t_6 (* (* (* 2.0 x1) t_4) (- t_4 3.0)))
        (t_7 (* 3.0 (/ (- (- t_3 (* 2.0 x2)) x1) t_2))))
   (if (<=
        (+
         x1
         (+
          (+
           (+ (+ (* (+ t_6 (* (* x1 x1) (- (* 4.0 t_4) 6.0))) t_2) t_5) t_0)
           x1)
          t_7))
        INFINITY)
     (+
      x1
      (+
       (+
        (+
         (+
          (*
           (+
            t_6
            (*
             (* x1 x1)
             (*
              x2
              (-
               (fma
                4.0
                (/ (- (* 3.0 (/ (* x1 x1) t_1)) (/ x1 t_1)) x2)
                (* 8.0 (pow t_1 -1.0)))
               (* 6.0 (pow x2 -1.0))))))
           t_2)
          t_5)
         t_0)
        x1)
       t_7))
     (+
      x1
      (*
       (pow x1 4.0)
       (+ 6.0 (* -1.0 (/ (+ 3.0 (* -1.0 (/ (- (* 8.0 x2) 3.0) x1))) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = 1.0 + (x1 * x1);
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = (3.0 * x1) * x1;
	double t_4 = ((t_3 + (2.0 * x2)) - x1) / t_2;
	double t_5 = t_3 * t_4;
	double t_6 = ((2.0 * x1) * t_4) * (t_4 - 3.0);
	double t_7 = 3.0 * (((t_3 - (2.0 * x2)) - x1) / t_2);
	double tmp;
	if ((x1 + ((((((t_6 + ((x1 * x1) * ((4.0 * t_4) - 6.0))) * t_2) + t_5) + t_0) + x1) + t_7)) <= ((double) INFINITY)) {
		tmp = x1 + ((((((t_6 + ((x1 * x1) * (x2 * (fma(4.0, (((3.0 * ((x1 * x1) / t_1)) - (x1 / t_1)) / x2), (8.0 * pow(t_1, -1.0))) - (6.0 * pow(x2, -1.0)))))) * t_2) + t_5) + t_0) + x1) + t_7);
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(1.0 + Float64(x1 * x1))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(3.0 * x1) * x1)
	t_4 = Float64(Float64(Float64(t_3 + Float64(2.0 * x2)) - x1) / t_2)
	t_5 = Float64(t_3 * t_4)
	t_6 = Float64(Float64(Float64(2.0 * x1) * t_4) * Float64(t_4 - 3.0))
	t_7 = Float64(3.0 * Float64(Float64(Float64(t_3 - Float64(2.0 * x2)) - x1) / t_2))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(t_6 + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_4) - 6.0))) * t_2) + t_5) + t_0) + x1) + t_7)) <= Inf)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(t_6 + Float64(Float64(x1 * x1) * Float64(x2 * Float64(fma(4.0, Float64(Float64(Float64(3.0 * Float64(Float64(x1 * x1) / t_1)) - Float64(x1 / t_1)) / x2), Float64(8.0 * (t_1 ^ -1.0))) - Float64(6.0 * (x2 ^ -1.0)))))) * t_2) + t_5) + t_0) + x1) + t_7));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(Float64(8.0 * x2) - 3.0) / x1))) / x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$3 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t$95$4 - 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(3.0 * N[(N[(N[(t$95$3 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(t$95$6 + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$4), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(N[(N[(N[(t$95$6 + N[(N[(x1 * x1), $MachinePrecision] * N[(x2 * N[(N[(4.0 * N[(N[(N[(3.0 * N[(N[(x1 * x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x1 / t$95$1), $MachinePrecision]), $MachinePrecision] / x2), $MachinePrecision] + N[(8.0 * N[Power[t$95$1, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(6.0 * N[Power[x2, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(N[(8.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 99.3%

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

   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 x2 around 0

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in x1 around -inf

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

   7. Applied rewrites 99.7%

      \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)\\ \end{array} \end{array} \]

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))
        (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))))))
   (if (<= t_3 INFINITY)
     t_3
     (+
      x1
      (*
       (pow x1 4.0)
       (+ 6.0 (* -1.0 (/ (+ 3.0 (* -1.0 (/ (- (* 8.0 x2) 3.0) x1))) 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 tmp;
	if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	}
	return tmp;
}

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;
	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 tmp;
	if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = x1 + (Math.pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	}
	return tmp;
}

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
	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)))
	tmp = 0
	if t_3 <= math.inf:
		tmp = t_3
	else:
		tmp = x1 + (math.pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / 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))))
	tmp = 0.0
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(Float64(8.0 * x2) - 3.0) / x1))) / x1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = 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;
	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)));
	tmp = 0.0;
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = x1 + ((x1 ^ 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	end
	tmp_2 = 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]}, If[LessEqual[t$95$3, Infinity], t$95$3, N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(N[(8.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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%

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

   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 x2 around 0

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in x1 around -inf

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

   7. Applied rewrites 99.7%

      \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(t\_2 + 2 \cdot x2\right) - x1}{t\_1}\\ t_4 := t\_2 \cdot t\_3\\ t_5 := \left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right)\\ t_6 := 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(t\_5 + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_1 + t\_4\right) + t\_0\right) + x1\right) + t\_6\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(t\_5 + \left(x1 \cdot x1\right) \cdot 6\right) \cdot t\_1 + t\_4\right) + t\_0\right) + x1\right) + t\_6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)\\ \end{array} \end{array} \]

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 (* (* (* 2.0 x1) t_3) (- t_3 3.0)))
        (t_6 (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_1))))
   (if (<=
        (+
         x1
         (+
          (+
           (+ (+ (* (+ t_5 (* (* x1 x1) (- (* 4.0 t_3) 6.0))) t_1) t_4) t_0)
           x1)
          t_6))
        INFINITY)
     (+ x1 (+ (+ (+ (+ (* (+ t_5 (* (* x1 x1) 6.0)) t_1) t_4) t_0) x1) t_6))
     (+
      x1
      (*
       (pow x1 4.0)
       (+ 6.0 (* -1.0 (/ (+ 3.0 (* -1.0 (/ (- (* 8.0 x2) 3.0) x1))) x1))))))))

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 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	double t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + ((((((t_5 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= ((double) INFINITY)) {
		tmp = x1 + ((((((t_5 + ((x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6);
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	}
	return tmp;
}

Java

public static 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 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	double t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + ((((((t_5 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= Double.POSITIVE_INFINITY) {
		tmp = x1 + ((((((t_5 + ((x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6);
	} else {
		tmp = x1 + (Math.pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	}
	return tmp;
}

Python

def code(x1, x2):
	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 = ((2.0 * x1) * t_3) * (t_3 - 3.0)
	t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1)
	tmp = 0
	if (x1 + ((((((t_5 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= math.inf:
		tmp = x1 + ((((((t_5 + ((x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6)
	else:
		tmp = x1 + (math.pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))))
	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(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0))
	t_6 = 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(t_5 + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= Inf)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(t_5 + Float64(Float64(x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(Float64(8.0 * x2) - 3.0) / x1))) / x1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	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 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	tmp = 0.0;
	if ((x1 + ((((((t_5 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= Inf)
		tmp = x1 + ((((((t_5 + ((x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6);
	else
		tmp = x1 + ((x1 ^ 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	end
	tmp_2 = 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[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = 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[(t$95$5 + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(N[(N[(N[(t$95$5 + N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(N[(8.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 inf

