Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.5% → 99.4%

Time: 25.9s

Alternatives: 20

Speedup: 6.5×

• 46.7% 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: 70.5% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(2.0 * N[(t$95$4 * N[(-3.0 + N[(t$95$3 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$3 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(3.0 * t$95$4 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(6.0 * N[Power[x1, 4.0], $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 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.5

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.6%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

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

Alternative 2: 41.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+304}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x2, 12, -18\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* 8.0 (* x1 (* x2 x2))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -1e+185)
     t_1
     (if (<= t_4 5e+304)
       (* x2 -6.0)
       (if (<= t_4 INFINITY) t_1 (+ x1 (* x1 (fma x2 12.0 -18.0))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = 8.0 * (x1 * (x2 * x2));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -1e+185) {
		tmp = t_1;
	} else if (t_4 <= 5e+304) {
		tmp = x2 * -6.0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + (x1 * fma(x2, 12.0, -18.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -1e+185)
		tmp = t_1;
	elseif (t_4 <= 5e+304)
		tmp = Float64(x2 * -6.0);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(x1 * fma(x2, 12.0, -18.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e+185], t$95$1, If[LessEqual[t$95$4, 5e+304], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 + N[(x1 * N[(x2 * 12.0 + -18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -9.9999999999999998e184 or 4.9999999999999997e304 < (+.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 99.9%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 6.5

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

   5. Applied rewrites 6.5%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 59.2

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

   10. Applied rewrites 59.2%

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

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 56.5%

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

   if -9.9999999999999998e184 < (+.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)))))) < 4.9999999999999997e304

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   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 49.3

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

   8. Applied rewrites 49.3%

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

   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

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 98.7%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 62.3%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + x1 \cdot \left(12 \cdot x2 - 18\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 22.9%

       \[\leadsto x1 + x1 \cdot \mathsf{fma}\left(x2, 12, -18\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -1 \cdot 10^{+185}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x2, 12, -18\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_1
               (+
                (* (* (* x1 2.0) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* t_2 4.0) 6.0))))
              (* t_0 t_2))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1)))))
        (t_4 (fma 2.0 x2 (* x1 (fma x1 3.0 -1.0)))))
   (if (<= t_3 1e-6)
     (+
      x1
      (fma
       (/ (- (* 3.0 (* x1 x1)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
       3.0
       (fma (fma x1 x1 1.0) x1 (* (* x1 x2) (fma x2 8.0 -12.0)))))
     (if (<= t_3 INFINITY)
       (+
        x1
        (fma
         (fma x1 x1 1.0)
         (fma
          2.0
          (* (* (/ x1 (fma x1 x1 1.0)) t_4) (+ -3.0 (/ t_4 (fma x1 x1 1.0))))
          (* (* x1 x1) (fma t_4 (/ 4.0 (fma x1 x1 1.0)) -6.0)))
         (* x2 -6.0)))
       (+ x1 (* 6.0 (pow x1 4.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double t_4 = fma(2.0, x2, (x1 * fma(x1, 3.0, -1.0)));
	double tmp;
	if (t_3 <= 1e-6) {
		tmp = x1 + fma((((3.0 * (x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), x1, ((x1 * x2) * fma(x2, 8.0, -12.0))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = x1 + fma(fma(x1, x1, 1.0), fma(2.0, (((x1 / fma(x1, x1, 1.0)) * t_4) * (-3.0 + (t_4 / fma(x1, x1, 1.0)))), ((x1 * x1) * fma(t_4, (4.0 / fma(x1, x1, 1.0)), -6.0))), (x2 * -6.0));
	} else {
		tmp = x1 + (6.0 * pow(x1, 4.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	t_4 = fma(2.0, x2, Float64(x1 * fma(x1, 3.0, -1.0)))
	tmp = 0.0
	if (t_3 <= 1e-6)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(3.0 * Float64(x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), x1, Float64(Float64(x1 * x2) * fma(x2, 8.0, -12.0)))));
	elseif (t_3 <= Inf)
		tmp = Float64(x1 + fma(fma(x1, x1, 1.0), fma(2.0, Float64(Float64(Float64(x1 / fma(x1, x1, 1.0)) * t_4) * Float64(-3.0 + Float64(t_4 / fma(x1, x1, 1.0)))), Float64(Float64(x1 * x1) * fma(t_4, Float64(4.0 / fma(x1, x1, 1.0)), -6.0))), Float64(x2 * -6.0)));
	else
		tmp = Float64(x1 + Float64(6.0 * (x1 ^ 4.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1e-6], N[(x1 + N[(N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(x1 * x2), $MachinePrecision] * N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(x1 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(2.0 * N[(N[(N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(-3.0 + N[(t$95$4 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$4 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 9.99999999999999955e-7

