Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.9% → 98.6%

Time: 11.6s

Alternatives: 23

Speedup: 3.6×

• 46.9% 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

real(8) function code(x1, x2)
    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: 70.9% 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

real(8) function code(x1, x2)
    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: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \mathsf{fma}\left(x1 \cdot x1, 3, x2 \cdot 2\right) - x1\\ t_4 := \frac{t\_3}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_5 := \frac{t\_1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_6 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_7 := x1 \cdot x1 - -1\\ t_8 := \frac{t\_6}{t\_7}\\ t_9 := x1 - \left(\left(\left(\left(\frac{t\_6}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_8\right) \cdot \left(\left(2 \cdot x1\right) \cdot t\_8\right) - \left(4 \cdot t\_8 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_7} \cdot 3\right)\\ t_10 := \frac{t\_3 \cdot x1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;t\_9 \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_0\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_4, 4, -6\right) \cdot x1, x1, \mathsf{fma}\left(t\_4, 2, -6\right) \cdot t\_10\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(t\_10, 3, x1 \cdot x1\right), x1\right)\right) + x1\right)\\ \mathbf{elif}\;t\_9 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(t\_5, 4, -6\right), \left(t\_5 - 3\right) \cdot \frac{2 \cdot x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{t\_1}}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(t\_1 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 \cdot x1\right), x1, \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (fma (* x1 x1) 3.0 (fma 2.0 x2 (- x1))))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (fma (* x1 x1) 3.0 (* x2 2.0)) x1))
        (t_4 (/ t_3 (fma x1 x1 1.0)))
        (t_5 (/ t_1 (fma x1 x1 1.0)))
        (t_6 (- (+ (* x2 2.0) t_0) x1))
        (t_7 (- (* x1 x1) -1.0))
        (t_8 (/ t_6 t_7))
        (t_9
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_6 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_8) (* (* 2.0 x1) t_8))
                (* (- (* 4.0 t_8) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_7) 3.0))))
        (t_10 (/ (* t_3 x1) (fma x1 x1 1.0))))
   (if (<= t_9 2e+239)
     (fma
      (/ (- (fma -2.0 x2 t_0) x1) (fma x1 x1 1.0))
      3.0
      (+
       (fma
        (fma (* (fma t_4 4.0 -6.0) x1) x1 (* (fma t_4 2.0 -6.0) t_10))
        (fma x1 x1 1.0)
        (fma x1 (fma t_10 3.0 (* x1 x1)) x1))
       x1))
     (if (<= t_9 INFINITY)
       (+
        (fma
         (fma
          (* x1 x1)
          (fma t_5 4.0 -6.0)
          (* (- t_5 3.0) (/ (* 2.0 x1) (/ (fma x1 x1 1.0) t_1))))
         (fma x1 x1 1.0)
         (fma
          (fma (* t_1 (/ x1 (fma x1 x1 1.0))) 3.0 (* x1 x1))
          x1
          (fma 3.0 3.0 x1)))
        x1)
       (+ (* (pow x1 4.0) (- 6.0 (/ 3.0 x1))) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = fma((x1 * x1), 3.0, fma(2.0, x2, -x1));
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = fma((x1 * x1), 3.0, (x2 * 2.0)) - x1;
	double t_4 = t_3 / fma(x1, x1, 1.0);
	double t_5 = t_1 / fma(x1, x1, 1.0);
	double t_6 = ((x2 * 2.0) + t_0) - x1;
	double t_7 = (x1 * x1) - -1.0;
	double t_8 = t_6 / t_7;
	double t_9 = x1 - ((((((t_6 / t_2) * t_0) - (t_2 * (((3.0 - t_8) * ((2.0 * x1) * t_8)) - (((4.0 * t_8) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_7) * 3.0));
	double t_10 = (t_3 * x1) / fma(x1, x1, 1.0);
	double tmp;
	if (t_9 <= 2e+239) {
		tmp = fma(((fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, (fma(fma((fma(t_4, 4.0, -6.0) * x1), x1, (fma(t_4, 2.0, -6.0) * t_10)), fma(x1, x1, 1.0), fma(x1, fma(t_10, 3.0, (x1 * x1)), x1)) + x1));
	} else if (t_9 <= ((double) INFINITY)) {
		tmp = fma(fma((x1 * x1), fma(t_5, 4.0, -6.0), ((t_5 - 3.0) * ((2.0 * x1) / (fma(x1, x1, 1.0) / t_1)))), fma(x1, x1, 1.0), fma(fma((t_1 * (x1 / fma(x1, x1, 1.0))), 3.0, (x1 * x1)), x1, fma(3.0, 3.0, x1))) + x1;
	} else {
		tmp = (pow(x1, 4.0) * (6.0 - (3.0 / x1))) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = fma(Float64(x1 * x1), 3.0, fma(2.0, x2, Float64(-x1)))
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(fma(Float64(x1 * x1), 3.0, Float64(x2 * 2.0)) - x1)
	t_4 = Float64(t_3 / fma(x1, x1, 1.0))
	t_5 = Float64(t_1 / fma(x1, x1, 1.0))
	t_6 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_7 = Float64(Float64(x1 * x1) - -1.0)
	t_8 = Float64(t_6 / t_7)
	t_9 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_6 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_8) * Float64(Float64(2.0 * x1) * t_8)) - Float64(Float64(Float64(4.0 * t_8) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_7) * 3.0)))
	t_10 = Float64(Float64(t_3 * x1) / fma(x1, x1, 1.0))
	tmp = 0.0
	if (t_9 <= 2e+239)
		tmp = fma(Float64(Float64(fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, Float64(fma(fma(Float64(fma(t_4, 4.0, -6.0) * x1), x1, Float64(fma(t_4, 2.0, -6.0) * t_10)), fma(x1, x1, 1.0), fma(x1, fma(t_10, 3.0, Float64(x1 * x1)), x1)) + x1));
	elseif (t_9 <= Inf)
		tmp = Float64(fma(fma(Float64(x1 * x1), fma(t_5, 4.0, -6.0), Float64(Float64(t_5 - 3.0) * Float64(Float64(2.0 * x1) / Float64(fma(x1, x1, 1.0) / t_1)))), fma(x1, x1, 1.0), fma(fma(Float64(t_1 * Float64(x1 / fma(x1, x1, 1.0))), 3.0, Float64(x1 * x1)), x1, fma(3.0, 3.0, x1))) + x1);
	else
		tmp = Float64(Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1))) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$6 / t$95$7), $MachinePrecision]}, Block[{t$95$9 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$6 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$8), $MachinePrecision] * N[(N[(2.0 * x1), $MachinePrecision] * t$95$8), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$8), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$7), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(t$95$3 * x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$9, 2e+239], N[(N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(N[(N[(t$95$4 * 4.0 + -6.0), $MachinePrecision] * x1), $MachinePrecision] * x1 + N[(N[(t$95$4 * 2.0 + -6.0), $MachinePrecision] * t$95$10), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(x1 * N[(t$95$10 * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$9, Infinity], N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$5 * 4.0 + -6.0), $MachinePrecision] + N[(N[(t$95$5 - 3.0), $MachinePrecision] * N[(N[(2.0 * x1), $MachinePrecision] / N[(N[(x1 * x1 + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(t$95$1 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(3.0 * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.99999999999999998e239

   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. Add Preprocessing

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

   if 1.99999999999999998e239 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.9%

