Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.7% → 99.7%

Time: 20.9s

Alternatives: 19

Speedup: 7.3×

• 46.6% 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: 70.7% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (- -1.0 (* x1 x1)))
        (t_2 (/ (fma x1 (fma x1 3.0 -1.0) (* x2 2.0)) (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
      (/ (fma x1 (fma x1 3.0 -1.0) (* -2.0 x2)) (fma x1 x1 1.0))
      3.0
      (+
       (fma
        (fma (* x1 x1) (fma t_2 4.0 -6.0) (* (- t_2 3.0) (* (* 2.0 x1) t_2)))
        (fma x1 x1 1.0)
        (fma x1 (fma (* t_2 x1) 3.0 (* x1 x1)) x1))
       x1))
     (*
      (pow x1 4.0)
      (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) 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(x1, fma(x1, 3.0, -1.0), (x2 * 2.0)) / 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((fma(x1, fma(x1, 3.0, -1.0), (-2.0 * x2)) / fma(x1, x1, 1.0)), 3.0, (fma(fma((x1 * x1), fma(t_2, 4.0, -6.0), ((t_2 - 3.0) * ((2.0 * x1) * t_2))), fma(x1, x1, 1.0), fma(x1, fma((t_2 * x1), 3.0, (x1 * x1)), x1)) + x1));
	} else {
		tmp = pow(x1, 4.0) * (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / 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(fma(x1, fma(x1, 3.0, -1.0), Float64(x2 * 2.0)) / 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 = fma(Float64(fma(x1, fma(x1, 3.0, -1.0), Float64(-2.0 * x2)) / fma(x1, x1, 1.0)), 3.0, Float64(fma(fma(Float64(x1 * x1), fma(t_2, 4.0, -6.0), Float64(Float64(t_2 - 3.0) * Float64(Float64(2.0 * x1) * t_2))), fma(x1, x1, 1.0), fma(x1, fma(Float64(t_2 * x1), 3.0, Float64(x1 * x1)), x1)) + x1));
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / 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[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(x2 * 2.0), $MachinePrecision]), $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 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$2 * 4.0 + -6.0), $MachinePrecision] + N[(N[(t$95$2 - 3.0), $MachinePrecision] * N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(x1 * N[(N[(t$95$2 * x1), $MachinePrecision] * 3.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 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]), $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 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternative 2: 94.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_2 := \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(t\_1, 4, -6\right), \left(t\_1 - 3\right) \cdot \left(\left(2 \cdot x1\right) \cdot t\_1\right)\right)\\ t_3 := -1 - x1 \cdot x1\\ t_4 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_5 := x1 \cdot x1 - -1\\ t_6 := \frac{t\_4}{t\_5}\\ t_7 := x1 - \left(\left(\left(\left(\frac{t\_4}{t\_3} \cdot t\_0 - t\_3 \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) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_5} \cdot 3\right)\\ t_8 := \mathsf{fma}\left(-2, x2, 3\right) \cdot 2\\ \mathbf{if}\;t\_7 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \mathsf{fma}\left(t\_1 \cdot \left(x1 \cdot x1\right), 3, \mathsf{fma}\left(t\_2, \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right)\right)\\ \mathbf{elif}\;t\_7 \leq -5 \cdot 10^{+171}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(3 - -2 \cdot x2, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;t\_7 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2, x2, \mathsf{fma}\left(-2, x2, 3\right)\right), 2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(t\_8, x2, \mathsf{fma}\left(t\_8, 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, -4\right)\right) \cdot x1\right)\right)\right) - 6, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right)\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(t\_2 \cdot x1, x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (/ (fma x1 (fma x1 3.0 -1.0) (* x2 2.0)) (fma x1 x1 1.0)))
        (t_2
         (fma (* x1 x1) (fma t_1 4.0 -6.0) (* (- t_1 3.0) (* (* 2.0 x1) t_1))))
        (t_3 (- -1.0 (* x1 x1)))
        (t_4 (- (+ (* x2 2.0) t_0) x1))
        (t_5 (- (* x1 x1) -1.0))
        (t_6 (/ t_4 t_5))
        (t_7
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_4 t_3) t_0)
              (*
               t_3
               (-
                (* (- 3.0 t_6) (* (* 2.0 x1) t_6))
                (* (- (* 4.0 t_6) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_5) 3.0))))
        (t_8 (* (fma -2.0 x2 3.0) 2.0)))
   (if (<= t_7 -2e+262)
     (fma
      (fma x1 x1 1.0)
      x1
      (fma (* t_1 (* x1 x1)) 3.0 (fma t_2 (fma x1 x1 1.0) (fma 3.0 3.0 x1))))
     (if (<= t_7 -5e+171)
       (+
        (+
         (* (fma (fma (- 3.0 (* -2.0 x2)) x1 -1.0) x1 (* -2.0 x2)) 3.0)
         (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2))
        x1)
       (if (<= t_7 5e+37)
         (+
          (fma
           (* x1 x1)
           x1
           (fma
            (fma
             (-
              (fma
               (fma -2.0 x2 (fma -2.0 x2 3.0))
               2.0
               (fma
                (fma 2.0 x2 3.0)
                3.0
                (fma
                 14.0
                 x2
                 (*
                  (fma
                   (fma t_8 x2 (fma t_8 x2 (fma (fma 2.0 x2 -3.0) 3.0 1.0)))
                   2.0
                   (fma (* (fma 2.0 x2 -3.0) 4.0) x2 -4.0))
                  x1))))
              6.0)
             x1
             (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0))
            x1
            (* -6.0 x2)))
          x1)
         (if (<= t_7 INFINITY)
           (+ (fma (* x1 x1) x1 (fma (* t_2 x1) x1 (* -6.0 x2))) x1)
           (*
            (pow x1 4.0)
            (-
             6.0
             (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1)))))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = fma(x1, fma(x1, 3.0, -1.0), (x2 * 2.0)) / fma(x1, x1, 1.0);
	double t_2 = fma((x1 * x1), fma(t_1, 4.0, -6.0), ((t_1 - 3.0) * ((2.0 * x1) * t_1)));
	double t_3 = -1.0 - (x1 * x1);
	double t_4 = ((x2 * 2.0) + t_0) - x1;
	double t_5 = (x1 * x1) - -1.0;
	double t_6 = t_4 / t_5;
	double t_7 = x1 - ((((((t_4 / t_3) * t_0) - (t_3 * (((3.0 - t_6) * ((2.0 * x1) * t_6)) - (((4.0 * t_6) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_5) * 3.0));
	double t_8 = fma(-2.0, x2, 3.0) * 2.0;
	double tmp;
	if (t_7 <= -2e+262) {
		tmp = fma(fma(x1, x1, 1.0), x1, fma((t_1 * (x1 * x1)), 3.0, fma(t_2, fma(x1, x1, 1.0), fma(3.0, 3.0, x1))));
	} else if (t_7 <= -5e+171) {
		tmp = ((fma(fma((3.0 - (-2.0 * x2)), x1, -1.0), x1, (-2.0 * x2)) * 3.0) + (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (t_7 <= 5e+37) {
		tmp = fma((x1 * x1), x1, fma(fma((fma(fma(-2.0, x2, fma(-2.0, x2, 3.0)), 2.0, fma(fma(2.0, x2, 3.0), 3.0, fma(14.0, x2, (fma(fma(t_8, x2, fma(t_8, x2, fma(fma(2.0, x2, -3.0), 3.0, 1.0))), 2.0, fma((fma(2.0, x2, -3.0) * 4.0), x2, -4.0)) * x1)))) - 6.0), x1, fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0)), x1, (-6.0 * x2))) + x1;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = fma((x1 * x1), x1, fma((t_2 * x1), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = pow(x1, 4.0) * (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(fma(x1, fma(x1, 3.0, -1.0), Float64(x2 * 2.0)) / fma(x1, x1, 1.0))
	t_2 = fma(Float64(x1 * x1), fma(t_1, 4.0, -6.0), Float64(Float64(t_1 - 3.0) * Float64(Float64(2.0 * x1) * t_1)))
	t_3 = Float64(-1.0 - Float64(x1 * x1))
	t_4 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_5 = Float64(Float64(x1 * x1) - -1.0)
	t_6 = Float64(t_4 / t_5)
	t_7 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 / t_3) * t_0) - Float64(t_3 * 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) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_5) * 3.0)))
	t_8 = Float64(fma(-2.0, x2, 3.0) * 2.0)
	tmp = 0.0
	if (t_7 <= -2e+262)
		tmp = fma(fma(x1, x1, 1.0), x1, fma(Float64(t_1 * Float64(x1 * x1)), 3.0, fma(t_2, fma(x1, x1, 1.0), fma(3.0, 3.0, x1))));
	elseif (t_7 <= -5e+171)
		tmp = Float64(Float64(Float64(fma(fma(Float64(3.0 - Float64(-2.0 * x2)), x1, -1.0), x1, Float64(-2.0 * x2)) * 3.0) + Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (t_7 <= 5e+37)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(fma(Float64(fma(fma(-2.0, x2, fma(-2.0, x2, 3.0)), 2.0, fma(fma(2.0, x2, 3.0), 3.0, fma(14.0, x2, Float64(fma(fma(t_8, x2, fma(t_8, x2, fma(fma(2.0, x2, -3.0), 3.0, 1.0))), 2.0, fma(Float64(fma(2.0, x2, -3.0) * 4.0), x2, -4.0)) * x1)))) - 6.0), x1, fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -2.0)), x1, Float64(-6.0 * x2))) + x1);
	elseif (t_7 <= Inf)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(Float64(t_2 * x1), x1, Float64(-6.0 * x2))) + x1);
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / 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 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$1 * 4.0 + -6.0), $MachinePrecision] + N[(N[(t$95$1 - 3.0), $MachinePrecision] * N[(N[(2.0 * x1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$4 / t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$3 * 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] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[t$95$7, -2e+262], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(t$95$1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(t$95$2 * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(3.0 * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, -5e+171], N[(N[(N[(N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[t$95$7, 5e+37], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(N[(-2.0 * x2 + N[(-2.0 * x2 + 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(2.0 * x2 + 3.0), $MachinePrecision] * 3.0 + N[(14.0 * x2 + N[(N[(N[(t$95$8 * x2 + N[(t$95$8 * x2 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision] * x2 + -4.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(t$95$2 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 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]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 5 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)))))) < -2e262