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

   4. Applied rewrites 96.0%

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

   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 x2 around 0

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in x1 around -inf

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

   7. Applied rewrites 99.7%

      \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(t\_2 + 2 \cdot x2\right) - x1}{t\_1}\\ t_4 := \left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right)\\ t_5 := 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(t\_4 + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_1 + t\_2 \cdot t\_3\right) + t\_0\right) + x1\right) + t\_5\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(t\_4 + \left(x1 \cdot x1\right) \cdot 6\right) \cdot t\_1 + 9 \cdot \left(x1 \cdot x1\right)\right) + t\_0\right) + x1\right) + t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)\\ \end{array} \end{array} \]

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 (* (* (* 2.0 x1) t_3) (- t_3 3.0)))
        (t_5 (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_1))))
   (if (<=
        (+
         x1
         (+
          (+
           (+
            (+ (* (+ t_4 (* (* x1 x1) (- (* 4.0 t_3) 6.0))) t_1) (* t_2 t_3))
            t_0)
           x1)
          t_5))
        INFINITY)
     (+
      x1
      (+
       (+ (+ (+ (* (+ t_4 (* (* x1 x1) 6.0)) t_1) (* 9.0 (* x1 x1))) t_0) x1)
       t_5))
     (+
      x1
      (*
       (pow x1 4.0)
       (+ 6.0 (* -1.0 (/ (+ 3.0 (* -1.0 (/ (- (* 8.0 x2) 3.0) x1))) x1))))))))

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 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	double t_5 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + ((((((t_4 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_5)) <= ((double) INFINITY)) {
		tmp = x1 + ((((((t_4 + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_5);
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	}
	return tmp;
}

Java

public static 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 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	double t_5 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + ((((((t_4 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_5)) <= Double.POSITIVE_INFINITY) {
		tmp = x1 + ((((((t_4 + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_5);
	} else {
		tmp = x1 + (Math.pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	}
	return tmp;
}

Python

def code(x1, x2):
	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 = ((2.0 * x1) * t_3) * (t_3 - 3.0)
	t_5 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1)
	tmp = 0
	if (x1 + ((((((t_4 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_5)) <= math.inf:
		tmp = x1 + ((((((t_4 + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_5)
	else:
		tmp = x1 + (math.pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))))
	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(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0))
	t_5 = 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(t_4 + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_1) + Float64(t_2 * t_3)) + t_0) + x1) + t_5)) <= Inf)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(t_4 + Float64(Float64(x1 * x1) * 6.0)) * t_1) + Float64(9.0 * Float64(x1 * x1))) + t_0) + x1) + t_5));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(Float64(8.0 * x2) - 3.0) / x1))) / x1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	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 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	t_5 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	tmp = 0.0;
	if ((x1 + ((((((t_4 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_5)) <= Inf)
		tmp = x1 + ((((((t_4 + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_5);
	else
		tmp = x1 + ((x1 ^ 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	end
	tmp_2 = 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[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = 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[(t$95$4 + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(N[(N[(N[(t$95$4 + N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(N[(8.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 96.0%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 95.9

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

   10. Applied rewrites 95.9%

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

   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 x2 around 0

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

   4. Applied rewrites 0.3%

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

   5. Taylor expanded in x1 around -inf

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

   7. Applied rewrites 99.7%

      \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 70.9% accurate, 0.9× speedup

Math

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

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)))
   (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))))
        INFINITY)
     (+
      x1
      (fma
       -6.0
       x2
       (* x1 (- (fma x1 (+ 9.0 (* 12.0 x2)) (* x2 (- (* 8.0 x2) 12.0))) 2.0))))
     (+
      x1
      (+
       (+ (* -12.0 (* x1 x2)) x1)
       (fma -6.0 x2 (* x1 (- (* 3.0 (* x1 3.0)) 3.0))))))))

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)))) <= ((double) INFINITY)) {
		tmp = x1 + fma(-6.0, x2, (x1 * (fma(x1, (9.0 + (12.0 * x2)), (x2 * ((8.0 * x2) - 12.0))) - 2.0)));
	} else {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + fma(-6.0, x2, (x1 * ((3.0 * (x1 * 3.0)) - 3.0))));
	}
	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)))) <= Inf)
		tmp = Float64(x1 + fma(-6.0, x2, Float64(x1 * Float64(fma(x1, Float64(9.0 + Float64(12.0 * x2)), Float64(x2 * Float64(Float64(8.0 * x2) - 12.0))) - 2.0))));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + fma(-6.0, x2, Float64(x1 * Float64(Float64(3.0 * Float64(x1 * 3.0)) - 3.0)))));
	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], Infinity], N[(x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x1 * N[(9.0 + N[(12.0 * x2), $MachinePrecision]), $MachinePrecision] + N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * N[(N[(3.0 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 x2 around 0

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot x2 + \color{blue}{x1 \cdot \left(\left(x1 \cdot \left(9 + 12 \cdot x2\right) + x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 70.3%

      \[\leadsto x1 + \mathsf{fma}\left(-6, \color{blue}{x2}, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right) \]

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.2

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

   4. Applied rewrites 0.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 60.4

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 60.4%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 72.4%

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

Alternative 6: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right)\\ \end{array} \end{array} \]

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)))
   (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))))
        INFINITY)
     (fma -6.0 x2 (* x1 (- (* 4.0 (* x2 (- (* 2.0 x2) 3.0))) 1.0)))
     (+
      x1
      (+
       (+ (* -12.0 (* x1 x2)) x1)
       (fma -6.0 x2 (* x1 (- (* 3.0 (* x1 3.0)) 3.0))))))))

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)))) <= ((double) INFINITY)) {
		tmp = fma(-6.0, x2, (x1 * ((4.0 * (x2 * ((2.0 * x2) - 3.0))) - 1.0)));
	} else {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + fma(-6.0, x2, (x1 * ((3.0 * (x1 * 3.0)) - 3.0))));
	}
	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)))) <= Inf)
		tmp = fma(-6.0, x2, Float64(x1 * Float64(Float64(4.0 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0))) - 1.0)));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + fma(-6.0, x2, Float64(x1 * Float64(Float64(3.0 * Float64(x1 * 3.0)) - 3.0)))));
	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], Infinity], N[(-6.0 * x2 + N[(x1 * N[(N[(4.0 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * N[(N[(3.0 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 69.3%

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

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.2

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

   4. Applied rewrites 0.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 60.4

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 60.4%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 72.4%

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

Alternative 7: 70.2% accurate, 0.9× speedup

Math

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

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)))
   (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))))
        INFINITY)
     (fma -6.0 x2 (* x1 (- (* 4.0 (* x2 (- (* 2.0 x2) 3.0))) 1.0)))
     (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) (* x1 (- (* 9.0 x1) 3.0)))))))

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)))) <= ((double) INFINITY)) {
		tmp = fma(-6.0, x2, (x1 * ((4.0 * (x2 * ((2.0 * x2) - 3.0))) - 1.0)));
	} else {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + (x1 * ((9.0 * x1) - 3.0)));
	}
	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)))) <= Inf)
		tmp = fma(-6.0, x2, Float64(x1 * Float64(Float64(4.0 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0))) - 1.0)));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))));
	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], Infinity], N[(-6.0 * x2 + N[(x1 * N[(N[(4.0 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 69.3%