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

   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. Add Preprocessing

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, x2 \cdot 2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right) \cdot x1, x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, x2 \cdot 2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 2, -6\right) \cdot \frac{\left(\mathsf{fma}\left(x1 \cdot x1, 3, x2 \cdot 2\right) - x1\right) \cdot x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(x1 \cdot x1, 3, x2 \cdot 2\right) - x1\right) \cdot x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 \cdot x1\right), x1\right)\right) + x1\right)\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \frac{2 \cdot x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right) \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 \cdot x1\right), x1, \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \frac{\mathsf{fma}\left(x2, 2, t\_0\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\\ t_6 := \frac{t\_3}{t\_4}\\ t_7 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_6\right) \cdot \left(\left(2 \cdot x1\right) \cdot t\_6\right) - \left(4 \cdot t\_6 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - t\_5\right)\\ \mathbf{if}\;t\_7 \leq 0.05:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + t\_5\right) + x1\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\frac{x2}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot -2, 3, x1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4, t\_1, -6\right), x1 \cdot x1, \left(\left(2 \cdot x1\right) \cdot t\_1\right) \cdot \left(t\_1 - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), t\_1 \cdot t\_0\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (/ (- (fma x2 2.0 t_0) x1) (fma x1 x1 1.0)))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (+ (* x2 2.0) t_0) x1))
        (t_4 (- (* x1 x1) -1.0))
        (t_5 (* (/ (- (- t_0 (* x2 2.0)) x1) t_4) 3.0))
        (t_6 (/ t_3 t_4))
        (t_7
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_3 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_6) (* (* 2.0 x1) t_6))
                (* (- (* 4.0 t_6) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           t_5))))
   (if (<= t_7 0.05)
     (+ (+ (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1) t_5) x1)
     (if (<= t_7 INFINITY)
       (+
        (fma
         (* x1 x1)
         x1
         (+
          (fma (* (/ x2 (fma x1 x1 1.0)) -2.0) 3.0 x1)
          (fma
           (fma
            (fma 4.0 t_1 -6.0)
            (* x1 x1)
            (* (* (* 2.0 x1) t_1) (- t_1 3.0)))
           (fma x1 x1 1.0)
           (* t_1 t_0))))
        x1)
       (+ (* (pow x1 4.0) (- 6.0 (/ 3.0 x1))) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (fma(x2, 2.0, t_0) - x1) / fma(x1, x1, 1.0);
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = (((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0;
	double t_6 = t_3 / t_4;
	double t_7 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_6) * ((2.0 * x1) * t_6)) - (((4.0 * t_6) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - t_5);
	double tmp;
	if (t_7 <= 0.05) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + t_5) + x1;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = fma((x1 * x1), x1, (fma(((x2 / fma(x1, x1, 1.0)) * -2.0), 3.0, x1) + fma(fma(fma(4.0, t_1, -6.0), (x1 * x1), (((2.0 * x1) * t_1) * (t_1 - 3.0))), fma(x1, x1, 1.0), (t_1 * t_0)))) + x1;
	} else {
		tmp = (pow(x1, 4.0) * (6.0 - (3.0 / x1))) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(fma(x2, 2.0, t_0) - x1) / fma(x1, x1, 1.0))
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)
	t_6 = Float64(t_3 / t_4)
	t_7 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_6) * Float64(Float64(2.0 * x1) * t_6)) - Float64(Float64(Float64(4.0 * t_6) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - t_5))
	tmp = 0.0
	if (t_7 <= 0.05)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + t_5) + x1);
	elseif (t_7 <= Inf)
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(fma(Float64(Float64(x2 / fma(x1, x1, 1.0)) * -2.0), 3.0, x1) + fma(fma(fma(4.0, t_1, -6.0), Float64(x1 * x1), Float64(Float64(Float64(2.0 * x1) * t_1) * Float64(t_1 - 3.0))), fma(x1, x1, 1.0), Float64(t_1 * t_0)))) + x1);
	else
		tmp = Float64(Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1))) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x2 * 2.0 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$6), $MachinePrecision] * N[(N[(2.0 * x1), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$6), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.05], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + t$95$5), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(x2 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 3.0 + x1), $MachinePrecision] + N[(N[(N[(4.0 * t$95$1 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$1 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 0.050000000000000003

   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. Add Preprocessing

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if 0.050000000000000003 < (+.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 99.5%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

   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. Add Preprocessing

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 0.05:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\frac{x2}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot -2, 3, x1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, -6\right), x1 \cdot x1, \left(\left(2 \cdot x1\right) \cdot \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(3 \cdot x1, x1, x2 \cdot 2\right) - x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot t\_0\\ t_4 := \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_5 := \mathsf{fma}\left(t\_4, 2, -6\right)\\ t_6 := \mathsf{fma}\left(t\_4, 4, -6\right)\\ t_7 := \left(x2 \cdot 2 + t\_1\right) - x1\\ t_8 := x1 \cdot x1 - -1\\ t_9 := \frac{t\_7}{t\_8}\\ \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{t\_7}{t\_2} \cdot t\_1 - t\_2 \cdot \left(\left(3 - t\_9\right) \cdot \left(\left(2 \cdot x1\right) \cdot t\_9\right) - \left(4 \cdot t\_9 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_1 - x2 \cdot 2\right) - x1}{t\_8} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(t\_6, x1 \cdot x1, t\_3 \cdot t\_5\right), x1 \cdot x1, \mathsf{fma}\left(t\_6, x1 \cdot x1, \mathsf{fma}\left(t\_5, t\_3, \mathsf{fma}\left(3 \cdot x1, t\_3, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \mathsf{fma}\left(3 \cdot x1, x1, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right)\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (fma (* 3.0 x1) x1 (* x2 2.0)) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (* (/ x1 (fma x1 x1 1.0)) t_0))
        (t_4 (/ t_0 (fma x1 x1 1.0)))
        (t_5 (fma t_4 2.0 -6.0))
        (t_6 (fma t_4 4.0 -6.0))
        (t_7 (- (+ (* x2 2.0) t_1) x1))
        (t_8 (- (* x1 x1) -1.0))
        (t_9 (/ t_7 t_8)))
   (if (<=
        (-
         x1
         (-
          (-
           (-
            (-
             (* (/ t_7 t_2) t_1)
             (*
              t_2
              (-
               (* (- 3.0 t_9) (* (* 2.0 x1) t_9))
               (* (- (* 4.0 t_9) 6.0) (* x1 x1)))))
            (* (* x1 x1) x1))
           x1)
          (* (/ (- (- t_1 (* x2 2.0)) x1) t_8) 3.0)))
        INFINITY)
     (+
      (fma
       (* x1 x1)
       x1
       (fma
        (fma t_6 (* x1 x1) (* t_3 t_5))
        (* x1 x1)
        (fma
         t_6
         (* x1 x1)
         (fma
          t_5
          t_3
          (fma
           (* 3.0 x1)
           t_3
           (fma
            (/ (fma -2.0 x2 (fma (* 3.0 x1) x1 (- x1))) (fma x1 x1 1.0))
            3.0
            x1))))))
      x1)
     (+ (* (pow x1 4.0) (- 6.0 (/ 3.0 x1))) x1))))

C

double code(double x1, double x2) {
	double t_0 = fma((3.0 * x1), x1, (x2 * 2.0)) - x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = (x1 / fma(x1, x1, 1.0)) * t_0;
	double t_4 = t_0 / fma(x1, x1, 1.0);
	double t_5 = fma(t_4, 2.0, -6.0);
	double t_6 = fma(t_4, 4.0, -6.0);
	double t_7 = ((x2 * 2.0) + t_1) - x1;
	double t_8 = (x1 * x1) - -1.0;
	double t_9 = t_7 / t_8;
	double tmp;
	if ((x1 - ((((((t_7 / t_2) * t_1) - (t_2 * (((3.0 - t_9) * ((2.0 * x1) * t_9)) - (((4.0 * t_9) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_1 - (x2 * 2.0)) - x1) / t_8) * 3.0))) <= ((double) INFINITY)) {
		tmp = fma((x1 * x1), x1, fma(fma(t_6, (x1 * x1), (t_3 * t_5)), (x1 * x1), fma(t_6, (x1 * x1), fma(t_5, t_3, fma((3.0 * x1), t_3, fma((fma(-2.0, x2, fma((3.0 * x1), x1, -x1)) / fma(x1, x1, 1.0)), 3.0, x1)))))) + x1;
	} else {
		tmp = (pow(x1, 4.0) * (6.0 - (3.0 / x1))) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(Float64(3.0 * x1), x1, Float64(x2 * 2.0)) - x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(x1 / fma(x1, x1, 1.0)) * t_0)
	t_4 = Float64(t_0 / fma(x1, x1, 1.0))
	t_5 = fma(t_4, 2.0, -6.0)
	t_6 = fma(t_4, 4.0, -6.0)
	t_7 = Float64(Float64(Float64(x2 * 2.0) + t_1) - x1)
	t_8 = Float64(Float64(x1 * x1) - -1.0)
	t_9 = Float64(t_7 / t_8)
	tmp = 0.0
	if (Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_7 / t_2) * t_1) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_9) * Float64(Float64(2.0 * x1) * t_9)) - Float64(Float64(Float64(4.0 * t_9) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_1 - Float64(x2 * 2.0)) - x1) / t_8) * 3.0))) <= Inf)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(fma(t_6, Float64(x1 * x1), Float64(t_3 * t_5)), Float64(x1 * x1), fma(t_6, Float64(x1 * x1), fma(t_5, t_3, fma(Float64(3.0 * x1), t_3, fma(Float64(fma(-2.0, x2, fma(Float64(3.0 * x1), x1, Float64(-x1))) / fma(x1, x1, 1.0)), 3.0, x1)))))) + x1);
	else
		tmp = Float64(Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1))) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(3.0 * x1), $MachinePrecision] * x1 + N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * 2.0 + -6.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * 4.0 + -6.0), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$1), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$8 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$7 / t$95$8), $MachinePrecision]}, If[LessEqual[N[(x1 - N[(N[(N[(N[(N[(N[(t$95$7 / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$9), $MachinePrecision] * N[(N[(2.0 * x1), $MachinePrecision] * t$95$9), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$9), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$1 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$8), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(t$95$6 * N[(x1 * x1), $MachinePrecision] + N[(t$95$3 * t$95$5), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(t$95$6 * N[(x1 * x1), $MachinePrecision] + N[(t$95$5 * t$95$3 + N[(N[(3.0 * x1), $MachinePrecision] * t$95$3 + N[(N[(N[(-2.0 * x2 + N[(N[(3.0 * x1), $MachinePrecision] * x1 + (-x1)), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 95.9%