   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%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x1 around inf

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

   9. Applied rewrites 95.0%

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

   if -2e262 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -5.0000000000000004e171

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 93.0

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

   8. Applied rewrites 93.0%

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

   if -5.0000000000000004e171 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999989e37

   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%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 95.0%

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

   if 4.99999999999999989e37 < (+.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 99.6%

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

   4. Applied rewrites 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 99.5%

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

   7. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.8

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

   9. Applied rewrites 87.8%

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

   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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.5%

   \[\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 -2 \cdot 10^{+262}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(x1 \cdot x1\right), 3, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \left(\frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(3, 3, x1\right)\right)\right)\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 -5 \cdot 10^{+171}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(3 - -2 \cdot x2, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\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 5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2, x2, \mathsf{fma}\left(-2, x2, 3\right)\right), 2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, \mathsf{fma}\left(14, x2, \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, -4\right)\right) \cdot x1\right)\right)\right) - 6, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right)\right), x1, -6 \cdot x2\right)\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(\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \left(\frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right) \cdot x1, x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \mathsf{fma}\left(-2, x2, 3\right) \cdot 2\\ t_2 := \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_3 := -1 - x1 \cdot x1\\ t_4 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_5 := x1 \cdot x1 - -1\\ t_6 := \frac{t\_4}{t\_5}\\ t_7 := x1 - \left(\left(\left(\left(\frac{t\_4}{t\_3} \cdot t\_0 - t\_3 \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) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_5} \cdot 3\right)\\ \mathbf{if}\;t\_7 \leq -10000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;t\_7 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2, x2, \mathsf{fma}\left(-2, x2, 3\right)\right), 2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(t\_1, x2, \mathsf{fma}\left(t\_1, 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, -4\right)\right) \cdot x1\right)\right)\right) - 6, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right)\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(t\_2, 4, -6\right), \left(t\_2 - 3\right) \cdot \left(\left(2 \cdot x1\right) \cdot t\_2\right)\right) \cdot x1, x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (* (fma -2.0 x2 3.0) 2.0))
        (t_2 (/ (fma x1 (fma x1 3.0 -1.0) (* x2 2.0)) (fma x1 x1 1.0)))
        (t_3 (- -1.0 (* x1 x1)))
        (t_4 (- (+ (* x2 2.0) t_0) x1))
        (t_5 (- (* x1 x1) -1.0))
        (t_6 (/ t_4 t_5))
        (t_7
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_4 t_3) t_0)
              (*
               t_3
               (-
                (* (- 3.0 t_6) (* (* 2.0 x1) t_6))
                (* (- (* 4.0 t_6) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_5) 3.0)))))
   (if (<= t_7 -10000000000000.0)
     (+
      (fma
       (fma (fma 3.0 x1 -1.0) x1 (* -2.0 x2))
       (/ 3.0 (fma x1 x1 1.0))
       (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2))
      x1)
     (if (<= t_7 5e+37)
       (+
        (fma
         (* x1 x1)
         x1
         (fma
          (fma
           (-
            (fma
             (fma -2.0 x2 (fma -2.0 x2 3.0))
             2.0
             (fma
              (fma 2.0 x2 3.0)
              3.0
              (fma
               14.0
               x2
               (*
                (fma
                 (fma t_1 x2 (fma t_1 x2 (fma (fma 2.0 x2 -3.0) 3.0 1.0)))
                 2.0
                 (fma (* (fma 2.0 x2 -3.0) 4.0) x2 -4.0))
                x1))))
            6.0)
           x1
           (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0))
          x1
          (* -6.0 x2)))
        x1)
       (if (<= t_7 INFINITY)
         (+
          (fma
           (* x1 x1)
           x1
           (fma
            (*
             (fma
              (* x1 x1)
              (fma t_2 4.0 -6.0)
              (* (- t_2 3.0) (* (* 2.0 x1) t_2)))
             x1)
            x1
            (* -6.0 x2)))
          x1)
         (*
          (pow x1 4.0)
          (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = fma(-2.0, x2, 3.0) * 2.0;
	double t_2 = fma(x1, fma(x1, 3.0, -1.0), (x2 * 2.0)) / fma(x1, x1, 1.0);
	double t_3 = -1.0 - (x1 * x1);
	double t_4 = ((x2 * 2.0) + t_0) - x1;
	double t_5 = (x1 * x1) - -1.0;
	double t_6 = t_4 / t_5;
	double t_7 = x1 - ((((((t_4 / t_3) * t_0) - (t_3 * (((3.0 - t_6) * ((2.0 * x1) * t_6)) - (((4.0 * t_6) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_5) * 3.0));
	double tmp;
	if (t_7 <= -10000000000000.0) {
		tmp = fma(fma(fma(3.0, x1, -1.0), x1, (-2.0 * x2)), (3.0 / fma(x1, x1, 1.0)), (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (t_7 <= 5e+37) {
		tmp = fma((x1 * x1), x1, fma(fma((fma(fma(-2.0, x2, fma(-2.0, x2, 3.0)), 2.0, fma(fma(2.0, x2, 3.0), 3.0, fma(14.0, x2, (fma(fma(t_1, x2, fma(t_1, x2, fma(fma(2.0, x2, -3.0), 3.0, 1.0))), 2.0, fma((fma(2.0, x2, -3.0) * 4.0), x2, -4.0)) * x1)))) - 6.0), x1, fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0)), x1, (-6.0 * x2))) + x1;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = fma((x1 * x1), x1, fma((fma((x1 * x1), fma(t_2, 4.0, -6.0), ((t_2 - 3.0) * ((2.0 * x1) * t_2))) * x1), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = pow(x1, 4.0) * (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(fma(-2.0, x2, 3.0) * 2.0)
	t_2 = Float64(fma(x1, fma(x1, 3.0, -1.0), Float64(x2 * 2.0)) / fma(x1, x1, 1.0))
	t_3 = Float64(-1.0 - Float64(x1 * x1))
	t_4 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_5 = Float64(Float64(x1 * x1) - -1.0)
	t_6 = Float64(t_4 / t_5)
	t_7 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 / t_3) * t_0) - Float64(t_3 * 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) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_5) * 3.0)))
	tmp = 0.0
	if (t_7 <= -10000000000000.0)
		tmp = Float64(fma(fma(fma(3.0, x1, -1.0), x1, Float64(-2.0 * x2)), Float64(3.0 / fma(x1, x1, 1.0)), Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (t_7 <= 5e+37)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(fma(Float64(fma(fma(-2.0, x2, fma(-2.0, x2, 3.0)), 2.0, fma(fma(2.0, x2, 3.0), 3.0, fma(14.0, x2, Float64(fma(fma(t_1, x2, fma(t_1, x2, fma(fma(2.0, x2, -3.0), 3.0, 1.0))), 2.0, fma(Float64(fma(2.0, x2, -3.0) * 4.0), x2, -4.0)) * x1)))) - 6.0), x1, fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -2.0)), x1, Float64(-6.0 * x2))) + x1);
	elseif (t_7 <= Inf)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(Float64(fma(Float64(x1 * x1), fma(t_2, 4.0, -6.0), Float64(Float64(t_2 - 3.0) * Float64(Float64(2.0 * x1) * t_2))) * x1), x1, Float64(-6.0 * x2))) + x1);
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / 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[(-2.0 * x2 + 3.0), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$4 / t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$3 * 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] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, -10000000000000.0], N[(N[(N[(N[(3.0 * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[t$95$7, 5e+37], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(N[(-2.0 * x2 + N[(-2.0 * x2 + 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(2.0 * x2 + 3.0), $MachinePrecision] * 3.0 + N[(14.0 * x2 + N[(N[(N[(t$95$1 * x2 + N[(t$95$1 * x2 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision] * x2 + -4.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$2 * 4.0 + -6.0), $MachinePrecision] + N[(N[(t$95$2 - 3.0), $MachinePrecision] * N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 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]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 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)))))) < -1e13