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

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.2

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

   4. Applied rewrites 0.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 60.4

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 60.4%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 72.3

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 72.3%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{8 \cdot x2 - 3}{x1}}{x1}\right)\\ \mathbf{if}\;x1 \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           (pow x1 4.0)
           (+
            6.0
            (* -1.0 (/ (+ 3.0 (* -1.0 (/ (- (* 8.0 x2) 3.0) x1))) x1)))))))
   (if (<= x1 -2e+42)
     t_0
     (if (<= x1 4.5e+14)
       (+
        x1
        (+
         (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))) x1)
         (*
          3.0
          (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * (((8.0 * x2) - 3.0) / x1))) / x1))));
	double tmp;
	if (x1 <= -2e+42) {
		tmp = t_0;
	} else if (x1 <= 4.5e+14) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(Float64(8.0 * x2) - 3.0) / x1))) / x1)))))
	tmp = 0.0
	if (x1 <= -2e+42)
		tmp = t_0;
	elseif (x1 <= 4.5e+14)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(N[(8.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -2e+42], t$95$0, If[LessEqual[x1, 4.5e+14], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.00000000000000009e42 or 4.5e14 < x1

   1. Initial program 35.3%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in x1 around -inf

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

   7. Applied rewrites 96.9%

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

   if -2.00000000000000009e42 < x1 < 4.5e14

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      4. lower-*.f64 92.8

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 92.8%

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(9 \cdot x1 - 3\right)\\ t_2 := 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{t\_0}\\ t_3 := x1 + \left(\left(\left(\left(\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot t\_0 + 9 \cdot \left(x1 \cdot x1\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + t\_2\right)\\ \mathbf{if}\;x1 \leq -4.1 \cdot 10^{+158}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + t\_1\right)\\ \mathbf{elif}\;x1 \leq -2.35 \cdot 10^{+76}:\\ \;\;\;\;x1 + \left(x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, -6 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x1 \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + t\_2\right)\\ \mathbf{elif}\;x1 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (- (* 9.0 x1) 3.0)))
        (t_2 (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) t_0)))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+ (* (* 6.0 (* x1 x1)) t_0) (* 9.0 (* x1 x1)))
             (* (* x1 x1) x1))
            x1)
           t_2))))
   (if (<= x1 -4.1e+158)
     (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) t_1))
     (if (<= x1 -2.35e+76)
       (+
        x1
        (+
         x1
         (fma
          3.0
          (/ (* (* x1 x1) (- (* 3.0 (* x1 x1)) x1)) (+ 1.0 (* x1 x1)))
          (* -6.0 x2))))
       (if (<= x1 -2e+42)
         t_3
         (if (<= x1 4.5e+14)
           (+ x1 (+ (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))) x1) t_2))
           (if (<= x1 5e+151)
             t_3
             (+
              x1
              (+ (+ (* 4.0 (* x1 (* x2 (- (* 2.0 x2) 3.0)))) x1) t_1)))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * ((9.0 * x1) - 3.0);
	double t_2 = 3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / t_0);
	double t_3 = x1 + ((((((6.0 * (x1 * x1)) * t_0) + (9.0 * (x1 * x1))) + ((x1 * x1) * x1)) + x1) + t_2);
	double tmp;
	if (x1 <= -4.1e+158) {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + t_1);
	} else if (x1 <= -2.35e+76) {
		tmp = x1 + (x1 + fma(3.0, (((x1 * x1) * ((3.0 * (x1 * x1)) - x1)) / (1.0 + (x1 * x1))), (-6.0 * x2)));
	} else if (x1 <= -2e+42) {
		tmp = t_3;
	} else if (x1 <= 4.5e+14) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))) + x1) + t_2);
	} else if (x1 <= 5e+151) {
		tmp = t_3;
	} else {
		tmp = x1 + (((4.0 * (x1 * (x2 * ((2.0 * x2) - 3.0)))) + x1) + t_1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))
	t_2 = Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / t_0))
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * t_0) + Float64(9.0 * Float64(x1 * x1))) + Float64(Float64(x1 * x1) * x1)) + x1) + t_2))
	tmp = 0.0
	if (x1 <= -4.1e+158)
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + t_1));
	elseif (x1 <= -2.35e+76)
		tmp = Float64(x1 + Float64(x1 + fma(3.0, Float64(Float64(Float64(x1 * x1) * Float64(Float64(3.0 * Float64(x1 * x1)) - x1)) / Float64(1.0 + Float64(x1 * x1))), Float64(-6.0 * x2))));
	elseif (x1 <= -2e+42)
		tmp = t_3;
	elseif (x1 <= 4.5e+14)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))) + x1) + t_2));
	elseif (x1 <= 5e+151)
		tmp = t_3;
	else
		tmp = Float64(x1 + Float64(Float64(Float64(4.0 * Float64(x1 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))) + x1) + t_1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -4.1e+158], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -2.35e+76], N[(x1 + N[(x1 + N[(3.0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -2e+42], t$95$3, If[LessEqual[x1, 4.5e+14], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 5e+151], t$95$3, N[(x1 + N[(N[(N[(4.0 * N[(x1 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -4.10000000000000004e158

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 51.6

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 74.4

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 74.4%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 80.1

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 80.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]

   if -4.10000000000000004e158 < x1 < -2.3500000000000002e76

   1. Initial program 33.8%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 27.8%

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

   5. Taylor expanded in x1 around 0

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

      1. lift-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

   if -2.3500000000000002e76 < x1 < -2.00000000000000009e42 or 4.5e14 < x1 < 5.0000000000000002e151

   1. Initial program 98.8%

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

   2. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 92.5%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 92.5

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

   10. Applied rewrites 92.5%

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

   11. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 80.0

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

   13. Applied rewrites 80.0%

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

   if -2.00000000000000009e42 < x1 < 4.5e14

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      4. lower-*.f64 92.8

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 92.8%

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   if 5.0000000000000002e151 < x1

   1. Initial program 1.1%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.5

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 74.4

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in x2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

         \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

      3. lower-*.f64 99.6

         \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   10. Applied rewrites 99.6%

       \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 93.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot {x1}^{4}\\ \mathbf{if}\;x1 \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 6.0 (pow x1 4.0))))
   (if (<= x1 -2e+42)
     t_0
     (if (<= x1 4.5e+14)
       (+
        x1
        (+
         (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))) x1)
         (*
          3.0
          (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = 6.0 * pow(x1, 4.0);
	double tmp;
	if (x1 <= -2e+42) {
		tmp = t_0;
	} else if (x1 <= 4.5e+14) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(6.0 * (x1 ^ 4.0))
	tmp = 0.0
	if (x1 <= -2e+42)
		tmp = t_0;
	elseif (x1 <= 4.5e+14)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -2e+42], t$95$0, If[LessEqual[x1, 4.5e+14], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.00000000000000009e42 or 4.5e14 < x1

   1. Initial program 35.3%

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

   2. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if -2.00000000000000009e42 < x1 < 4.5e14