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

   8. Applied rewrites 99.6%

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

   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. Add Preprocessing

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 \cdot 2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), x1 \cdot x1, \left(\frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(\mathsf{fma}\left(3 \cdot x1, x1, x2 \cdot 2\right) - x1\right)\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 \cdot 2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 2, -6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 \cdot 2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(3 \cdot x1, x1, x2 \cdot 2\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 2, -6\right), \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(\mathsf{fma}\left(3 \cdot x1, x1, x2 \cdot 2\right) - x1\right), \mathsf{fma}\left(3 \cdot x1, \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(\mathsf{fma}\left(3 \cdot x1, x1, x2 \cdot 2\right) - x1\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \mathsf{fma}\left(3 \cdot x1, x1, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right)\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   4. Applied rewrites 99.6%

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

   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. Add Preprocessing

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, -6\right), x1 \cdot x1, \left(\left(2 \cdot x1\right) \cdot \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4} + x1\\ t_1 := \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_2 := \mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)\\ t_3 := \frac{t\_2}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -4.2 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -0.52:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(t\_3, 4, -6\right), \left(t\_3 - 3\right) \cdot \frac{2 \cdot x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{t\_2}}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(t\_2 \cdot t\_1, 3, x1 \cdot x1\right), x1, \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot t\_1\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))
           (pow x1 4.0))
          x1))
        (t_1 (/ x1 (fma x1 x1 1.0)))
        (t_2 (fma (* x1 x1) 3.0 (fma 2.0 x2 (- x1))))
        (t_3 (/ t_2 (fma x1 x1 1.0))))
   (if (<= x1 -4.2e+59)
     t_0
     (if (<= x1 -0.52)
       (+
        (fma
         (fma
          (* x1 x1)
          (fma t_3 4.0 -6.0)
          (* (- t_3 3.0) (/ (* 2.0 x1) (/ (fma x1 x1 1.0) t_2))))
         (fma x1 x1 1.0)
         (fma (fma (* t_2 t_1) 3.0 (* x1 x1)) x1 (fma 3.0 3.0 x1)))
        x1)
       (if (<= x1 1150000.0)
         (+
          (+
           (+ (* (* (* 8.0 t_1) x2) x2) x1)
           (*
            (/ (- (- (* (* 3.0 x1) x1) (* x2 2.0)) x1) (- (* x1 x1) -1.0))
            3.0))
          x1)
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * pow(x1, 4.0)) + x1;
	double t_1 = x1 / fma(x1, x1, 1.0);
	double t_2 = fma((x1 * x1), 3.0, fma(2.0, x2, -x1));
	double t_3 = t_2 / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -4.2e+59) {
		tmp = t_0;
	} else if (x1 <= -0.52) {
		tmp = fma(fma((x1 * x1), fma(t_3, 4.0, -6.0), ((t_3 - 3.0) * ((2.0 * x1) / (fma(x1, x1, 1.0) / t_2)))), fma(x1, x1, 1.0), fma(fma((t_2 * t_1), 3.0, (x1 * x1)), x1, fma(3.0, 3.0, x1))) + x1;
	} else if (x1 <= 1150000.0) {
		tmp = (((((8.0 * t_1) * x2) * x2) + x1) + ((((((3.0 * x1) * x1) - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * (x1 ^ 4.0)) + x1)
	t_1 = Float64(x1 / fma(x1, x1, 1.0))
	t_2 = fma(Float64(x1 * x1), 3.0, fma(2.0, x2, Float64(-x1)))
	t_3 = Float64(t_2 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -4.2e+59)
		tmp = t_0;
	elseif (x1 <= -0.52)
		tmp = Float64(fma(fma(Float64(x1 * x1), fma(t_3, 4.0, -6.0), Float64(Float64(t_3 - 3.0) * Float64(Float64(2.0 * x1) / Float64(fma(x1, x1, 1.0) / t_2)))), fma(x1, x1, 1.0), fma(fma(Float64(t_2 * t_1), 3.0, Float64(x1 * x1)), x1, fma(3.0, 3.0, x1))) + x1);
	elseif (x1 <= 1150000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * t_1) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -4.2e+59], t$95$0, If[LessEqual[x1, -0.52], N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$3 * 4.0 + -6.0), $MachinePrecision] + N[(N[(t$95$3 - 3.0), $MachinePrecision] * N[(N[(2.0 * x1), $MachinePrecision] / N[(N[(x1 * x1 + 1.0), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(t$95$2 * t$95$1), $MachinePrecision] * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(3.0 * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1150000.0], N[(N[(N[(N[(N[(N[(8.0 * t$95$1), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.19999999999999968e59 or 1.15e6 < x1

   1. Initial program 38.1%

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

   4. Applied rewrites 46.5%

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

   5. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.1%

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

   if -4.19999999999999968e59 < x1 < -0.52000000000000002

   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. Add Preprocessing

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 96.5%

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

   9. Applied rewrites 96.8%

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

   if -0.52000000000000002 < x1 < 1.15e6

   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. Add Preprocessing

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.2 \cdot 10^{+59}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4} + x1\\ \mathbf{elif}\;x1 \leq -0.52:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \frac{2 \cdot x1}{\frac{\mathsf{fma}\left(x1, x1, 1\right)}{\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right)}}\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, 3, \mathsf{fma}\left(2, x2, -x1\right)\right) \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 \cdot x1\right), x1, \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4} + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4} + x1\\ \mathbf{if}\;x1 \leq -460000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))
           (pow x1 4.0))
          x1)))
   (if (<= x1 -460000.0)
     t_0
     (if (<= x1 1150000.0)
       (+
        (+
         (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
         (*
          (/ (- (- (* (* 3.0 x1) x1) (* x2 2.0)) x1) (- (* x1 x1) -1.0))
          3.0))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * pow(x1, 4.0)) + x1;
	double tmp;
	if (x1 <= -460000.0) {
		tmp = t_0;
	} else if (x1 <= 1150000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((((3.0 * x1) * x1) - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * (x1 ^ 4.0)) + x1)
	tmp = 0.0
	if (x1 <= -460000.0)
		tmp = t_0;
	elseif (x1 <= 1150000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -460000.0], t$95$0, If[LessEqual[x1, 1150000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -4.6e5 or 1.15e6 < x1