   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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.5%

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

   if -1e13 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999989e37

   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 99.6%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 94.4%

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

   if 4.99999999999999989e37 < (+.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 99.6%

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

   4. Applied rewrites 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 99.5%

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

   7. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.8

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

   9. Applied rewrites 87.8%

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

   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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.0%

   \[\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 -10000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\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 5 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2, x2, \mathsf{fma}\left(-2, x2, 3\right)\right), 2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, \mathsf{fma}\left(14, x2, \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, -4\right)\right) \cdot x1\right)\right)\right) - 6, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right)\right), x1, -6 \cdot x2\right)\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(\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \left(\frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)} - 3\right) \cdot \left(\left(2 \cdot x1\right) \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right)\right) \cdot x1, x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_2 := 6 \cdot \left(x1 \cdot x1\right)\\ t_3 := -1 - x1 \cdot x1\\ t_4 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_5 := x1 \cdot x1 - -1\\ t_6 := \frac{t\_4}{t\_5}\\ t_7 := x1 - \left(\left(\left(\left(\frac{t\_4}{t\_3} \cdot t\_0 - t\_3 \cdot \left(\left(3 - t\_6\right) \cdot \left(\left(2 \cdot x1\right) \cdot t\_6\right) - \left(4 \cdot t\_6 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_5} \cdot 3\right)\\ \mathbf{if}\;t\_7 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), t\_1, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), t\_1, \mathsf{fma}\left(t\_2, \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (/ 3.0 (fma x1 x1 1.0)))
        (t_2 (* 6.0 (* x1 x1)))
        (t_3 (- -1.0 (* x1 x1)))
        (t_4 (- (+ (* x2 2.0) t_0) x1))
        (t_5 (- (* x1 x1) -1.0))
        (t_6 (/ t_4 t_5))
        (t_7
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_4 t_3) t_0)
              (*
               t_3
               (-
                (* (- 3.0 t_6) (* (* 2.0 x1) t_6))
                (* (- (* 4.0 t_6) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_5) 3.0)))))
   (if (<= t_7 -0.02)
     (+
      (fma
       (fma (fma 3.0 x1 -1.0) x1 (* -2.0 x2))
       t_1
       (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2))
      x1)
     (if (<= t_7 INFINITY)
       (+
        (fma
         (fma x1 (fma x1 3.0 -1.0) (* -2.0 x2))
         t_1
         (fma t_2 (fma x1 x1 1.0) (fma x1 (* (* x2 x1) 6.0) x1)))
        x1)
       (+ (* t_2 (* x1 x1)) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = 3.0 / fma(x1, x1, 1.0);
	double t_2 = 6.0 * (x1 * x1);
	double t_3 = -1.0 - (x1 * x1);
	double t_4 = ((x2 * 2.0) + t_0) - x1;
	double t_5 = (x1 * x1) - -1.0;
	double t_6 = t_4 / t_5;
	double t_7 = x1 - ((((((t_4 / t_3) * t_0) - (t_3 * (((3.0 - t_6) * ((2.0 * x1) * t_6)) - (((4.0 * t_6) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_5) * 3.0));
	double tmp;
	if (t_7 <= -0.02) {
		tmp = fma(fma(fma(3.0, x1, -1.0), x1, (-2.0 * x2)), t_1, (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = fma(fma(x1, fma(x1, 3.0, -1.0), (-2.0 * x2)), t_1, fma(t_2, fma(x1, x1, 1.0), fma(x1, ((x2 * x1) * 6.0), x1))) + x1;
	} else {
		tmp = (t_2 * (x1 * x1)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(3.0 / fma(x1, x1, 1.0))
	t_2 = Float64(6.0 * Float64(x1 * x1))
	t_3 = Float64(-1.0 - Float64(x1 * x1))
	t_4 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_5 = Float64(Float64(x1 * x1) - -1.0)
	t_6 = Float64(t_4 / t_5)
	t_7 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 / t_3) * t_0) - Float64(t_3 * 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) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_5) * 3.0)))
	tmp = 0.0
	if (t_7 <= -0.02)
		tmp = Float64(fma(fma(fma(3.0, x1, -1.0), x1, Float64(-2.0 * x2)), t_1, Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (t_7 <= Inf)
		tmp = Float64(fma(fma(x1, fma(x1, 3.0, -1.0), Float64(-2.0 * x2)), t_1, fma(t_2, fma(x1, x1, 1.0), fma(x1, Float64(Float64(x2 * x1) * 6.0), x1))) + x1);
	else
		tmp = Float64(Float64(t_2 * Float64(x1 * 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[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$4 / t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$3 * 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] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, -0.02], N[(N[(N[(N[(3.0 * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(N[(N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(t$95$2 * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(x1 * N[(N[(x2 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(t$95$2 * N[(x1 * x1), $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)))))) < -0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 83.6