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      4. lower-*.f64 92.8

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 92.8%

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 84.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -4.1 \cdot 10^{+158}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right)\\ \mathbf{elif}\;x1 \leq -7.2 \cdot 10^{+76}:\\ \;\;\;\;x1 + \left(x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(t\_0 - x1\right)}{1 + x1 \cdot x1}, -6 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 \leq 3.6 \cdot 10^{-42}:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 3.0 (* x1 x1))))
   (if (<= x1 -4.1e+158)
     (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) (* x1 (- (* 9.0 x1) 3.0))))
     (if (<= x1 -7.2e+76)
       (+
        x1
        (+
         x1
         (fma 3.0 (/ (* (* x1 x1) (- t_0 x1)) (+ 1.0 (* x1 x1))) (* -6.0 x2))))
       (if (<= x1 3.6e-42)
         (+
          x1
          (+
           (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))) x1)
           (*
            3.0
            (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
         (+
          x1
          (+
           (+ (* 4.0 (* x1 (* x2 (- (* 2.0 x2) 3.0)))) x1)
           (fma -6.0 x2 (* x1 t_0)))))))))

C

double code(double x1, double x2) {
	double t_0 = 3.0 * (x1 * x1);
	double tmp;
	if (x1 <= -4.1e+158) {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + (x1 * ((9.0 * x1) - 3.0)));
	} else if (x1 <= -7.2e+76) {
		tmp = x1 + (x1 + fma(3.0, (((x1 * x1) * (t_0 - x1)) / (1.0 + (x1 * x1))), (-6.0 * x2)));
	} else if (x1 <= 3.6e-42) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
	} else {
		tmp = x1 + (((4.0 * (x1 * (x2 * ((2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, (x1 * t_0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(3.0 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -4.1e+158)
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))));
	elseif (x1 <= -7.2e+76)
		tmp = Float64(x1 + Float64(x1 + fma(3.0, Float64(Float64(Float64(x1 * x1) * Float64(t_0 - x1)) / Float64(1.0 + Float64(x1 * x1))), Float64(-6.0 * x2))));
	elseif (x1 <= 3.6e-42)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(4.0 * Float64(x1 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, Float64(x1 * t_0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -4.1e+158], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -7.2e+76], N[(x1 + N[(x1 + N[(3.0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$0 - x1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3.6e-42], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(4.0 * N[(x1 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -4.10000000000000004e158

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 51.6

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 74.4

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 74.4%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 80.1

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 80.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]

   if -4.10000000000000004e158 < x1 < -7.2000000000000006e76

   1. Initial program 33.2%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lift-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -7.2000000000000006e76 < x1 < 3.6000000000000002e-42

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 79.6

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      4. lower-*.f64 90.2

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 90.2%

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   if 3.6000000000000002e-42 < x1

   1. Initial program 54.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 23.5

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

   4. Applied rewrites 23.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 71.7

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 73.4

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

   10. Applied rewrites 73.4%

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

Alternative 12: 77.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -12 \cdot \left(x1 \cdot x2\right) + x1\\ t_1 := 3 \cdot \left(x1 \cdot x1\right)\\ t_2 := x1 \cdot \left(9 \cdot x1 - 3\right)\\ \mathbf{if}\;x1 \leq -4.1 \cdot 10^{+158}:\\ \;\;\;\;x1 + \left(t\_0 + t\_2\right)\\ \mathbf{elif}\;x1 \leq -7.2 \cdot 10^{+76}:\\ \;\;\;\;x1 + \left(x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(t\_1 - x1\right)}{1 + x1 \cdot x1}, -6 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-206}:\\ \;\;\;\;x1 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right)\\ \mathbf{elif}\;x1 \leq 2.75 \cdot 10^{-117}:\\ \;\;\;\;x1 + \left(t\_0 + x2 \cdot \left(\mathsf{fma}\left(6, x1 \cdot x1, \frac{t\_2}{x2}\right) - 6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* -12.0 (* x1 x2)) x1))
        (t_1 (* 3.0 (* x1 x1)))
        (t_2 (* x1 (- (* 9.0 x1) 3.0))))
   (if (<= x1 -4.1e+158)
     (+ x1 (+ t_0 t_2))
     (if (<= x1 -7.2e+76)
       (+
        x1
        (+
         x1
         (fma 3.0 (/ (* (* x1 x1) (- t_1 x1)) (+ 1.0 (* x1 x1))) (* -6.0 x2))))
       (if (<= x1 -1.3e-206)
         (+
          x1
          (fma
           -6.0
           x2
           (*
            x1
            (- (fma x1 (+ 9.0 (* 12.0 x2)) (* x2 (- (* 8.0 x2) 12.0))) 2.0))))
         (if (<= x1 2.75e-117)
           (+ x1 (+ t_0 (* x2 (- (fma 6.0 (* x1 x1) (/ t_2 x2)) 6.0))))
           (+
            x1
            (+
             (+ (* 4.0 (* x1 (* x2 (- (* 2.0 x2) 3.0)))) x1)
             (fma -6.0 x2 (* x1 t_1))))))))))

C

double code(double x1, double x2) {
	double t_0 = (-12.0 * (x1 * x2)) + x1;
	double t_1 = 3.0 * (x1 * x1);
	double t_2 = x1 * ((9.0 * x1) - 3.0);
	double tmp;
	if (x1 <= -4.1e+158) {
		tmp = x1 + (t_0 + t_2);
	} else if (x1 <= -7.2e+76) {
		tmp = x1 + (x1 + fma(3.0, (((x1 * x1) * (t_1 - x1)) / (1.0 + (x1 * x1))), (-6.0 * x2)));
	} else if (x1 <= -1.3e-206) {
		tmp = x1 + fma(-6.0, x2, (x1 * (fma(x1, (9.0 + (12.0 * x2)), (x2 * ((8.0 * x2) - 12.0))) - 2.0)));
	} else if (x1 <= 2.75e-117) {
		tmp = x1 + (t_0 + (x2 * (fma(6.0, (x1 * x1), (t_2 / x2)) - 6.0)));
	} else {
		tmp = x1 + (((4.0 * (x1 * (x2 * ((2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, (x1 * t_1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(-12.0 * Float64(x1 * x2)) + x1)
	t_1 = Float64(3.0 * Float64(x1 * x1))
	t_2 = Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))
	tmp = 0.0
	if (x1 <= -4.1e+158)
		tmp = Float64(x1 + Float64(t_0 + t_2));
	elseif (x1 <= -7.2e+76)
		tmp = Float64(x1 + Float64(x1 + fma(3.0, Float64(Float64(Float64(x1 * x1) * Float64(t_1 - x1)) / Float64(1.0 + Float64(x1 * x1))), Float64(-6.0 * x2))));
	elseif (x1 <= -1.3e-206)
		tmp = Float64(x1 + fma(-6.0, x2, Float64(x1 * Float64(fma(x1, Float64(9.0 + Float64(12.0 * x2)), Float64(x2 * Float64(Float64(8.0 * x2) - 12.0))) - 2.0))));
	elseif (x1 <= 2.75e-117)
		tmp = Float64(x1 + Float64(t_0 + Float64(x2 * Float64(fma(6.0, Float64(x1 * x1), Float64(t_2 / x2)) - 6.0))));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(4.0 * Float64(x1 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, Float64(x1 * t_1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -4.1e+158], N[(x1 + N[(t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -7.2e+76], N[(x1 + N[(x1 + N[(3.0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$1 - x1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.3e-206], N[(x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x1 * N[(9.0 + N[(12.0 * x2), $MachinePrecision]), $MachinePrecision] + N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.75e-117], N[(x1 + N[(t$95$0 + N[(x2 * N[(N[(6.0 * N[(x1 * x1), $MachinePrecision] + N[(t$95$2 / x2), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(4.0 * N[(x1 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -4.10000000000000004e158