   1. Initial program 44.5%

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

   4. Applied rewrites 52.0%

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

   5. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 96.5%

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

   if -4.6e5 < x1 < 1.15e6

   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. Add Preprocessing

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -460000:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4} + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4} + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \left(6 - \frac{\frac{3}{x1} + 4}{x1}\right) \cdot {x1}^{4}\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -3.1e+17)
   (+ (* (pow x1 4.0) (- 6.0 (/ 3.0 x1))) x1)
   (if (<= x1 1150000.0)
     (+
      (+
       (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
       (* (/ (- (- (* (* 3.0 x1) x1) (* x2 2.0)) x1) (- (* x1 x1) -1.0)) 3.0))
      x1)
     (+
      (fma (* x1 x1) x1 (* (- 6.0 (/ (+ (/ 3.0 x1) 4.0) x1)) (pow x1 4.0)))
      x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.1e+17) {
		tmp = (pow(x1, 4.0) * (6.0 - (3.0 / x1))) + x1;
	} else if (x1 <= 1150000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((((3.0 * x1) * x1) - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = fma((x1 * x1), x1, ((6.0 - (((3.0 / x1) + 4.0) / x1)) * pow(x1, 4.0))) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3.1e+17)
		tmp = Float64(Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1))) + x1);
	elseif (x1 <= 1150000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(Float64(6.0 - Float64(Float64(Float64(3.0 / x1) + 4.0) / x1)) * (x1 ^ 4.0))) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3.1e+17], N[(N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1150000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(6.0 - N[(N[(N[(3.0 / x1), $MachinePrecision] + 4.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.1e17

   1. Initial program 35.7%

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if -3.1e17 < x1 < 1.15e6

   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. Add Preprocessing

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 1.15e6 < x1

   1. Initial program 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x2 around 0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 95.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \left(6 - \frac{\frac{3}{x1} + 4}{x1}\right) \cdot {x1}^{4}\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_0\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 22500000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, {x1}^{4} \cdot 6\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1)))
   (if (<= x1 -2e+131)
     (+
      (-
       (* (fma (fma -4.0 x2 (* -22.0 x1)) x1 (fma -12.0 x2 1.0)) x1)
       (* (/ (* -2.0 x2) (- -1.0 (* x1 x1))) 3.0))
      x1)
     (if (<= x1 -3.1e+17)
       (+
        (fma
         (fma x1 x1 1.0)
         (* 6.0 (* x1 x1))
         (fma
          x1
          (* (* x2 x1) 6.0)
          (fma (/ (- (fma -2.0 x2 t_0) x1) (fma x1 x1 1.0)) 3.0 x1)))
        x1)
       (if (<= x1 22500000.0)
         (+
          (+
           (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) (- (* x1 x1) -1.0)) 3.0))
          x1)
         (+ (fma (* x1 x1) x1 (* (pow x1 4.0) 6.0)) x1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double tmp;
	if (x1 <= -2e+131) {
		tmp = ((fma(fma(-4.0, x2, (-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - (((-2.0 * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= -3.1e+17) {
		tmp = fma(fma(x1, x1, 1.0), (6.0 * (x1 * x1)), fma(x1, ((x2 * x1) * 6.0), fma(((fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, x1))) + x1;
	} else if (x1 <= 22500000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((t_0 - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = fma((x1 * x1), x1, (pow(x1, 4.0) * 6.0)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	tmp = 0.0
	if (x1 <= -2e+131)
		tmp = Float64(Float64(Float64(fma(fma(-4.0, x2, Float64(-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - Float64(Float64(Float64(-2.0 * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= -3.1e+17)
		tmp = Float64(fma(fma(x1, x1, 1.0), Float64(6.0 * Float64(x1 * x1)), fma(x1, Float64(Float64(x2 * x1) * 6.0), fma(Float64(Float64(fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, x1))) + x1);
	elseif (x1 <= 22500000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64((x1 ^ 4.0) * 6.0)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, If[LessEqual[x1, -2e+131], N[(N[(N[(N[(N[(-4.0 * x2 + N[(-22.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(-12.0 * x2 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -3.1e+17], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x2 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + N[(N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 22500000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.9999999999999998e131

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.8%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 78.4

          \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   11. Applied rewrites 78.4%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, 1 + -12 \cdot x2\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.3%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]

   if -1.9999999999999998e131 < x1 < -3.1e17

   1. Initial program 84.6%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 78.9

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

   10. Applied rewrites 78.9%

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

   if -3.1e17 < x1 < 2.25e7

   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. Add Preprocessing

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 2.25e7 < x1

   1. Initial program 51.2%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 94.4

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

   7. Applied rewrites 94.4%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 22500000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, {x1}^{4} \cdot 6\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \mathbf{if}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (pow x1 4.0) (- 6.0 (/ 3.0 x1))) x1)))
   (if (<= x1 -3.1e+17)
     t_0
     (if (<= x1 1150000.0)
       (+
        (+
         (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
         (*
          (/ (- (- (* (* 3.0 x1) x1) (* x2 2.0)) x1) (- (* x1 x1) -1.0))
          3.0))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (pow(x1, 4.0) * (6.0 - (3.0 / x1))) + x1;
	double tmp;
	if (x1 <= -3.1e+17) {
		tmp = t_0;
	} else if (x1 <= 1150000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((((3.0 * x1) * x1) - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / x1))) + x1)
	tmp = 0.0
	if (x1 <= -3.1e+17)
		tmp = t_0;
	elseif (x1 <= 1150000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -3.1e+17], t$95$0, If[LessEqual[x1, 1150000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.1e17 or 1.15e6 < x1

   1. Initial program 43.7%

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 92.9

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

   7. Applied rewrites 92.9%

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

   if -3.1e17 < x1 < 1.15e6

   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. Add Preprocessing

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3}{x1}\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x1 \cdot x1\right)\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\\ \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), t\_0, \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, t\_2\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 22500000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(t\_1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), t\_0, \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x2, -2 \cdot x1\right) \cdot x1, t\_2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x1 x1)))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (fma (/ (- (fma -2.0 x2 t_1) x1) (fma x1 x1 1.0)) 3.0 x1)))
   (if (<= x1 -2e+131)
     (+
      (-
       (* (fma (fma -4.0 x2 (* -22.0 x1)) x1 (fma -12.0 x2 1.0)) x1)
       (* (/ (* -2.0 x2) (- -1.0 (* x1 x1))) 3.0))
      x1)
     (if (<= x1 -3.1e+17)
       (+ (fma (fma x1 x1 1.0) t_0 (fma x1 (* (* x2 x1) 6.0) t_2)) x1)
       (if (<= x1 22500000.0)
         (+
          (+
           (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
           (* (/ (- (- t_1 (* x2 2.0)) x1) (- (* x1 x1) -1.0)) 3.0))
          x1)
         (if (<= x1 5.6e+102)
           (+
            (fma
             (fma x1 x1 1.0)
             t_0
             (fma x1 (* (fma 6.0 x2 (* -2.0 x1)) x1) t_2))
            x1)
           (fma (* x1 x1) x1 (* -6.0 x2))))))))

C

double code(double x1, double x2) {
	double t_0 = 6.0 * (x1 * x1);
	double t_1 = (3.0 * x1) * x1;
	double t_2 = fma(((fma(-2.0, x2, t_1) - x1) / fma(x1, x1, 1.0)), 3.0, x1);
	double tmp;
	if (x1 <= -2e+131) {
		tmp = ((fma(fma(-4.0, x2, (-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - (((-2.0 * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= -3.1e+17) {
		tmp = fma(fma(x1, x1, 1.0), t_0, fma(x1, ((x2 * x1) * 6.0), t_2)) + x1;
	} else if (x1 <= 22500000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((t_1 - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else if (x1 <= 5.6e+102) {
		tmp = fma(fma(x1, x1, 1.0), t_0, fma(x1, (fma(6.0, x2, (-2.0 * x1)) * x1), t_2)) + x1;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(6.0 * Float64(x1 * x1))
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = fma(Float64(Float64(fma(-2.0, x2, t_1) - x1) / fma(x1, x1, 1.0)), 3.0, x1)
	tmp = 0.0
	if (x1 <= -2e+131)
		tmp = Float64(Float64(Float64(fma(fma(-4.0, x2, Float64(-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - Float64(Float64(Float64(-2.0 * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= -3.1e+17)
		tmp = Float64(fma(fma(x1, x1, 1.0), t_0, fma(x1, Float64(Float64(x2 * x1) * 6.0), t_2)) + x1);
	elseif (x1 <= 22500000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(t_1 - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	elseif (x1 <= 5.6e+102)
		tmp = Float64(fma(fma(x1, x1, 1.0), t_0, fma(x1, Float64(fma(6.0, x2, Float64(-2.0 * x1)) * x1), t_2)) + x1);
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-2.0 * x2 + t$95$1), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]}, If[LessEqual[x1, -2e+131], N[(N[(N[(N[(N[(-4.0 * x2 + N[(-22.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(-12.0 * x2 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -3.1e+17], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * t$95$0 + N[(x1 * N[(N[(x2 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 22500000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(t$95$1 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 5.6e+102], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * t$95$0 + N[(x1 * N[(N[(6.0 * x2 + N[(-2.0 * x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -1.9999999999999998e131