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

   5. Applied rewrites 83.6%

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

   7. Applied rewrites 83.7%

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

   if -0.0200000000000000004 < (+.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 99.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.6

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

   10. Applied rewrites 69.6%

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

   11. Taylor expanded in x1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 84.6

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

   13. Applied rewrites 84.6%

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

   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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 98.7

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

4. Final simplification 88.7%

   \[\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 -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\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(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(6 \cdot \left(x1 \cdot x1\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, \left(x2 \cdot x1\right) \cdot 6, x1\right)\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.6× speedup

Math

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

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) (* (* t_2 (* 2.0 x1)) (- t_2 3.0)))
         (fma x1 x1 1.0)
         (* 3.0 t_0))))
      x1)
     (*
      (pow x1 4.0)
      (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) 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), ((t_2 * (2.0 * x1)) * (t_2 - 3.0))), fma(x1, x1, 1.0), (3.0 * t_0)))) + x1;
	} else {
		tmp = pow(x1, 4.0) * (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / 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(t_2 * Float64(2.0 * x1)) * Float64(t_2 - 3.0))), fma(x1, x1, 1.0), Float64(3.0 * t_0)))) + x1);
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / 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[(t$95$2 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(3.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * 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]), $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%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 97.1%

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

   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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

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

Alternative 6: 91.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_1 := {x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ t_2 := \mathsf{fma}\left(2, x2, -3\right) \cdot x2\\ \mathbf{if}\;x1 \leq -500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(t\_2, 4, -2\right)\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 1.26 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), t\_0, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 31000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), t\_0, \mathsf{fma}\left(t\_2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (/ 3.0 (fma x1 x1 1.0)))
        (t_1
         (*
          (pow x1 4.0)
          (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))))
        (t_2 (* (fma 2.0 x2 -3.0) x2)))
   (if (<= x1 -500000000.0)
     t_1
     (if (<= x1 -1.35e-184)
       (+
        (fma
         (fma
          (fma
           -4.0
           x2
           (fma
            (fma -2.0 x2 3.0)
            2.0
            (fma 14.0 x2 (fma (fma 2.0 x2 3.0) 3.0 -6.0))))
          x1
          (fma t_2 4.0 -2.0))
         x1
         (* -6.0 x2))
        x1)
       (if (<= x1 1.26e-139)
         (+
          (fma
           (fma (fma 3.0 x1 -1.0) x1 (* -2.0 x2))
           t_0
           (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2))
          x1)
         (if (<= x1 31000000.0)
           (+
            (fma
             (fma x1 (fma x1 3.0 -1.0) (* -2.0 x2))
             t_0
             (* (fma t_2 4.0 1.0) x1))
            x1)
           (+ t_1 x1)))))))

C

double code(double x1, double x2) {
	double t_0 = 3.0 / fma(x1, x1, 1.0);
	double t_1 = pow(x1, 4.0) * (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1));
	double t_2 = fma(2.0, x2, -3.0) * x2;
	double tmp;
	if (x1 <= -500000000.0) {
		tmp = t_1;
	} else if (x1 <= -1.35e-184) {
		tmp = fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(14.0, x2, fma(fma(2.0, x2, 3.0), 3.0, -6.0)))), x1, fma(t_2, 4.0, -2.0)), x1, (-6.0 * x2)) + x1;
	} else if (x1 <= 1.26e-139) {
		tmp = fma(fma(fma(3.0, x1, -1.0), x1, (-2.0 * x2)), t_0, (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (x1 <= 31000000.0) {
		tmp = fma(fma(x1, fma(x1, 3.0, -1.0), (-2.0 * x2)), t_0, (fma(t_2, 4.0, 1.0) * x1)) + x1;
	} else {
		tmp = t_1 + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(3.0 / fma(x1, x1, 1.0))
	t_1 = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)))
	t_2 = Float64(fma(2.0, x2, -3.0) * x2)
	tmp = 0.0
	if (x1 <= -500000000.0)
		tmp = t_1;
	elseif (x1 <= -1.35e-184)
		tmp = Float64(fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(14.0, x2, fma(fma(2.0, x2, 3.0), 3.0, -6.0)))), x1, fma(t_2, 4.0, -2.0)), x1, Float64(-6.0 * x2)) + x1);
	elseif (x1 <= 1.26e-139)
		tmp = Float64(fma(fma(fma(3.0, x1, -1.0), x1, Float64(-2.0 * x2)), t_0, Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (x1 <= 31000000.0)
		tmp = Float64(fma(fma(x1, fma(x1, 3.0, -1.0), Float64(-2.0 * x2)), t_0, Float64(fma(t_2, 4.0, 1.0) * x1)) + x1);
	else
		tmp = Float64(t_1 + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x1, 4.0], $MachinePrecision] * 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]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision]}, If[LessEqual[x1, -500000000.0], t$95$1, If[LessEqual[x1, -1.35e-184], N[(N[(N[(N[(-4.0 * x2 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * 2.0 + N[(14.0 * x2 + N[(N[(2.0 * x2 + 3.0), $MachinePrecision] * 3.0 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(t$95$2 * 4.0 + -2.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.26e-139], N[(N[(N[(N[(3.0 * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 31000000.0], N[(N[(N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$2 * 4.0 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(t$95$1 + x1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -5e8

   1. Initial program 37.6%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 1.3

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

   5. Applied rewrites 1.3%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 94.5%

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

   if -5e8 < x1 < -1.35e-184

   1. Initial program 98.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 81.0%

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

   if -1.35e-184 < x1 < 1.26000000000000001e-139

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 95.5

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

   5. Applied rewrites 95.5%

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

   7. Applied rewrites 95.6%

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

   if 1.26000000000000001e-139 < x1 < 3.1e7

   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

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 27.0

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

   5. Applied rewrites 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 91.0

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

   10. Applied rewrites 91.0%

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

   if 3.1e7 < x1

   1. Initial program 42.5%

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

   3. Taylor expanded in x1 around -inf

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

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

   5. Applied rewrites 98.9%

      \[\leadsto x1 + \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -500000000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right)\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 1.26 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 31000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_1 := {x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ t_2 := \mathsf{fma}\left(2, x2, -3\right) \cdot x2\\ \mathbf{if}\;x1 \leq -500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(t\_2, 4, -2\right)\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 1.26 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), t\_0, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 31000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), t\_0, \mathsf{fma}\left(t\_2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (/ 3.0 (fma x1 x1 1.0)))
        (t_1
         (*
          (pow x1 4.0)
          (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))))
        (t_2 (* (fma 2.0 x2 -3.0) x2)))
   (if (<= x1 -500000000.0)
     t_1
     (if (<= x1 -1.35e-184)
       (+
        (fma
         (fma
          (fma
           -4.0
           x2
           (fma
            (fma -2.0 x2 3.0)
            2.0
            (fma 14.0 x2 (fma (fma 2.0 x2 3.0) 3.0 -6.0))))
          x1
          (fma t_2 4.0 -2.0))
         x1
         (* -6.0 x2))
        x1)
       (if (<= x1 1.26e-139)
         (+
          (fma
           (fma (fma 3.0 x1 -1.0) x1 (* -2.0 x2))
           t_0
           (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2))
          x1)
         (if (<= x1 31000000.0)
           (+
            (fma
             (fma x1 (fma x1 3.0 -1.0) (* -2.0 x2))
             t_0
             (* (fma t_2 4.0 1.0) x1))
            x1)
           t_1))))))