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 51.6

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 74.4

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 74.4%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 80.1

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 80.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]

   if -4.10000000000000004e158 < x1 < -7.2000000000000006e76

   1. Initial program 33.2%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lift-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -7.2000000000000006e76 < x1 < -1.3e-206

   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 x2 around 0

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot x2 + \color{blue}{x1 \cdot \left(\left(x1 \cdot \left(9 + 12 \cdot x2\right) + x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 74.0%

      \[\leadsto x1 + \mathsf{fma}\left(-6, \color{blue}{x2}, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right) \]

   if -1.3e-206 < x1 < 2.75000000000000013e-117

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 86.1

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 86.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around inf

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \color{blue}{\left(\left(6 \cdot {x1}^{2} + \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\left(6 \cdot {x1}^{2} + \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - \color{blue}{6}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\left(6 \cdot {x1}^{2} + \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\mathsf{fma}\left(6, {x1}^{2}, \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)\right) \]

       4. pow2 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\mathsf{fma}\left(6, x1 \cdot x1, \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\mathsf{fma}\left(6, x1 \cdot x1, \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)\right) \]

       6. lower-/.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\mathsf{fma}\left(6, x1 \cdot x1, \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\mathsf{fma}\left(6, x1 \cdot x1, \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)\right) \]

       8. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\mathsf{fma}\left(6, x1 \cdot x1, \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)\right) \]

       9. lower-*.f64 86.0

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \left(\mathsf{fma}\left(6, x1 \cdot x1, \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)\right) \]

   13. Applied rewrites 86.0%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x2 \cdot \color{blue}{\left(\mathsf{fma}\left(6, x1 \cdot x1, \frac{x1 \cdot \left(9 \cdot x1 - 3\right)}{x2}\right) - 6\right)}\right) \]

   if 2.75000000000000013e-117 < x1

   1. Initial program 63.0%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 70.7

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

   10. Applied rewrites 70.7%

       \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot x1\right)\right)\right)\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 13: 84.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -4.1 \cdot 10^{+158}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right)\\ \mathbf{elif}\;x1 \leq -7 \cdot 10^{+76}:\\ \;\;\;\;x1 + \left(x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(t\_0 - x1\right)}{1 + x1 \cdot x1}, -6 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 \leq 3.6 \cdot 10^{-42}:\\ \;\;\;\;x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 3.0 (* x1 x1))))
   (if (<= x1 -4.1e+158)
     (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) (* x1 (- (* 9.0 x1) 3.0))))
     (if (<= x1 -7e+76)
       (+
        x1
        (+
         x1
         (fma 3.0 (/ (* (* x1 x1) (- t_0 x1)) (+ 1.0 (* x1 x1))) (* -6.0 x2))))
       (if (<= x1 3.6e-42)
         (+
          x1
          (+
           (+ (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))) x1)
           (fma -6.0 x2 (* x1 (- (* 3.0 (* x1 (- 3.0 (* -2.0 x2)))) 3.0)))))
         (+
          x1
          (+
           (+ (* 4.0 (* x1 (* x2 (- (* 2.0 x2) 3.0)))) x1)
           (fma -6.0 x2 (* x1 t_0)))))))))

C

double code(double x1, double x2) {
	double t_0 = 3.0 * (x1 * x1);
	double tmp;
	if (x1 <= -4.1e+158) {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + (x1 * ((9.0 * x1) - 3.0)));
	} else if (x1 <= -7e+76) {
		tmp = x1 + (x1 + fma(3.0, (((x1 * x1) * (t_0 - x1)) / (1.0 + (x1 * x1))), (-6.0 * x2)));
	} else if (x1 <= 3.6e-42) {
		tmp = x1 + (((x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))) + x1) + fma(-6.0, x2, (x1 * ((3.0 * (x1 * (3.0 - (-2.0 * x2)))) - 3.0))));
	} else {
		tmp = x1 + (((4.0 * (x1 * (x2 * ((2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, (x1 * t_0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(3.0 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -4.1e+158)
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))));
	elseif (x1 <= -7e+76)
		tmp = Float64(x1 + Float64(x1 + fma(3.0, Float64(Float64(Float64(x1 * x1) * Float64(t_0 - x1)) / Float64(1.0 + Float64(x1 * x1))), Float64(-6.0 * x2))));
	elseif (x1 <= 3.6e-42)
		tmp = Float64(x1 + Float64(Float64(Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))) + x1) + fma(-6.0, x2, Float64(x1 * Float64(Float64(3.0 * Float64(x1 * Float64(3.0 - Float64(-2.0 * x2)))) - 3.0)))));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(4.0 * Float64(x1 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, Float64(x1 * t_0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -4.1e+158], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -7e+76], N[(x1 + N[(x1 + N[(3.0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$0 - x1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3.6e-42], N[(x1 + N[(N[(N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * N[(N[(3.0 * N[(x1 * N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(4.0 * N[(x1 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -4.10000000000000004e158

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 51.6

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 74.4

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 74.4%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 80.1

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 80.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]

   if -4.10000000000000004e158 < x1 < -7.00000000000000001e76

   1. Initial program 33.2%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lift-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -7.00000000000000001e76 < x1 < 3.6000000000000002e-42

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 79.6

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 79.4

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \left(-12 \cdot x1 + \color{blue}{8 \cdot \left(x1 \cdot x2\right)}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      4. lift-*.f64 89.9

         \[\leadsto x1 + \left(\left(x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 89.9%

       \[\leadsto x1 + \left(\left(x2 \cdot \color{blue}{\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   if 3.6000000000000002e-42 < x1

   1. Initial program 54.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 23.5

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

   4. Applied rewrites 23.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 71.7

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 73.4

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

   10. Applied rewrites 73.4%

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

Alternative 14: 77.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -12 \cdot \left(x1 \cdot x2\right) + x1\\ t_1 := 3 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -4.1 \cdot 10^{+158}:\\ \;\;\;\;x1 + \left(t\_0 + x1 \cdot \left(9 \cdot x1 - 3\right)\right)\\ \mathbf{elif}\;x1 \leq -7.2 \cdot 10^{+76}:\\ \;\;\;\;x1 + \left(x1 + \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(t\_1 - x1\right)}{1 + x1 \cdot x1}, -6 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-206}:\\ \;\;\;\;x1 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right)\\ \mathbf{elif}\;x1 \leq 3.8 \cdot 10^{-117}:\\ \;\;\;\;x1 + \left(t\_0 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* -12.0 (* x1 x2)) x1)) (t_1 (* 3.0 (* x1 x1))))
   (if (<= x1 -4.1e+158)
     (+ x1 (+ t_0 (* x1 (- (* 9.0 x1) 3.0))))
     (if (<= x1 -7.2e+76)
       (+
        x1
        (+
         x1
         (fma 3.0 (/ (* (* x1 x1) (- t_1 x1)) (+ 1.0 (* x1 x1))) (* -6.0 x2))))
       (if (<= x1 -1.3e-206)
         (+
          x1
          (fma
           -6.0
           x2
           (*
            x1
            (- (fma x1 (+ 9.0 (* 12.0 x2)) (* x2 (- (* 8.0 x2) 12.0))) 2.0))))
         (if (<= x1 3.8e-117)
           (+ x1 (+ t_0 (fma -6.0 x2 (* x1 (- (* 3.0 (* x1 3.0)) 3.0)))))
           (+
            x1
            (+
             (+ (* 4.0 (* x1 (* x2 (- (* 2.0 x2) 3.0)))) x1)
             (fma -6.0 x2 (* x1 t_1))))))))))