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.8%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 78.4

          \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   11. Applied rewrites 78.4%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, 1 + -12 \cdot x2\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.3%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]

   if -1.9999999999999998e131 < x1 < -3.1e17

   1. Initial program 84.6%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 78.9

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

   10. Applied rewrites 78.9%

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

   if -3.1e17 < x1 < 2.25e7

   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. Add Preprocessing

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 2.25e7 < x1 < 5.60000000000000037e102

   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. Add Preprocessing

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

   if 5.60000000000000037e102 < x1

   1. Initial program 17.5%

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot x1}, x1, -6 \cdot x2\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 22500000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(6, x2, -2 \cdot x1\right) \cdot x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_0\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 22500000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1
         (+
          (fma
           (fma x1 x1 1.0)
           (* 6.0 (* x1 x1))
           (fma
            x1
            (* (* x2 x1) 6.0)
            (fma (/ (- (fma -2.0 x2 t_0) x1) (fma x1 x1 1.0)) 3.0 x1)))
          x1)))
   (if (<= x1 -2e+131)
     (+
      (-
       (* (fma (fma -4.0 x2 (* -22.0 x1)) x1 (fma -12.0 x2 1.0)) x1)
       (* (/ (* -2.0 x2) (- -1.0 (* x1 x1))) 3.0))
      x1)
     (if (<= x1 -3.1e+17)
       t_1
       (if (<= x1 22500000.0)
         (+
          (+
           (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) (- (* x1 x1) -1.0)) 3.0))
          x1)
         (if (<= x1 1e+110) t_1 (fma (* x1 x1) x1 (* -6.0 x2))))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = fma(fma(x1, x1, 1.0), (6.0 * (x1 * x1)), fma(x1, ((x2 * x1) * 6.0), fma(((fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, x1))) + x1;
	double tmp;
	if (x1 <= -2e+131) {
		tmp = ((fma(fma(-4.0, x2, (-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - (((-2.0 * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= -3.1e+17) {
		tmp = t_1;
	} else if (x1 <= 22500000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((t_0 - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else if (x1 <= 1e+110) {
		tmp = t_1;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(fma(fma(x1, x1, 1.0), Float64(6.0 * Float64(x1 * x1)), fma(x1, Float64(Float64(x2 * x1) * 6.0), fma(Float64(Float64(fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, x1))) + x1)
	tmp = 0.0
	if (x1 <= -2e+131)
		tmp = Float64(Float64(Float64(fma(fma(-4.0, x2, Float64(-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - Float64(Float64(Float64(-2.0 * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= -3.1e+17)
		tmp = t_1;
	elseif (x1 <= 22500000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	elseif (x1 <= 1e+110)
		tmp = t_1;
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	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[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x2 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + N[(N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2e+131], N[(N[(N[(N[(N[(-4.0 * x2 + N[(-22.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(-12.0 * x2 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -3.1e+17], t$95$1, If[LessEqual[x1, 22500000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1e+110], t$95$1, N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.9999999999999998e131

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.8%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 78.4

          \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   11. Applied rewrites 78.4%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, 1 + -12 \cdot x2\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.3%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]

   if -1.9999999999999998e131 < x1 < -3.1e17 or 2.25e7 < x1 < 1e110

   1. Initial program 92.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 85.6%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 84.5

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

   10. Applied rewrites 84.5%

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

   if -3.1e17 < x1 < 2.25e7

   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. Add Preprocessing

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 1e110 < x1

   1. Initial program 17.5%

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 22500000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_1 := \mathsf{fma}\left(-4, x2, -22 \cdot x1\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(t\_0, 3, x1\right)\right)\right) + x1\\ \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_1, x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 3, \mathsf{fma}\left(t\_1, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (/ (- (fma -2.0 x2 (* (* 3.0 x1) x1)) x1) (fma x1 x1 1.0)))
        (t_1 (fma -4.0 x2 (* -22.0 x1)))
        (t_2
         (+
          (fma
           (fma x1 x1 1.0)
           (* 6.0 (* x1 x1))
           (fma x1 (* (* x2 x1) 6.0) (fma t_0 3.0 x1)))
          x1)))
   (if (<= x1 -2e+131)
     (+
      (-
       (* (fma t_1 x1 (fma -12.0 x2 1.0)) x1)
       (* (/ (* -2.0 x2) (- -1.0 (* x1 x1))) 3.0))
      x1)
     (if (<= x1 -480000.0)
       t_2
       (if (<= x1 1150000.0)
         (+
          (fma
           t_0
           3.0
           (* (fma t_1 x1 (fma (* (fma 2.0 x2 -3.0) x2) 4.0 1.0)) x1))
          x1)
         (if (<= x1 1e+110) t_2 (fma (* x1 x1) x1 (* -6.0 x2))))))))

C

double code(double x1, double x2) {
	double t_0 = (fma(-2.0, x2, ((3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0);
	double t_1 = fma(-4.0, x2, (-22.0 * x1));
	double t_2 = fma(fma(x1, x1, 1.0), (6.0 * (x1 * x1)), fma(x1, ((x2 * x1) * 6.0), fma(t_0, 3.0, x1))) + x1;
	double tmp;
	if (x1 <= -2e+131) {
		tmp = ((fma(t_1, x1, fma(-12.0, x2, 1.0)) * x1) - (((-2.0 * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= -480000.0) {
		tmp = t_2;
	} else if (x1 <= 1150000.0) {
		tmp = fma(t_0, 3.0, (fma(t_1, x1, fma((fma(2.0, x2, -3.0) * x2), 4.0, 1.0)) * x1)) + x1;
	} else if (x1 <= 1e+110) {
		tmp = t_2;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(-2.0, x2, Float64(Float64(3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0))
	t_1 = fma(-4.0, x2, Float64(-22.0 * x1))
	t_2 = Float64(fma(fma(x1, x1, 1.0), Float64(6.0 * Float64(x1 * x1)), fma(x1, Float64(Float64(x2 * x1) * 6.0), fma(t_0, 3.0, x1))) + x1)
	tmp = 0.0
	if (x1 <= -2e+131)
		tmp = Float64(Float64(Float64(fma(t_1, x1, fma(-12.0, x2, 1.0)) * x1) - Float64(Float64(Float64(-2.0 * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= -480000.0)
		tmp = t_2;
	elseif (x1 <= 1150000.0)
		tmp = Float64(fma(t_0, 3.0, Float64(fma(t_1, x1, fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, 1.0)) * x1)) + x1);
	elseif (x1 <= 1e+110)
		tmp = t_2;
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(-2.0 * x2 + N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * x2 + N[(-22.0 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x2 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + N[(t$95$0 * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2e+131], N[(N[(N[(N[(t$95$1 * x1 + N[(-12.0 * x2 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -480000.0], t$95$2, If[LessEqual[x1, 1150000.0], N[(N[(t$95$0 * 3.0 + N[(N[(t$95$1 * x1 + N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1e+110], t$95$2, N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.9999999999999998e131

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.8%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 78.4

          \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   11. Applied rewrites 78.4%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, 1 + -12 \cdot x2\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.3%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]

   if -1.9999999999999998e131 < x1 < -4.8e5 or 1.15e6 < x1 < 1e110

   1. Initial program 92.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 83.5%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 82.4

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

   10. Applied rewrites 82.4%

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

   if -4.8e5 < x1 < 1.15e6

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.6%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   10. Applied rewrites 85.0%