C

double code(double x1, double x2) {
	double t_0 = 3.0 / fma(x1, x1, 1.0);
	double t_1 = pow(x1, 4.0) * (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1));
	double t_2 = fma(2.0, x2, -3.0) * x2;
	double tmp;
	if (x1 <= -500000000.0) {
		tmp = t_1;
	} else if (x1 <= -1.35e-184) {
		tmp = fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(14.0, x2, fma(fma(2.0, x2, 3.0), 3.0, -6.0)))), x1, fma(t_2, 4.0, -2.0)), x1, (-6.0 * x2)) + x1;
	} else if (x1 <= 1.26e-139) {
		tmp = fma(fma(fma(3.0, x1, -1.0), x1, (-2.0 * x2)), t_0, (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (x1 <= 31000000.0) {
		tmp = fma(fma(x1, fma(x1, 3.0, -1.0), (-2.0 * x2)), t_0, (fma(t_2, 4.0, 1.0) * x1)) + x1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(3.0 / fma(x1, x1, 1.0))
	t_1 = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)))
	t_2 = Float64(fma(2.0, x2, -3.0) * x2)
	tmp = 0.0
	if (x1 <= -500000000.0)
		tmp = t_1;
	elseif (x1 <= -1.35e-184)
		tmp = Float64(fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(14.0, x2, fma(fma(2.0, x2, 3.0), 3.0, -6.0)))), x1, fma(t_2, 4.0, -2.0)), x1, Float64(-6.0 * x2)) + x1);
	elseif (x1 <= 1.26e-139)
		tmp = Float64(fma(fma(fma(3.0, x1, -1.0), x1, Float64(-2.0 * x2)), t_0, Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (x1 <= 31000000.0)
		tmp = Float64(fma(fma(x1, fma(x1, 3.0, -1.0), Float64(-2.0 * x2)), t_0, Float64(fma(t_2, 4.0, 1.0) * x1)) + x1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x1, 4.0], $MachinePrecision] * 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]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision]}, If[LessEqual[x1, -500000000.0], t$95$1, If[LessEqual[x1, -1.35e-184], N[(N[(N[(N[(-4.0 * x2 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * 2.0 + N[(14.0 * x2 + N[(N[(2.0 * x2 + 3.0), $MachinePrecision] * 3.0 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(t$95$2 * 4.0 + -2.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.26e-139], N[(N[(N[(N[(3.0 * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 31000000.0], N[(N[(N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$2 * 4.0 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -5e8 or 3.1e7 < x1

   1. Initial program 39.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 96.3%

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

   if -5e8 < x1 < -1.35e-184

   1. Initial program 98.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 81.0%

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

   if -1.35e-184 < x1 < 1.26000000000000001e-139

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 95.5

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

   5. Applied rewrites 95.5%

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

   7. Applied rewrites 95.6%

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

   if 1.26000000000000001e-139 < x1 < 3.1e7

   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

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 27.0

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

   5. Applied rewrites 27.0%

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 91.0

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

   10. Applied rewrites 91.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -500000000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right)\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 1.26 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 31000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := \mathsf{fma}\left(2, x2, -3\right) \cdot x2\\ \mathbf{if}\;x1 \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(t\_1, 4, -2\right)\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 1.12 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, \mathsf{fma}\left(14, x2, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(t\_1, 4, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1))
        (t_1 (* (fma 2.0 x2 -3.0) x2)))
   (if (<= x1 -3800000000.0)
     t_0
     (if (<= x1 -1.35e-184)
       (+
        (fma
         (fma
          (fma
           -4.0
           x2
           (fma
            (fma -2.0 x2 3.0)
            2.0
            (fma 14.0 x2 (fma (fma 2.0 x2 3.0) 3.0 -6.0))))
          x1
          (fma t_1 4.0 -2.0))
         x1
         (* -6.0 x2))
        x1)
       (if (<= x1 1.12e-181)
         (+
          (fma
           (fma (fma 3.0 x1 -1.0) x1 (* -2.0 x2))
           (/ 3.0 (fma x1 x1 1.0))
           (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2))
          x1)
         (if (<= x1 2.7e+26)
           (fma
            (fma
             (fma
              -4.0
              x2
              (fma
               (fma -2.0 x2 3.0)
               2.0
               (fma (fma 2.0 x2 3.0) 3.0 (fma 14.0 x2 -6.0))))
             x1
             (fma t_1 4.0 -1.0))
            x1
            (* -6.0 x2))
           t_0))))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double t_1 = fma(2.0, x2, -3.0) * x2;
	double tmp;
	if (x1 <= -3800000000.0) {
		tmp = t_0;
	} else if (x1 <= -1.35e-184) {
		tmp = fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(14.0, x2, fma(fma(2.0, x2, 3.0), 3.0, -6.0)))), x1, fma(t_1, 4.0, -2.0)), x1, (-6.0 * x2)) + x1;
	} else if (x1 <= 1.12e-181) {
		tmp = fma(fma(fma(3.0, x1, -1.0), x1, (-2.0 * x2)), (3.0 / fma(x1, x1, 1.0)), (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (x1 <= 2.7e+26) {
		tmp = fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(fma(2.0, x2, 3.0), 3.0, fma(14.0, x2, -6.0)))), x1, fma(t_1, 4.0, -1.0)), x1, (-6.0 * x2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	t_1 = Float64(fma(2.0, x2, -3.0) * x2)
	tmp = 0.0
	if (x1 <= -3800000000.0)
		tmp = t_0;
	elseif (x1 <= -1.35e-184)
		tmp = Float64(fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(14.0, x2, fma(fma(2.0, x2, 3.0), 3.0, -6.0)))), x1, fma(t_1, 4.0, -2.0)), x1, Float64(-6.0 * x2)) + x1);
	elseif (x1 <= 1.12e-181)
		tmp = Float64(fma(fma(fma(3.0, x1, -1.0), x1, Float64(-2.0 * x2)), Float64(3.0 / fma(x1, x1, 1.0)), Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (x1 <= 2.7e+26)
		tmp = fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(fma(2.0, x2, 3.0), 3.0, fma(14.0, x2, -6.0)))), x1, fma(t_1, 4.0, -1.0)), x1, Float64(-6.0 * x2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision]}, If[LessEqual[x1, -3800000000.0], t$95$0, If[LessEqual[x1, -1.35e-184], N[(N[(N[(N[(-4.0 * x2 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * 2.0 + N[(14.0 * x2 + N[(N[(2.0 * x2 + 3.0), $MachinePrecision] * 3.0 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(t$95$1 * 4.0 + -2.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.12e-181], N[(N[(N[(N[(3.0 * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 2.7e+26], N[(N[(N[(-4.0 * x2 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * 2.0 + N[(N[(2.0 * x2 + 3.0), $MachinePrecision] * 3.0 + N[(14.0 * x2 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(t$95$1 * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -3.8e9 or 2.7e26 < x1