C

double code(double x1, double x2) {
	double t_0 = (-12.0 * (x1 * x2)) + x1;
	double t_1 = 3.0 * (x1 * x1);
	double tmp;
	if (x1 <= -4.1e+158) {
		tmp = x1 + (t_0 + (x1 * ((9.0 * x1) - 3.0)));
	} else if (x1 <= -7.2e+76) {
		tmp = x1 + (x1 + fma(3.0, (((x1 * x1) * (t_1 - x1)) / (1.0 + (x1 * x1))), (-6.0 * x2)));
	} else if (x1 <= -1.3e-206) {
		tmp = x1 + fma(-6.0, x2, (x1 * (fma(x1, (9.0 + (12.0 * x2)), (x2 * ((8.0 * x2) - 12.0))) - 2.0)));
	} else if (x1 <= 3.8e-117) {
		tmp = x1 + (t_0 + fma(-6.0, x2, (x1 * ((3.0 * (x1 * 3.0)) - 3.0))));
	} else {
		tmp = x1 + (((4.0 * (x1 * (x2 * ((2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, (x1 * t_1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(-12.0 * Float64(x1 * x2)) + x1)
	t_1 = Float64(3.0 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -4.1e+158)
		tmp = Float64(x1 + Float64(t_0 + Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))));
	elseif (x1 <= -7.2e+76)
		tmp = Float64(x1 + Float64(x1 + fma(3.0, Float64(Float64(Float64(x1 * x1) * Float64(t_1 - x1)) / Float64(1.0 + Float64(x1 * x1))), Float64(-6.0 * x2))));
	elseif (x1 <= -1.3e-206)
		tmp = Float64(x1 + fma(-6.0, x2, Float64(x1 * Float64(fma(x1, Float64(9.0 + Float64(12.0 * x2)), Float64(x2 * Float64(Float64(8.0 * x2) - 12.0))) - 2.0))));
	elseif (x1 <= 3.8e-117)
		tmp = Float64(x1 + Float64(t_0 + fma(-6.0, x2, Float64(x1 * Float64(Float64(3.0 * Float64(x1 * 3.0)) - 3.0)))));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(4.0 * Float64(x1 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, Float64(x1 * t_1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -4.1e+158], N[(x1 + N[(t$95$0 + N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -7.2e+76], N[(x1 + N[(x1 + N[(3.0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$1 - x1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.3e-206], N[(x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x1 * N[(9.0 + N[(12.0 * x2), $MachinePrecision]), $MachinePrecision] + N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3.8e-117], N[(x1 + N[(t$95$0 + N[(-6.0 * x2 + N[(x1 * N[(N[(3.0 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(4.0 * N[(x1 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -4.10000000000000004e158

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 51.6

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 74.4

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 74.4%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 80.1

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 80.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]

   if -4.10000000000000004e158 < x1 < -7.2000000000000006e76

   1. Initial program 33.2%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lift-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -7.2000000000000006e76 < x1 < -1.3e-206

   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 x2 around 0

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot x2 + \color{blue}{x1 \cdot \left(\left(x1 \cdot \left(9 + 12 \cdot x2\right) + x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 74.0%

      \[\leadsto x1 + \mathsf{fma}\left(-6, \color{blue}{x2}, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right) \]

   if -1.3e-206 < x1 < 3.79999999999999972e-117

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 86.1

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 86.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 86.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right) \]

   if 3.79999999999999972e-117 < x1

   1. Initial program 63.0%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 70.7

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

   10. Applied rewrites 70.7%

       \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot x1\right)\right)\right)\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 15: 73.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -12 \cdot \left(x1 \cdot x2\right) + x1\\ \mathbf{if}\;x1 \leq -5.2 \cdot 10^{+87}:\\ \;\;\;\;x1 + \left(t\_0 + x1 \cdot \left(9 \cdot x1 - 3\right)\right)\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-206}:\\ \;\;\;\;x1 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right)\\ \mathbf{elif}\;x1 \leq 3.8 \cdot 10^{-117}:\\ \;\;\;\;x1 + \left(t\_0 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot x1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* -12.0 (* x1 x2)) x1)))
   (if (<= x1 -5.2e+87)
     (+ x1 (+ t_0 (* x1 (- (* 9.0 x1) 3.0))))
     (if (<= x1 -1.3e-206)
       (+
        x1
        (fma
         -6.0
         x2
         (*
          x1
          (- (fma x1 (+ 9.0 (* 12.0 x2)) (* x2 (- (* 8.0 x2) 12.0))) 2.0))))
       (if (<= x1 3.8e-117)
         (+ x1 (+ t_0 (fma -6.0 x2 (* x1 (- (* 3.0 (* x1 3.0)) 3.0)))))
         (+
          x1
          (+
           (+ (* 4.0 (* x1 (* x2 (- (* 2.0 x2) 3.0)))) x1)
           (fma -6.0 x2 (* x1 (* 3.0 (* x1 x1)))))))))))

C

double code(double x1, double x2) {
	double t_0 = (-12.0 * (x1 * x2)) + x1;
	double tmp;
	if (x1 <= -5.2e+87) {
		tmp = x1 + (t_0 + (x1 * ((9.0 * x1) - 3.0)));
	} else if (x1 <= -1.3e-206) {
		tmp = x1 + fma(-6.0, x2, (x1 * (fma(x1, (9.0 + (12.0 * x2)), (x2 * ((8.0 * x2) - 12.0))) - 2.0)));
	} else if (x1 <= 3.8e-117) {
		tmp = x1 + (t_0 + fma(-6.0, x2, (x1 * ((3.0 * (x1 * 3.0)) - 3.0))));
	} else {
		tmp = x1 + (((4.0 * (x1 * (x2 * ((2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, (x1 * (3.0 * (x1 * x1)))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(-12.0 * Float64(x1 * x2)) + x1)
	tmp = 0.0
	if (x1 <= -5.2e+87)
		tmp = Float64(x1 + Float64(t_0 + Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))));
	elseif (x1 <= -1.3e-206)
		tmp = Float64(x1 + fma(-6.0, x2, Float64(x1 * Float64(fma(x1, Float64(9.0 + Float64(12.0 * x2)), Float64(x2 * Float64(Float64(8.0 * x2) - 12.0))) - 2.0))));
	elseif (x1 <= 3.8e-117)
		tmp = Float64(x1 + Float64(t_0 + fma(-6.0, x2, Float64(x1 * Float64(Float64(3.0 * Float64(x1 * 3.0)) - 3.0)))));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(4.0 * Float64(x1 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))) + x1) + fma(-6.0, x2, Float64(x1 * Float64(3.0 * Float64(x1 * x1))))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -5.2e+87], N[(x1 + N[(t$95$0 + N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.3e-206], N[(x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x1 * N[(9.0 + N[(12.0 * x2), $MachinePrecision]), $MachinePrecision] + N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3.8e-117], N[(x1 + N[(t$95$0 + N[(-6.0 * x2 + N[(x1 * N[(N[(3.0 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(4.0 * N[(x1 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -5.19999999999999997e87

   1. Initial program 7.3%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 1.2

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 39.4

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

   7. Applied rewrites 39.4%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 61.6

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 61.6%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 63.2