       \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(x1 \cdot 3\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1\right)} \]

   if 1e110 < x1

   1. Initial program 17.5%

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(3 - -2 \cdot x2, 3, \mathsf{fma}\left(14, x2, -6\right)\right) + \mathsf{fma}\left(-4, x2, 6\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (fma
           (fma x1 x1 1.0)
           (* 6.0 (* x1 x1))
           (fma
            x1
            (* (* x2 x1) 6.0)
            (fma
             (/ (- (fma -2.0 x2 (* (* 3.0 x1) x1)) x1) (fma x1 x1 1.0))
             3.0
             x1)))
          x1)))
   (if (<= x1 -2e+131)
     (+
      (-
       (* (fma (fma -4.0 x2 (* -22.0 x1)) x1 (fma -12.0 x2 1.0)) x1)
       (* (/ (* -2.0 x2) (- -1.0 (* x1 x1))) 3.0))
      x1)
     (if (<= x1 -480000.0)
       t_0
       (if (<= x1 1150000.0)
         (fma
          (fma
           (fma
            -4.0
            x2
            (+
             (fma (- 3.0 (* -2.0 x2)) 3.0 (fma 14.0 x2 -6.0))
             (fma -4.0 x2 6.0)))
           x1
           (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0))
          x1
          (* -6.0 x2))
         (if (<= x1 1e+110) t_0 (fma (* x1 x1) x1 (* -6.0 x2))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(x1, x1, 1.0), (6.0 * (x1 * x1)), fma(x1, ((x2 * x1) * 6.0), fma(((fma(-2.0, x2, ((3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, x1))) + x1;
	double tmp;
	if (x1 <= -2e+131) {
		tmp = ((fma(fma(-4.0, x2, (-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - (((-2.0 * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= -480000.0) {
		tmp = t_0;
	} else if (x1 <= 1150000.0) {
		tmp = fma(fma(fma(-4.0, x2, (fma((3.0 - (-2.0 * x2)), 3.0, fma(14.0, x2, -6.0)) + fma(-4.0, x2, 6.0))), x1, fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0)), x1, (-6.0 * x2));
	} else if (x1 <= 1e+110) {
		tmp = t_0;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(fma(x1, x1, 1.0), Float64(6.0 * Float64(x1 * x1)), fma(x1, Float64(Float64(x2 * x1) * 6.0), fma(Float64(Float64(fma(-2.0, x2, Float64(Float64(3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, x1))) + x1)
	tmp = 0.0
	if (x1 <= -2e+131)
		tmp = Float64(Float64(Float64(fma(fma(-4.0, x2, Float64(-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - Float64(Float64(Float64(-2.0 * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= -480000.0)
		tmp = t_0;
	elseif (x1 <= 1150000.0)
		tmp = fma(fma(fma(-4.0, x2, Float64(fma(Float64(3.0 - Float64(-2.0 * x2)), 3.0, fma(14.0, x2, -6.0)) + fma(-4.0, x2, 6.0))), x1, fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0)), x1, Float64(-6.0 * x2));
	elseif (x1 <= 1e+110)
		tmp = t_0;
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x2 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + N[(N[(N[(N[(-2.0 * x2 + N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2e+131], N[(N[(N[(N[(N[(-4.0 * x2 + N[(-22.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(-12.0 * x2 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -480000.0], t$95$0, If[LessEqual[x1, 1150000.0], N[(N[(N[(-4.0 * x2 + N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(14.0 * x2 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * x2 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1e+110], t$95$0, N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.9999999999999998e131

   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. Add Preprocessing

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.8%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 78.4

          \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   11. Applied rewrites 78.4%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, 1 + -12 \cdot x2\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.3%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]

   if -1.9999999999999998e131 < x1 < -4.8e5 or 1.15e6 < x1 < 1e110

   1. Initial program 92.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 83.5%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 82.4

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

   10. Applied rewrites 82.4%

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

   if -4.8e5 < x1 < 1.15e6

   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. Add Preprocessing

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 85.0%

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

   if 1e110 < x1

   1. Initial program 17.5%

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2 \cdot 10^{+131}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(3 - -2 \cdot x2, 3, \mathsf{fma}\left(14, x2, -6\right)\right) + \mathsf{fma}\left(-4, x2, 6\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x1 \cdot x1\right)\\ t_1 := \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -1.25 \cdot 10^{+98}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), t\_0, \mathsf{fma}\left(x1, t\_1, \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(3 - -2 \cdot x2, 3, \mathsf{fma}\left(14, x2, -6\right)\right) + \mathsf{fma}\left(-4, x2, 6\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), t\_0, \mathsf{fma}\left(x1, t\_1, \mathsf{fma}\left(\frac{-2 \cdot x2}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x1 x1))) (t_1 (fma (* (* x2 x1) 2.0) 3.0 (* x1 x1))))
   (if (<= x1 -1.25e+98)
     (+
      (-
       (* (fma (fma -4.0 x2 (* -22.0 x1)) x1 (fma -12.0 x2 1.0)) x1)
       (* (/ (* -2.0 x2) (- -1.0 (* x1 x1))) 3.0))
      x1)
     (if (<= x1 -480000.0)
       (+ (fma (fma x1 x1 1.0) t_0 (fma x1 t_1 (fma 3.0 3.0 x1))) x1)
       (if (<= x1 1150000.0)
         (fma
          (fma
           (fma
            -4.0
            x2
            (+
             (fma (- 3.0 (* -2.0 x2)) 3.0 (fma 14.0 x2 -6.0))
             (fma -4.0 x2 6.0)))
           x1
           (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0))
          x1
          (* -6.0 x2))
         (if (<= x1 1e+110)
           (+
            (fma
             (fma x1 x1 1.0)
             t_0
             (fma x1 t_1 (fma (/ (* -2.0 x2) (fma x1 x1 1.0)) 3.0 x1)))
            x1)
           (fma (* x1 x1) x1 (* -6.0 x2))))))))

C

double code(double x1, double x2) {
	double t_0 = 6.0 * (x1 * x1);
	double t_1 = fma(((x2 * x1) * 2.0), 3.0, (x1 * x1));
	double tmp;
	if (x1 <= -1.25e+98) {
		tmp = ((fma(fma(-4.0, x2, (-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - (((-2.0 * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= -480000.0) {
		tmp = fma(fma(x1, x1, 1.0), t_0, fma(x1, t_1, fma(3.0, 3.0, x1))) + x1;
	} else if (x1 <= 1150000.0) {
		tmp = fma(fma(fma(-4.0, x2, (fma((3.0 - (-2.0 * x2)), 3.0, fma(14.0, x2, -6.0)) + fma(-4.0, x2, 6.0))), x1, fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0)), x1, (-6.0 * x2));
	} else if (x1 <= 1e+110) {
		tmp = fma(fma(x1, x1, 1.0), t_0, fma(x1, t_1, fma(((-2.0 * x2) / fma(x1, x1, 1.0)), 3.0, x1))) + x1;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(6.0 * Float64(x1 * x1))
	t_1 = fma(Float64(Float64(x2 * x1) * 2.0), 3.0, Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -1.25e+98)
		tmp = Float64(Float64(Float64(fma(fma(-4.0, x2, Float64(-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - Float64(Float64(Float64(-2.0 * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= -480000.0)
		tmp = Float64(fma(fma(x1, x1, 1.0), t_0, fma(x1, t_1, fma(3.0, 3.0, x1))) + x1);
	elseif (x1 <= 1150000.0)
		tmp = fma(fma(fma(-4.0, x2, Float64(fma(Float64(3.0 - Float64(-2.0 * x2)), 3.0, fma(14.0, x2, -6.0)) + fma(-4.0, x2, 6.0))), x1, fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0)), x1, Float64(-6.0 * x2));
	elseif (x1 <= 1e+110)
		tmp = Float64(fma(fma(x1, x1, 1.0), t_0, fma(x1, t_1, fma(Float64(Float64(-2.0 * x2) / fma(x1, x1, 1.0)), 3.0, x1))) + x1);
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x2 * x1), $MachinePrecision] * 2.0), $MachinePrecision] * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.25e+98], N[(N[(N[(N[(N[(-4.0 * x2 + N[(-22.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(-12.0 * x2 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -480000.0], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * t$95$0 + N[(x1 * t$95$1 + N[(3.0 * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1150000.0], N[(N[(N[(-4.0 * x2 + N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(14.0 * x2 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * x2 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1e+110], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * t$95$0 + N[(x1 * t$95$1 + N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -1.25e98

   1. Initial program 4.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 18.6%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 76.7