   1. Initial program 38.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.7%

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

   if -3.8e9 < x1 < -1.35e-184

   1. Initial program 98.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 81.0%

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

   if -1.35e-184 < x1 < 1.11999999999999997e-181

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 96.4%

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

   if 1.11999999999999997e-181 < x1 < 2.7e26

   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

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

      1. lower-*.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, \mathsf{fma}\left(14, x2, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right)\right), x1, x2 \cdot -6\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3800000000:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right)\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 1.12 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3, x1, -1\right), x1, -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, \mathsf{fma}\left(14, x2, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := \mathsf{fma}\left(2, x2, -3\right) \cdot x2\\ \mathbf{if}\;x1 \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(t\_1, 4, -2\right)\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 1.26 \cdot 10^{-139}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(3 - -2 \cdot x2, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(t\_1, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1))
        (t_1 (* (fma 2.0 x2 -3.0) x2)))
   (if (<= x1 -3800000000.0)
     t_0
     (if (<= x1 -1.35e-184)
       (+
        (fma
         (fma
          (fma
           -4.0
           x2
           (fma
            (fma -2.0 x2 3.0)
            2.0
            (fma 14.0 x2 (fma (fma 2.0 x2 3.0) 3.0 -6.0))))
          x1
          (fma t_1 4.0 -2.0))
         x1
         (* -6.0 x2))
        x1)
       (if (<= x1 1.26e-139)
         (+
          (+
           (* (fma (fma (- 3.0 (* -2.0 x2)) x1 -1.0) x1 (* -2.0 x2)) 3.0)
           (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2))
          x1)
         (if (<= x1 2.7e+26)
           (+
            (fma
             (fma x1 (fma x1 3.0 -1.0) (* -2.0 x2))
             (/ 3.0 (fma x1 x1 1.0))
             (* (fma t_1 4.0 1.0) x1))
            x1)
           t_0))))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double t_1 = fma(2.0, x2, -3.0) * x2;
	double tmp;
	if (x1 <= -3800000000.0) {
		tmp = t_0;
	} else if (x1 <= -1.35e-184) {
		tmp = fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(14.0, x2, fma(fma(2.0, x2, 3.0), 3.0, -6.0)))), x1, fma(t_1, 4.0, -2.0)), x1, (-6.0 * x2)) + x1;
	} else if (x1 <= 1.26e-139) {
		tmp = ((fma(fma((3.0 - (-2.0 * x2)), x1, -1.0), x1, (-2.0 * x2)) * 3.0) + (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (x1 <= 2.7e+26) {
		tmp = fma(fma(x1, fma(x1, 3.0, -1.0), (-2.0 * x2)), (3.0 / fma(x1, x1, 1.0)), (fma(t_1, 4.0, 1.0) * x1)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	t_1 = Float64(fma(2.0, x2, -3.0) * x2)
	tmp = 0.0
	if (x1 <= -3800000000.0)
		tmp = t_0;
	elseif (x1 <= -1.35e-184)
		tmp = Float64(fma(fma(fma(-4.0, x2, fma(fma(-2.0, x2, 3.0), 2.0, fma(14.0, x2, fma(fma(2.0, x2, 3.0), 3.0, -6.0)))), x1, fma(t_1, 4.0, -2.0)), x1, Float64(-6.0 * x2)) + x1);
	elseif (x1 <= 1.26e-139)
		tmp = Float64(Float64(Float64(fma(fma(Float64(3.0 - Float64(-2.0 * x2)), x1, -1.0), x1, Float64(-2.0 * x2)) * 3.0) + Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (x1 <= 2.7e+26)
		tmp = Float64(fma(fma(x1, fma(x1, 3.0, -1.0), Float64(-2.0 * x2)), Float64(3.0 / fma(x1, x1, 1.0)), Float64(fma(t_1, 4.0, 1.0) * x1)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision]}, If[LessEqual[x1, -3800000000.0], t$95$0, If[LessEqual[x1, -1.35e-184], N[(N[(N[(N[(-4.0 * x2 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * 2.0 + N[(14.0 * x2 + N[(N[(2.0 * x2 + 3.0), $MachinePrecision] * 3.0 + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(t$95$1 * 4.0 + -2.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 1.26e-139], N[(N[(N[(N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 2.7e+26], N[(N[(N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * 4.0 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -3.8e9 or 2.7e26 < x1

   1. Initial program 38.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.7%

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

   if -3.8e9 < x1 < -1.35e-184

   1. Initial program 98.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 81.0%

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

   if -1.35e-184 < x1 < 1.26000000000000001e-139

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 95.5

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

   8. Applied rewrites 95.5%

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

   if 1.26000000000000001e-139 < x1 < 2.7e26

   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

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 28.6%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 88.9

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

   10. Applied rewrites 88.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3800000000:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, 3\right), 2, \mathsf{fma}\left(14, x2, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, 3\right), 3, -6\right)\right)\right)\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right)\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 1.26 \cdot 10^{-139}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(3 - -2 \cdot x2, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 1.26 \cdot 10^{-139}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(3 - -2 \cdot x2, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1))
        (t_1
         (+
          (fma
           (fma x1 (fma x1 3.0 -1.0) (* -2.0 x2))
           (/ 3.0 (fma x1 x1 1.0))
           (* (fma (* (fma 2.0 x2 -3.0) x2) 4.0 1.0) x1))
          x1)))
   (if (<= x1 -3800000000.0)
     t_0
     (if (<= x1 -1.35e-184)
       t_1
       (if (<= x1 1.26e-139)
         (+
          (+
           (* (fma (fma (- 3.0 (* -2.0 x2)) x1 -1.0) x1 (* -2.0 x2)) 3.0)
           (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2))
          x1)
         (if (<= x1 2.7e+26) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double t_1 = fma(fma(x1, fma(x1, 3.0, -1.0), (-2.0 * x2)), (3.0 / fma(x1, x1, 1.0)), (fma((fma(2.0, x2, -3.0) * x2), 4.0, 1.0) * x1)) + x1;
	double tmp;
	if (x1 <= -3800000000.0) {
		tmp = t_0;
	} else if (x1 <= -1.35e-184) {
		tmp = t_1;
	} else if (x1 <= 1.26e-139) {
		tmp = ((fma(fma((3.0 - (-2.0 * x2)), x1, -1.0), x1, (-2.0 * x2)) * 3.0) + (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (x1 <= 2.7e+26) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	t_1 = Float64(fma(fma(x1, fma(x1, 3.0, -1.0), Float64(-2.0 * x2)), Float64(3.0 / fma(x1, x1, 1.0)), Float64(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, 1.0) * x1)) + x1)
	tmp = 0.0
	if (x1 <= -3800000000.0)
		tmp = t_0;
	elseif (x1 <= -1.35e-184)
		tmp = t_1;
	elseif (x1 <= 1.26e-139)
		tmp = Float64(Float64(Float64(fma(fma(Float64(3.0 - Float64(-2.0 * x2)), x1, -1.0), x1, Float64(-2.0 * x2)) * 3.0) + Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (x1 <= 2.7e+26)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -3800000000.0], t$95$0, If[LessEqual[x1, -1.35e-184], t$95$1, If[LessEqual[x1, 1.26e-139], N[(N[(N[(N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 2.7e+26], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.8e9 or 2.7e26 < x1