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 63.2%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]

   if -5.19999999999999997e87 < x1 < -1.3e-206

   1. Initial program 99.1%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot x2 + \color{blue}{x1 \cdot \left(\left(x1 \cdot \left(9 + 12 \cdot x2\right) + x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 71.7%

      \[\leadsto x1 + \mathsf{fma}\left(-6, \color{blue}{x2}, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right) \]

   if -1.3e-206 < x1 < 3.79999999999999972e-117

   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

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 86.1

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 86.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 86.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot 3\right) - 3\right)\right)\right) \]

   if 3.79999999999999972e-117 < x1

   1. Initial program 63.0%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 70.7

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

   10. Applied rewrites 70.7%

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

Alternative 16: 76.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -5.2 \cdot 10^{+87}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right)\\ \mathbf{elif}\;x1 \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;x1 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(x1 \cdot \left(9 + 3 \cdot x1\right) - 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -5.2e+87)
   (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) (* x1 (- (* 9.0 x1) 3.0))))
   (if (<= x1 4.2e-19)
     (+
      x1
      (fma
       -6.0
       x2
       (* x1 (- (fma x1 (+ 9.0 (* 12.0 x2)) (* x2 (- (* 8.0 x2) 12.0))) 2.0))))
     (+
      x1
      (+
       (+ (* 4.0 (* x1 (* x2 (- (* 2.0 x2) 3.0)))) x1)
       (* x1 (- (* x1 (+ 9.0 (* 3.0 x1))) 3.0)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -5.2e+87) {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + (x1 * ((9.0 * x1) - 3.0)));
	} else if (x1 <= 4.2e-19) {
		tmp = x1 + fma(-6.0, x2, (x1 * (fma(x1, (9.0 + (12.0 * x2)), (x2 * ((8.0 * x2) - 12.0))) - 2.0)));
	} else {
		tmp = x1 + (((4.0 * (x1 * (x2 * ((2.0 * x2) - 3.0)))) + x1) + (x1 * ((x1 * (9.0 + (3.0 * x1))) - 3.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -5.2e+87)
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))));
	elseif (x1 <= 4.2e-19)
		tmp = Float64(x1 + fma(-6.0, x2, Float64(x1 * Float64(fma(x1, Float64(9.0 + Float64(12.0 * x2)), Float64(x2 * Float64(Float64(8.0 * x2) - 12.0))) - 2.0))));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(4.0 * Float64(x1 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))) + x1) + Float64(x1 * Float64(Float64(x1 * Float64(9.0 + Float64(3.0 * x1))) - 3.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -5.2e+87], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.2e-19], N[(x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x1 * N[(9.0 + N[(12.0 * x2), $MachinePrecision]), $MachinePrecision] + N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(4.0 * N[(x1 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(x1 * N[(N[(x1 * N[(9.0 + N[(3.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5.19999999999999997e87

   1. Initial program 7.3%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 1.2

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 39.4

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

   7. Applied rewrites 39.4%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 61.6

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 61.6%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 63.2

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 63.2%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]

   if -5.19999999999999997e87 < x1 < 4.1999999999999998e-19

   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 x2 around 0

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot x2 + \color{blue}{x1 \cdot \left(\left(x1 \cdot \left(9 + 12 \cdot x2\right) + x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 79.9%

      \[\leadsto x1 + \mathsf{fma}\left(-6, \color{blue}{x2}, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right) \]

   if 4.1999999999999998e-19 < x1

   1. Initial program 51.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 18.4

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 69.9

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

   7. Applied rewrites 69.9%

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

   8. Taylor expanded in x2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

         \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(x1 \cdot \left(9 + 3 \cdot x1\right) - 3\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(x1 \cdot \left(9 + 3 \cdot x1\right) - 3\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(x1 \cdot \left(9 + 3 \cdot x1\right) - 3\right)\right) \]

      5. lift-*.f64 76.3

         \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(x1 \cdot \left(9 + 3 \cdot x1\right) - 3\right)\right) \]

   10. Applied rewrites 76.3%

       \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + 3 \cdot x1\right) - 3\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 73.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(9 \cdot x1 - 3\right)\\ \mathbf{if}\;x1 \leq -5.2 \cdot 10^{+87}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + t\_0\right)\\ \mathbf{elif}\;x1 \leq 4.2 \cdot 10^{-19}:\\ \;\;\;\;x1 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (- (* 9.0 x1) 3.0))))
   (if (<= x1 -5.2e+87)
     (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) t_0))
     (if (<= x1 4.2e-19)
       (+
        x1
        (fma
         -6.0
         x2
         (*
          x1
          (- (fma x1 (+ 9.0 (* 12.0 x2)) (* x2 (- (* 8.0 x2) 12.0))) 2.0))))
       (+ x1 (+ (+ (* 4.0 (* x1 (* x2 (- (* 2.0 x2) 3.0)))) x1) t_0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * ((9.0 * x1) - 3.0);
	double tmp;
	if (x1 <= -5.2e+87) {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + t_0);
	} else if (x1 <= 4.2e-19) {
		tmp = x1 + fma(-6.0, x2, (x1 * (fma(x1, (9.0 + (12.0 * x2)), (x2 * ((8.0 * x2) - 12.0))) - 2.0)));
	} else {
		tmp = x1 + (((4.0 * (x1 * (x2 * ((2.0 * x2) - 3.0)))) + x1) + t_0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(Float64(9.0 * x1) - 3.0))
	tmp = 0.0
	if (x1 <= -5.2e+87)
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + t_0));
	elseif (x1 <= 4.2e-19)
		tmp = Float64(x1 + fma(-6.0, x2, Float64(x1 * Float64(fma(x1, Float64(9.0 + Float64(12.0 * x2)), Float64(x2 * Float64(Float64(8.0 * x2) - 12.0))) - 2.0))));
	else
		tmp = Float64(x1 + Float64(Float64(Float64(4.0 * Float64(x1 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0)))) + x1) + t_0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(N[(9.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -5.2e+87], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.2e-19], N[(x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x1 * N[(9.0 + N[(12.0 * x2), $MachinePrecision]), $MachinePrecision] + N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(N[(4.0 * N[(x1 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5.19999999999999997e87

   1. Initial program 7.3%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 1.2

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 39.4

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

   7. Applied rewrites 39.4%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 61.6

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 61.6%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - \color{blue}{3}\right)\right) \]

       2. lower--.f64 N/A

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

       3. lower-*.f64 63.2

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   13. Applied rewrites 63.2%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]

   if -5.19999999999999997e87 < x1 < 4.1999999999999998e-19

   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 x2 around 0

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot x2 + \color{blue}{x1 \cdot \left(\left(x1 \cdot \left(9 + 12 \cdot x2\right) + x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 79.9%

      \[\leadsto x1 + \mathsf{fma}\left(-6, \color{blue}{x2}, x1 \cdot \left(\mathsf{fma}\left(x1, 9 + 12 \cdot x2, x2 \cdot \left(8 \cdot x2 - 12\right)\right) - 2\right)\right) \]

   if 4.1999999999999998e-19 < x1

   1. Initial program 51.8%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 18.4

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 51.0

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

   7. Applied rewrites 51.0%

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

   8. Taylor expanded in x2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

         \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

      3. lower-*.f64 66.3

         \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \left(9 \cdot x1 - 3\right)\right) \]