          \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   11. Applied rewrites 76.7%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, 1 + -12 \cdot x2\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 93.0%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]

   if -1.25e98 < x1 < -4.8e5

   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. Add Preprocessing

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 73.1

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

   10. Applied rewrites 73.1%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 73.1%

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

   if -4.8e5 < x1 < 1.15e6

   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. Add Preprocessing

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 85.0%

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

   if 1.15e6 < x1 < 1e110

   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. Add Preprocessing

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 89.9

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

   10. Applied rewrites 89.9%

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

   11. Taylor expanded in x1 around 0

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

       1. lower-*.f64 89.9

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

   13. Applied rewrites 89.9%

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

   if 1e110 < x1

   1. Initial program 17.5%

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot x1}, x1, -6 \cdot x2\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.25 \cdot 10^{+98}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(3 - -2 \cdot x2, 3, \mathsf{fma}\left(14, x2, -6\right)\right) + \mathsf{fma}\left(-4, x2, 6\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(\frac{-2 \cdot x2}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{if}\;x1 \leq -1.25 \cdot 10^{+98}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(3 - -2 \cdot x2, 3, \mathsf{fma}\left(14, x2, -6\right)\right) + \mathsf{fma}\left(-4, x2, 6\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (fma
           (fma x1 x1 1.0)
           (* 6.0 (* x1 x1))
           (fma x1 (fma (* (* x2 x1) 2.0) 3.0 (* x1 x1)) (fma 3.0 3.0 x1)))
          x1)))
   (if (<= x1 -1.25e+98)
     (+
      (-
       (* (fma (fma -4.0 x2 (* -22.0 x1)) x1 (fma -12.0 x2 1.0)) x1)
       (* (/ (* -2.0 x2) (- -1.0 (* x1 x1))) 3.0))
      x1)
     (if (<= x1 -480000.0)
       t_0
       (if (<= x1 1150000.0)
         (fma
          (fma
           (fma
            -4.0
            x2
            (+
             (fma (- 3.0 (* -2.0 x2)) 3.0 (fma 14.0 x2 -6.0))
             (fma -4.0 x2 6.0)))
           x1
           (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0))
          x1
          (* -6.0 x2))
         (if (<= x1 1e+110) t_0 (fma (* x1 x1) x1 (* -6.0 x2))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(x1, x1, 1.0), (6.0 * (x1 * x1)), fma(x1, fma(((x2 * x1) * 2.0), 3.0, (x1 * x1)), fma(3.0, 3.0, x1))) + x1;
	double tmp;
	if (x1 <= -1.25e+98) {
		tmp = ((fma(fma(-4.0, x2, (-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - (((-2.0 * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= -480000.0) {
		tmp = t_0;
	} else if (x1 <= 1150000.0) {
		tmp = fma(fma(fma(-4.0, x2, (fma((3.0 - (-2.0 * x2)), 3.0, fma(14.0, x2, -6.0)) + fma(-4.0, x2, 6.0))), x1, fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0)), x1, (-6.0 * x2));
	} else if (x1 <= 1e+110) {
		tmp = t_0;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(fma(x1, x1, 1.0), Float64(6.0 * Float64(x1 * x1)), fma(x1, fma(Float64(Float64(x2 * x1) * 2.0), 3.0, Float64(x1 * x1)), fma(3.0, 3.0, x1))) + x1)
	tmp = 0.0
	if (x1 <= -1.25e+98)
		tmp = Float64(Float64(Float64(fma(fma(-4.0, x2, Float64(-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - Float64(Float64(Float64(-2.0 * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= -480000.0)
		tmp = t_0;
	elseif (x1 <= 1150000.0)
		tmp = fma(fma(fma(-4.0, x2, Float64(fma(Float64(3.0 - Float64(-2.0 * x2)), 3.0, fma(14.0, x2, -6.0)) + fma(-4.0, x2, 6.0))), x1, fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0)), x1, Float64(-6.0 * x2));
	elseif (x1 <= 1e+110)
		tmp = t_0;
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(N[(x2 * x1), $MachinePrecision] * 2.0), $MachinePrecision] * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(3.0 * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.25e+98], N[(N[(N[(N[(N[(-4.0 * x2 + N[(-22.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(-12.0 * x2 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -480000.0], t$95$0, If[LessEqual[x1, 1150000.0], N[(N[(N[(-4.0 * x2 + N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(14.0 * x2 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * x2 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1e+110], t$95$0, N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.25e98

   1. Initial program 4.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 18.6%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 76.7

          \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   11. Applied rewrites 76.7%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, 1 + -12 \cdot x2\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 93.0%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]

   if -1.25e98 < x1 < -4.8e5 or 1.15e6 < x1 < 1e110

   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. Add Preprocessing

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 82.3%

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

   if -4.8e5 < x1 < 1.15e6

   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. Add Preprocessing

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 85.0%

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

   if 1e110 < x1

   1. Initial program 17.5%

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.25 \cdot 10^{+98}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(3 - -2 \cdot x2, 3, \mathsf{fma}\left(14, x2, -6\right)\right) + \mathsf{fma}\left(-4, x2, 6\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 86.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{if}\;x1 \leq -1.25 \cdot 10^{+98}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (fma
           (fma x1 x1 1.0)
           (* 6.0 (* x1 x1))
           (fma x1 (fma (* (* x2 x1) 2.0) 3.0 (* x1 x1)) (fma 3.0 3.0 x1)))
          x1)))
   (if (<= x1 -1.25e+98)
     (+
      (-
       (* (fma (fma -4.0 x2 (* -22.0 x1)) x1 (fma -12.0 x2 1.0)) x1)
       (* (/ (* -2.0 x2) (- -1.0 (* x1 x1))) 3.0))
      x1)
     (if (<= x1 -480000.0)
       t_0
       (if (<= x1 1150000.0)
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
         (if (<= x1 1e+110) t_0 (fma (* x1 x1) x1 (* -6.0 x2))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(x1, x1, 1.0), (6.0 * (x1 * x1)), fma(x1, fma(((x2 * x1) * 2.0), 3.0, (x1 * x1)), fma(3.0, 3.0, x1))) + x1;
	double tmp;
	if (x1 <= -1.25e+98) {
		tmp = ((fma(fma(-4.0, x2, (-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - (((-2.0 * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= -480000.0) {
		tmp = t_0;
	} else if (x1 <= 1150000.0) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else if (x1 <= 1e+110) {
		tmp = t_0;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(fma(x1, x1, 1.0), Float64(6.0 * Float64(x1 * x1)), fma(x1, fma(Float64(Float64(x2 * x1) * 2.0), 3.0, Float64(x1 * x1)), fma(3.0, 3.0, x1))) + x1)
	tmp = 0.0
	if (x1 <= -1.25e+98)
		tmp = Float64(Float64(Float64(fma(fma(-4.0, x2, Float64(-22.0 * x1)), x1, fma(-12.0, x2, 1.0)) * x1) - Float64(Float64(Float64(-2.0 * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= -480000.0)
		tmp = t_0;
	elseif (x1 <= 1150000.0)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	elseif (x1 <= 1e+110)
		tmp = t_0;
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(N[(x2 * x1), $MachinePrecision] * 2.0), $MachinePrecision] * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(3.0 * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.25e+98], N[(N[(N[(N[(N[(-4.0 * x2 + N[(-22.0 * x1), $MachinePrecision]), $MachinePrecision] * x1 + N[(-12.0 * x2 + 1.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] - N[(N[(N[(-2.0 * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -480000.0], t$95$0, If[LessEqual[x1, 1150000.0], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1e+110], t$95$0, N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.25e98

   1. Initial program 4.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 18.6%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 18.6%

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 76.7

          \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   11. Applied rewrites 76.7%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right)\right) \cdot x1 + 3 \cdot \frac{\color{blue}{-2 \cdot x2}}{x1 \cdot x1 + 1}\right) \]