   1. Initial program 38.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.7%

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

   if -3.8e9 < x1 < -1.35e-184 or 1.26000000000000001e-139 < x1 < 2.7e26

   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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 28.6%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 85.1

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

   10. Applied rewrites 85.1%

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

   if -1.35e-184 < x1 < 1.26000000000000001e-139

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 95.5

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

   8. Applied rewrites 95.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3800000000:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -1.35 \cdot 10^{-184}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 1.26 \cdot 10^{-139}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(3 - -2 \cdot x2, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -9.4 \cdot 10^{-185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 3.4 \cdot 10^{-202}:\\ \;\;\;\;\left(-6 \cdot x2 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1))
        (t_1
         (+
          (fma
           (fma x1 (fma x1 3.0 -1.0) (* -2.0 x2))
           (/ 3.0 (fma x1 x1 1.0))
           (* (fma (* (fma 2.0 x2 -3.0) x2) 4.0 1.0) x1))
          x1)))
   (if (<= x1 -3800000000.0)
     t_0
     (if (<= x1 -9.4e-185)
       t_1
       (if (<= x1 3.4e-202)
         (+ (+ (* -6.0 x2) (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2)) x1)
         (if (<= x1 2.7e+26) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double t_1 = fma(fma(x1, fma(x1, 3.0, -1.0), (-2.0 * x2)), (3.0 / fma(x1, x1, 1.0)), (fma((fma(2.0, x2, -3.0) * x2), 4.0, 1.0) * x1)) + x1;
	double tmp;
	if (x1 <= -3800000000.0) {
		tmp = t_0;
	} else if (x1 <= -9.4e-185) {
		tmp = t_1;
	} else if (x1 <= 3.4e-202) {
		tmp = ((-6.0 * x2) + (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (x1 <= 2.7e+26) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	t_1 = Float64(fma(fma(x1, fma(x1, 3.0, -1.0), Float64(-2.0 * x2)), Float64(3.0 / fma(x1, x1, 1.0)), Float64(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, 1.0) * x1)) + x1)
	tmp = 0.0
	if (x1 <= -3800000000.0)
		tmp = t_0;
	elseif (x1 <= -9.4e-185)
		tmp = t_1;
	elseif (x1 <= 3.4e-202)
		tmp = Float64(Float64(Float64(-6.0 * x2) + Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (x1 <= 2.7e+26)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -3800000000.0], t$95$0, If[LessEqual[x1, -9.4e-185], t$95$1, If[LessEqual[x1, 3.4e-202], N[(N[(N[(-6.0 * x2), $MachinePrecision] + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 2.7e+26], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.8e9 or 2.7e26 < x1

   1. Initial program 38.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 88.7%

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

   if -3.8e9 < x1 < -9.4000000000000004e-185 or 3.40000000000000012e-202 < x1 < 2.7e26

   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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 34.4

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

   5. Applied rewrites 34.4%

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 85.3

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

   10. Applied rewrites 85.3%

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

   if -9.4000000000000004e-185 < x1 < 3.40000000000000012e-202

   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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 97.4

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

   8. Applied rewrites 97.4%

      \[\leadsto x1 + \left(\left(\left(\frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot 8\right) \cdot x2\right) \cdot x2 + \color{blue}{-6 \cdot x2}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3800000000:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -9.4 \cdot 10^{-185}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 3.4 \cdot 10^{-202}:\\ \;\;\;\;\left(-6 \cdot x2 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 3, -1\right), -2 \cdot x2\right), \frac{3}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, 1\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x2, -3\right) \cdot x2\\ t_1 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq -9.4 \cdot 10^{-185}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 3.4 \cdot 10^{-202}:\\ \;\;\;\;\left(-6 \cdot x2 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(t\_0, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (fma 2.0 x2 -3.0) x2))
        (t_1 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1)))
   (if (<= x1 -0.8)
     t_1
     (if (<= x1 -9.4e-185)
       (fma (fma t_0 4.0 -1.0) x1 (* -6.0 x2))
       (if (<= x1 3.4e-202)
         (+ (+ (* -6.0 x2) (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2)) x1)
         (if (<= x1 2.7e+26)
           (+ (fma (* x1 x1) x1 (fma (fma t_0 4.0 -2.0) x1 (* -6.0 x2))) x1)
           t_1))))))

C

double code(double x1, double x2) {
	double t_0 = fma(2.0, x2, -3.0) * x2;
	double t_1 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double tmp;
	if (x1 <= -0.8) {
		tmp = t_1;
	} else if (x1 <= -9.4e-185) {
		tmp = fma(fma(t_0, 4.0, -1.0), x1, (-6.0 * x2));
	} else if (x1 <= 3.4e-202) {
		tmp = ((-6.0 * x2) + (((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1;
	} else if (x1 <= 2.7e+26) {
		tmp = fma((x1 * x1), x1, fma(fma(t_0, 4.0, -2.0), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(2.0, x2, -3.0) * x2)
	t_1 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -0.8)
		tmp = t_1;
	elseif (x1 <= -9.4e-185)
		tmp = fma(fma(t_0, 4.0, -1.0), x1, Float64(-6.0 * x2));
	elseif (x1 <= 3.4e-202)
		tmp = Float64(Float64(Float64(-6.0 * x2) + Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2)) + x1);
	elseif (x1 <= 2.7e+26)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(fma(t_0, 4.0, -2.0), x1, Float64(-6.0 * x2))) + x1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -0.8], t$95$1, If[LessEqual[x1, -9.4e-185], N[(N[(t$95$0 * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3.4e-202], N[(N[(N[(-6.0 * x2), $MachinePrecision] + N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 2.7e+26], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(t$95$0 * 4.0 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -0.80000000000000004 or 2.7e26 < x1

   1. Initial program 39.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 88.1%

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

   if -0.80000000000000004 < x1 < -9.4000000000000004e-185

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 23.2

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

   5. Applied rewrites 23.2%

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

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   7. 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. *-commutative N/A

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

      14. lower-*.f64 81.5

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

   8. Applied rewrites 81.5%

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

   if -9.4000000000000004e-185 < x1 < 3.40000000000000012e-202

   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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 97.4

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

   8. Applied rewrites 97.4%

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

   if 3.40000000000000012e-202 < x1 < 2.7e26

   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 99.7%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -0.8:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -9.4 \cdot 10^{-185}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 3.4 \cdot 10^{-202}:\\ \;\;\;\;\left(-6 \cdot x2 + \left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -0.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1)))
   (if (<= x1 -0.8)
     t_0
     (if (<= x1 2.7e+26)
       (+
        (fma
         (* x1 x1)
         x1
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0) x1 (* -6.0 x2)))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double tmp;
	if (x1 <= -0.8) {
		tmp = t_0;
	} else if (x1 <= 2.7e+26) {
		tmp = fma((x1 * x1), x1, fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -0.8)
		tmp = t_0;
	elseif (x1 <= 2.7e+26)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, Float64(-6.0 * x2))) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -0.8], t$95$0, If[LessEqual[x1, 2.7e+26], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -0.80000000000000004 or 2.7e26 < x1