   10. Applied rewrites 66.3%

       \[\leadsto x1 + \left(\left(4 \cdot \left(x1 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right)\right) + x1\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - 3\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 58.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.75 \cdot 10^{+62}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.75e+62)
   (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) (fma -6.0 x2 (* x1 -3.0))))
   (fma -6.0 x2 (* x1 (- (* 4.0 (* x2 (- (* 2.0 x2) 3.0))) 1.0)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.75e+62) {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + fma(-6.0, x2, (x1 * -3.0)));
	} else {
		tmp = fma(-6.0, x2, (x1 * ((4.0 * (x2 * ((2.0 * x2) - 3.0))) - 1.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.75e+62)
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + fma(-6.0, x2, Float64(x1 * -3.0))));
	else
		tmp = fma(-6.0, x2, Float64(x1 * Float64(Float64(4.0 * Float64(x2 * Float64(Float64(2.0 * x2) - 3.0))) - 1.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.75e+62], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-6.0 * x2 + N[(x1 * N[(N[(4.0 * N[(x2 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.74999999999999992e62

   1. Initial program 16.7%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 37.0

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 57.5

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 57.5%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]

   if -1.74999999999999992e62 < x1

   1. Initial program 83.5%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-6, x2, x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 58.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.75 \cdot 10^{+62}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.75e+62)
   (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) (fma -6.0 x2 (* x1 -3.0))))
   (+ x1 (fma -6.0 x2 (* x1 (- (* x2 (- (* 8.0 x2) 12.0)) 2.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.75e+62) {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + fma(-6.0, x2, (x1 * -3.0)));
	} else {
		tmp = x1 + fma(-6.0, x2, (x1 * ((x2 * ((8.0 * x2) - 12.0)) - 2.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.75e+62)
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + fma(-6.0, x2, Float64(x1 * -3.0))));
	else
		tmp = Float64(x1 + fma(-6.0, x2, Float64(x1 * Float64(Float64(x2 * Float64(Float64(8.0 * x2) - 12.0)) - 2.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.75e+62], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2 + N[(x1 * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.74999999999999992e62

   1. Initial program 16.7%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 37.0

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 57.5

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 57.5%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.1%

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]

   if -1.74999999999999992e62 < x1

   1. Initial program 83.5%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 83.5%

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

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot x2 + \color{blue}{x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 2\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 67.4%

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

Alternative 20: 57.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{+121}:\\ \;\;\;\;x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.8e+121)
   (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) (* -6.0 x2)))
   (+ x1 (fma -6.0 x2 (* x1 (- (* x2 (- (* 8.0 x2) 12.0)) 2.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.8e+121) {
		tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + (-6.0 * x2));
	} else {
		tmp = x1 + fma(-6.0, x2, (x1 * ((x2 * ((8.0 * x2) - 12.0)) - 2.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.8e+121)
		tmp = Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + Float64(-6.0 * x2)));
	else
		tmp = Float64(x1 + fma(-6.0, x2, Float64(x1 * Float64(Float64(x2 * Float64(Float64(8.0 * x2) - 12.0)) - 2.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.8e+121], N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.80000000000000006e121

   1. Initial program 0.2%

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

   2. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 0.2

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

   4. Applied rewrites 0.2%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 44.7

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

   7. Applied rewrites 44.7%

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

   8. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

      2. lift-*.f64 66.8

         \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   10. Applied rewrites 66.8%

       \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   11. Taylor expanded in x1 around 0

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

       1. lift-*.f64 19.3

          \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + -6 \cdot x2\right) \]

   13. Applied rewrites 19.3%

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

   if -2.80000000000000006e121 < x1

   1. Initial program 82.7%

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

   2. Taylor expanded in x2 around 0

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot x2 + \color{blue}{x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 2\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 64.3%

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

Alternative 21: 32.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + -6 \cdot x2\right) \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (+ x1 (+ (+ (* -12.0 (* x1 x2)) x1) (* -6.0 x2))))

C

double code(double x1, double x2) {
	return x1 + (((-12.0 * (x1 * x2)) + x1) + (-6.0 * x2));
}

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 + ((((-12.0d0) * (x1 * x2)) + x1) + ((-6.0d0) * x2))
end function

Java

public static double code(double x1, double x2) {
	return x1 + (((-12.0 * (x1 * x2)) + x1) + (-6.0 * x2));
}

Python

def code(x1, x2):
	return x1 + (((-12.0 * (x1 * x2)) + x1) + (-6.0 * x2))

Julia

function code(x1, x2)
	return Float64(x1 + Float64(Float64(Float64(-12.0 * Float64(x1 * x2)) + x1) + Float64(-6.0 * x2)))
end

MATLAB

function tmp = code(x1, x2)
	tmp = x1 + (((-12.0 * (x1 * x2)) + x1) + (-6.0 * x2));
end

Wolfram

code[x1_, x2_] := N[(x1 + N[(N[(N[(-12.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.1%

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

2. Taylor expanded in x1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lift-*.f64 49.1

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

4. Applied rewrites 49.1%

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

5. Taylor expanded in x1 around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 64.4

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

7. Applied rewrites 64.4%

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

8. Taylor expanded in x2 around 0

   \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot \color{blue}{x2}\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

   2. lift-*.f64 57.4

      \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

10. Applied rewrites 57.4%

    \[\leadsto x1 + \left(\left(-12 \cdot \color{blue}{\left(x1 \cdot x2\right)} + x1\right) + \mathsf{fma}\left(-6, x2, x1 \cdot \left(3 \cdot \left(x1 \cdot \left(3 - -2 \cdot x2\right)\right) - 3\right)\right)\right) \]

11. Taylor expanded in x1 around 0

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

    1. lift-*.f64 32.0

       \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + -6 \cdot x2\right) \]

13. Applied rewrites 32.0%

    \[\leadsto x1 + \left(\left(-12 \cdot \left(x1 \cdot x2\right) + x1\right) + -6 \cdot \color{blue}{x2}\right) \]
14. Add Preprocessing

Alternative 22: 26.0% accurate, 33.1× speedup

Math

\[\begin{array}{l} \\ x1 + -6 \cdot x2 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (+ x1 (* -6.0 x2)))

C

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

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 + ((-6.0d0) * x2)
end function

Java

public static double code(double x1, double x2) {
	return x1 + (-6.0 * x2);
}

Python

def code(x1, x2):
	return x1 + (-6.0 * x2)

Julia

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

MATLAB

function tmp = code(x1, x2)
	tmp = x1 + (-6.0 * x2);
end

Wolfram

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

Derivation

1. Initial program 70.1%

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

2. Taylor expanded in x1 around 0

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

   1. lower-*.f64 26.0

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

4. Applied rewrites 26.0%

   \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
5. Add Preprocessing

Alternative 23: 25.8% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ -6 \cdot x2 \end{array} \]

FPCore

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

C

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

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 = (-6.0d0) * x2
end function

Java

public static double code(double x1, double x2) {
	return -6.0 * x2;
}

Python

def code(x1, x2):
	return -6.0 * x2

Julia

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

MATLAB

function tmp = code(x1, x2)
	tmp = -6.0 * x2;
end

Wolfram

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

Derivation

1. Initial program 70.1%

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

2. Taylor expanded in x1 around 0

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

   1. lower-*.f64 25.8

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

4. Applied rewrites 25.8%

   \[\leadsto \color{blue}{-6 \cdot x2} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(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))))))