   12. Taylor expanded in x2 around 0

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, 1 + -12 \cdot x2\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 93.0%

       \[\leadsto x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 + 3 \cdot \frac{-2 \cdot x2}{x1 \cdot x1 + 1}\right) \]

   if -1.25e98 < x1 < -4.8e5 or 1.15e6 < x1 < 1e110

   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. Add Preprocessing

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 82.3%

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

   if -4.8e5 < x1 < 1.15e6

   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. Add Preprocessing

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

   if 1e110 < x1

   1. Initial program 17.5%

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.25 \cdot 10^{+98}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, -22 \cdot x1\right), x1, \mathsf{fma}\left(-12, x2, 1\right)\right) \cdot x1 - \frac{-2 \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{if}\;x1 \leq -1.25 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-20, x1, 9\right), x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (fma
           (fma x1 x1 1.0)
           (* 6.0 (* x1 x1))
           (fma x1 (fma (* (* x2 x1) 2.0) 3.0 (* x1 x1)) (fma 3.0 3.0 x1)))
          x1)))
   (if (<= x1 -1.25e+98)
     (+ (fma (* x1 x1) x1 (* (fma (fma -20.0 x1 9.0) x1 -2.0) x1)) x1)
     (if (<= x1 -480000.0)
       t_0
       (if (<= x1 1150000.0)
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
         (if (<= x1 1e+110) t_0 (fma (* x1 x1) x1 (* -6.0 x2))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(x1, x1, 1.0), (6.0 * (x1 * x1)), fma(x1, fma(((x2 * x1) * 2.0), 3.0, (x1 * x1)), fma(3.0, 3.0, x1))) + x1;
	double tmp;
	if (x1 <= -1.25e+98) {
		tmp = fma((x1 * x1), x1, (fma(fma(-20.0, x1, 9.0), x1, -2.0) * x1)) + x1;
	} else if (x1 <= -480000.0) {
		tmp = t_0;
	} else if (x1 <= 1150000.0) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else if (x1 <= 1e+110) {
		tmp = t_0;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(fma(x1, x1, 1.0), Float64(6.0 * Float64(x1 * x1)), fma(x1, fma(Float64(Float64(x2 * x1) * 2.0), 3.0, Float64(x1 * x1)), fma(3.0, 3.0, x1))) + x1)
	tmp = 0.0
	if (x1 <= -1.25e+98)
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(fma(fma(-20.0, x1, 9.0), x1, -2.0) * x1)) + x1);
	elseif (x1 <= -480000.0)
		tmp = t_0;
	elseif (x1 <= 1150000.0)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	elseif (x1 <= 1e+110)
		tmp = t_0;
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(N[(x2 * x1), $MachinePrecision] * 2.0), $MachinePrecision] * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(3.0 * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.25e+98], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(-20.0 * x1 + 9.0), $MachinePrecision] * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -480000.0], t$95$0, If[LessEqual[x1, 1150000.0], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1e+110], t$95$0, N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.25e98

   1. Initial program 4.7%

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

   4. Applied rewrites 27.9%

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

   5. Taylor expanded in x2 around 0

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 23.5%

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-20, x1, 9\right), x1, -2\right) \cdot \color{blue}{x1}\right) \]

   if -1.25e98 < x1 < -4.8e5 or 1.15e6 < x1 < 1e110

   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. Add Preprocessing

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 82.3%

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

   if -4.8e5 < x1 < 1.15e6

   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. Add Preprocessing

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

   if 1e110 < x1

   1. Initial program 17.5%

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

   4. Applied rewrites 17.5%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.25 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-20, x1, 9\right), x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -480000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 1150000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), 6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(\left(x2 \cdot x1\right) \cdot 2, 3, x1 \cdot x1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 67.4% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-20, x1, 9\right), x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 7 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4.5e+68)
   (+ (fma (* x1 x1) x1 (* (fma (fma -20.0 x1 9.0) x1 -2.0) x1)) x1)
   (if (<= x1 7e+96)
     (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
     (fma (* x1 x1) x1 (* -6.0 x2)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4.5e+68) {
		tmp = fma((x1 * x1), x1, (fma(fma(-20.0, x1, 9.0), x1, -2.0) * x1)) + x1;
	} else if (x1 <= 7e+96) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4.5e+68)
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(fma(fma(-20.0, x1, 9.0), x1, -2.0) * x1)) + x1);
	elseif (x1 <= 7e+96)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4.5e+68], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(-20.0 * x1 + 9.0), $MachinePrecision] * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 7e+96], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.5000000000000003e68

   1. Initial program 16.3%

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in x2 around 0

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -20 \cdot x1\right) - 2\right)}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.6%

       \[\leadsto x1 + \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-20, x1, 9\right), x1, -2\right) \cdot \color{blue}{x1}\right) \]

   if -4.5000000000000003e68 < x1 < 6.9999999999999998e96

   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. Add Preprocessing

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if 6.9999999999999998e96 < x1

   1. Initial program 23.3%

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 93.8

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

   7. Applied rewrites 93.8%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 93.8%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 93.8

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

   12. Applied rewrites 93.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-20, x1, 9\right), x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 7 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.9 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (fma (* x1 x1) x1 (* (fma 9.0 x1 -2.0) x1)) x1)))
   (if (<= x1 -3.1e-145)
     t_0
     (if (<= x1 3.9e-76) (fma (* x1 x1) x1 (* -6.0 x2)) t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), x1, (fma(9.0, x1, -2.0) * x1)) + x1;
	double tmp;
	if (x1 <= -3.1e-145) {
		tmp = t_0;
	} else if (x1 <= 3.9e-76) {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(Float64(x1 * x1), x1, Float64(fma(9.0, x1, -2.0) * x1)) + x1)
	tmp = 0.0
	if (x1 <= -3.1e-145)
		tmp = t_0;
	elseif (x1 <= 3.9e-76)
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -3.1e-145], t$95$0, If[LessEqual[x1, 3.9e-76], N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.1e-145 or 3.90000000000000025e-76 < x1

   1. Initial program 60.5%

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in x2 around 0

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

   7. Applied rewrites 35.2%

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

   8. Taylor expanded in x1 around 0

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

   10. Applied rewrites 35.5%

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

   if -3.1e-145 < x1 < 3.90000000000000025e-76

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 78.1

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

   7. Applied rewrites 78.1%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 78.1%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 78.2

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

   12. Applied rewrites 78.2%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 3.9 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq 7 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 7e+96)
   (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
   (fma (* x1 x1) x1 (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= 7e+96) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= 7e+96)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, 7e+96], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < 6.9999999999999998e96

   1. Initial program 80.2%

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   if 6.9999999999999998e96 < x1

   1. Initial program 23.3%

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

   4. Applied rewrites 23.3%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 93.8

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

   7. Applied rewrites 93.8%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 93.8%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 93.8

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

   12. Applied rewrites 93.8%

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

Alternative 21: 45.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, x1, -2 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.9 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (fma (* x1 x1) x1 (* -2.0 x1)) x1)))
   (if (<= x1 -3.1e-145)
     t_0
     (if (<= x1 3.9e-76) (fma (* x1 x1) x1 (* -6.0 x2)) t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), x1, (-2.0 * x1)) + x1;
	double tmp;
	if (x1 <= -3.1e-145) {
		tmp = t_0;
	} else if (x1 <= 3.9e-76) {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(Float64(x1 * x1), x1, Float64(-2.0 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -3.1e-145)
		tmp = t_0;
	elseif (x1 <= 3.9e-76)
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-2.0 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -3.1e-145], t$95$0, If[LessEqual[x1, 3.9e-76], N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.1e-145 or 3.90000000000000025e-76 < x1

   1. Initial program 60.5%

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in x2 around 0

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

   7. Applied rewrites 35.2%

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

   8. Taylor expanded in x1 around 0

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

   10. Applied rewrites 35.3%

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

   if -3.1e-145 < x1 < 3.90000000000000025e-76

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 78.1

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

   7. Applied rewrites 78.1%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 78.1%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 78.2

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

   12. Applied rewrites 78.2%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -2 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 3.9 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -2 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.7% accurate, 17.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

4. Applied rewrites 74.7%

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

5. Taylor expanded in x1 around 0

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

   1. lower-*.f64 40.2

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

7. Applied rewrites 40.2%

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

   1. lift-+.f64 N/A

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

9. Applied rewrites 40.2%

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

10. Taylor expanded in x1 around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 40.4

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

12. Applied rewrites 40.4%

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

Alternative 23: 26.3% accurate, 33.1× speedup

Math

\[\begin{array}{l} \\ -6 \cdot x2 + x1 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (+ (* -6.0 x2) x1))

C

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

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = ((-6.0d0) * x2) + x1
end function

Java

public static double code(double x1, double x2) {
	return (-6.0 * x2) + x1;
}

Python

def code(x1, x2):
	return (-6.0 * x2) + x1

Julia

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

MATLAB

function tmp = code(x1, x2)
	tmp = (-6.0 * x2) + x1;
end

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 25.1

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

5. Applied rewrites 25.1%

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

6. Final simplification 25.1%

   \[\leadsto -6 \cdot x2 + x1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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))))))