   1. Initial program 39.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 88.1%

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

   if -0.80000000000000004 < x1 < 2.7e26

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 81.2

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

   7. Applied rewrites 81.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -0.8:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -1.25:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -9.5 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 1.15 \cdot 10^{-102}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.7:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1))
        (t_1 (+ (fma (* x1 x1) x1 (* (fma 9.0 x1 -2.0) x1)) x1)))
   (if (<= x1 -1.25)
     t_0
     (if (<= x1 -9.5e-141)
       t_1
       (if (<= x1 1.15e-102) (* -6.0 x2) (if (<= x1 1.7) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double t_1 = fma((x1 * x1), x1, (fma(9.0, x1, -2.0) * x1)) + x1;
	double tmp;
	if (x1 <= -1.25) {
		tmp = t_0;
	} else if (x1 <= -9.5e-141) {
		tmp = t_1;
	} else if (x1 <= 1.15e-102) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	t_1 = Float64(fma(Float64(x1 * x1), x1, Float64(fma(9.0, x1, -2.0) * x1)) + x1)
	tmp = 0.0
	if (x1 <= -1.25)
		tmp = t_0;
	elseif (x1 <= -9.5e-141)
		tmp = t_1;
	elseif (x1 <= 1.15e-102)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 1.7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(9.0 * x1 + -2.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -1.25], t$95$0, If[LessEqual[x1, -9.5e-141], t$95$1, If[LessEqual[x1, 1.15e-102], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 1.7], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.25 or 1.69999999999999996 < x1

   1. Initial program 41.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 86.2

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

   5. Applied rewrites 86.2%

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

   7. Applied rewrites 86.2%

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

   if -1.25 < x1 < -9.49999999999999996e-141 or 1.14999999999999993e-102 < x1 < 1.69999999999999996

   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 99.7%

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

   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 44.8%

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

   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 37.4%

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

   if -9.49999999999999996e-141 < x1 < 1.14999999999999993e-102

   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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.3

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

   8. Applied rewrites 77.3%

      \[\leadsto \color{blue}{x2 \cdot -6} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.25:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -9.5 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 1.15 \cdot 10^{-102}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.7:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(9, x1, -2\right) \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ t_1 := \mathsf{fma}\left(x1 \cdot x1, x1, -2 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -0.62:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -9.5 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 1.15 \cdot 10^{-102}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 0.19:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1))
        (t_1 (+ (fma (* x1 x1) x1 (* -2.0 x1)) x1)))
   (if (<= x1 -0.62)
     t_0
     (if (<= x1 -9.5e-141)
       t_1
       (if (<= x1 1.15e-102) (* -6.0 x2) (if (<= x1 0.19) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double t_1 = fma((x1 * x1), x1, (-2.0 * x1)) + x1;
	double tmp;
	if (x1 <= -0.62) {
		tmp = t_0;
	} else if (x1 <= -9.5e-141) {
		tmp = t_1;
	} else if (x1 <= 1.15e-102) {
		tmp = -6.0 * x2;
	} else if (x1 <= 0.19) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	t_1 = Float64(fma(Float64(x1 * x1), x1, Float64(-2.0 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -0.62)
		tmp = t_0;
	elseif (x1 <= -9.5e-141)
		tmp = t_1;
	elseif (x1 <= 1.15e-102)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 0.19)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-2.0 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -0.62], t$95$0, If[LessEqual[x1, -9.5e-141], t$95$1, If[LessEqual[x1, 1.15e-102], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 0.19], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -0.619999999999999996 or 0.19 < x1

   1. Initial program 41.9%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 85.1

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

   5. Applied rewrites 85.1%

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

   7. Applied rewrites 85.1%

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

   if -0.619999999999999996 < x1 < -9.49999999999999996e-141 or 1.14999999999999993e-102 < x1 < 0.19

   1. Initial program 99.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. 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 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 44.6%

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

   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 37.0%

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

   if -9.49999999999999996e-141 < x1 < 1.14999999999999993e-102

   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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.3

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

   8. Applied rewrites 77.3%

      \[\leadsto \color{blue}{x2 \cdot -6} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -0.62:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq -9.5 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -2 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 1.15 \cdot 10^{-102}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 0.19:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -2 \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \mathbf{if}\;x2 \cdot 2 \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x2 \cdot 2 \leq 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -2 \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (* x1 x1) x1 (* -6.0 x2))))
   (if (<= (* x2 2.0) -1e-123)
     t_0
     (if (<= (* x2 2.0) 1e-144) (+ (fma (* x1 x1) x1 (* -2.0 x1)) x1) t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), x1, (-6.0 * x2));
	double tmp;
	if ((x2 * 2.0) <= -1e-123) {
		tmp = t_0;
	} else if ((x2 * 2.0) <= 1e-144) {
		tmp = fma((x1 * x1), x1, (-2.0 * x1)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2))
	tmp = 0.0
	if (Float64(x2 * 2.0) <= -1e-123)
		tmp = t_0;
	elseif (Float64(x2 * 2.0) <= 1e-144)
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-2.0 * x1)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x2 * 2.0), $MachinePrecision], -1e-123], t$95$0, If[LessEqual[N[(x2 * 2.0), $MachinePrecision], 1e-144], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-2.0 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -1.0000000000000001e-123 or 9.9999999999999995e-145 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 69.6%

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

   4. Applied rewrites 75.6%

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in x1 around 0

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

      1. lower-*.f64 45.0

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

   9. Applied rewrites 45.0%

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

   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 45.1

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

   12. Applied rewrites 45.1%

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

   if -1.0000000000000001e-123 < (*.f64 #s(literal 2 binary64) x2) < 9.9999999999999995e-145

   1. Initial program 69.2%

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in 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 59.6%

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

   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 51.9%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -1 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x2 \cdot 2 \leq 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -2 \cdot x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 86.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -0.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1450000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* (* 6.0 (* x1 x1)) (* x1 x1)) x1)))
   (if (<= x1 -0.8)
     t_0
     (if (<= x1 1450000000.0)
       (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = ((6.0 * (x1 * x1)) * (x1 * x1)) + x1;
	double tmp;
	if (x1 <= -0.8) {
		tmp = t_0;
	} else if (x1 <= 1450000000.0) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -0.8)
		tmp = t_0;
	elseif (x1 <= 1450000000.0)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -0.8], t$95$0, If[LessEqual[x1, 1450000000.0], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -0.80000000000000004 or 1.45e9 < x1

   1. Initial program 40.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.5%

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

   if -0.80000000000000004 < x1 < 1.45e9

   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

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

      1. lower-*.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   7. 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. *-commutative N/A

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

      14. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -0.8:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 1450000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.4% 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 69.5%

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

4. Applied rewrites 74.8%

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

6. Applied rewrites 74.7%

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

7. Taylor expanded in x1 around 0

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

   1. lower-*.f64 41.9

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

9. Applied rewrites 41.9%

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

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 42.1

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

12. Applied rewrites 42.1%

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

Alternative 19: 25.9% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ -6 \cdot x2 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = (-6.0d0) * x2
end function

Java

public static double code(double x1, double x2) {
	return -6.0 * x2;
}

Python

def code(x1, x2):
	return -6.0 * x2

Julia

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

MATLAB

function tmp = code(x1, x2)
	tmp = -6.0 * x2;
end

Wolfram

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

Derivation

1. Initial program 69.5%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 27.1

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

5. Applied rewrites 27.1%

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

6. Taylor expanded in x1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 27.2

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

8. Applied rewrites 27.2%

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

9. Final simplification 27.2%

   \[\leadsto -6 \cdot x2 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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))))))