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 88.2

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 95.7%

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

   if 9.99999999999999955e-7 < (+.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
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.6

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.9

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

   9. Applied rewrites 98.9%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 97.7%

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

Alternative 4: 98.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(2.0 * N[(N[(N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(-3.0 + N[(1.0 / N[(N[(x1 * x1 + 1.0), $MachinePrecision] / N[(x1 * N[(x1 * 3.0 + -1.0), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(t$95$3 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] * N[(3.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(6.0 * N[Power[x1, 4.0], $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 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.5

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. associate-+r+ N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. *-commutative N/A

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

      17. neg-mul-1 N/A

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

      18. distribute-rgt-in N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 99.4

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

   11. Applied rewrites 99.4%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

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

Alternative 5: 32.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x2, 12, -18\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_1
              (+
               (* (* (* x1 2.0) t_2) (- t_2 3.0))
               (* (* x1 x1) (- (* t_2 4.0) 6.0))))
             (* t_0 t_2))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))
        5e+304)
     (* x2 -6.0)
     (+ x1 (* x1 (fma x2 12.0 -18.0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+304) {
		tmp = x2 * -6.0;
	} else {
		tmp = x1 + (x1 * fma(x2, 12.0, -18.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= 5e+304)
		tmp = Float64(x2 * -6.0);
	else
		tmp = Float64(x1 + Float64(x1 * fma(x2, 12.0, -18.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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], 5e+304], N[(x2 * -6.0), $MachinePrecision], N[(x1 + N[(x1 * N[(x2 * 12.0 + -18.0), $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)))))) < 4.9999999999999997e304

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.3

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

   5. Applied rewrites 43.3%

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

   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 43.8

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

   8. Applied rewrites 43.8%

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

   if 4.9999999999999997e304 < (+.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 31.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 86.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 47.6%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + x1 \cdot \left(12 \cdot x2 - 18\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 18.2%

       \[\leadsto x1 + x1 \cdot \mathsf{fma}\left(x2, 12, -18\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x2, 12, -18\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x2\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_1
              (+
               (* (* (* x1 2.0) t_2) (- t_2 3.0))
               (* (* x1 x1) (- (* t_2 4.0) 6.0))))
             (* t_0 t_2))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))
        5e+304)
     (* x2 -6.0)
     (+ x1 (* (* x1 x2) 12.0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+304) {
		tmp = x2 * -6.0;
	} else {
		tmp = x1 + ((x1 * x2) * 12.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = x1 * (x1 * 3.0d0)
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0d0) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((t_2 * 4.0d0) - 6.0d0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))) <= 5d+304) then
        tmp = x2 * (-6.0d0)
    else
        tmp = x1 + ((x1 * x2) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+304) {
		tmp = x2 * -6.0;
	} else {
		tmp = x1 + ((x1 * x2) * 12.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 3.0)
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	tmp = 0
	if (x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+304:
		tmp = x2 * -6.0
	else:
		tmp = x1 + ((x1 * x2) * 12.0)
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= 5e+304)
		tmp = Float64(x2 * -6.0);
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x2) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 3.0);
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = 0.0;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+304)
		tmp = x2 * -6.0;
	else
		tmp = x1 + ((x1 * x2) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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], 5e+304], N[(x2 * -6.0), $MachinePrecision], N[(x1 + N[(N[(x1 * x2), $MachinePrecision] * 12.0), $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)))))) < 4.9999999999999997e304

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.3

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

   5. Applied rewrites 43.3%

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

   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 43.8

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

   8. Applied rewrites 43.8%

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

   if 4.9999999999999997e304 < (+.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 31.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 86.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 18.2%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 17.1%

       \[\leadsto x1 + \left(x1 \cdot x2\right) \cdot 12 \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x2\right) \cdot 12\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(x1 \cdot x2\right) \cdot \mathsf{fma}\left(x2, 8, -12\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3 + \frac{\mathsf{fma}\left(x2, 8, -3\right) - \frac{\mathsf{fma}\left(2 \cdot x2, -6, 18\right)}{x1}}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.4e+19)
   (+
    x1
    (*
     (pow x1 4.0)
     (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))
   (if (<= x1 1.6e+41)
     (+
      x1
      (fma
       (/ (- (* 3.0 (* x1 x1)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
       3.0
       (fma (fma x1 x1 1.0) x1 (* (* x1 x2) (fma x2 8.0 -12.0)))))
     (+
      x1
      (*
       (* x1 (* x1 (* x1 x1)))
       (+
        6.0
        (/
         (+
          -3.0
          (/ (- (fma x2 8.0 -3.0) (/ (fma (* 2.0 x2) -6.0 18.0) x1)) x1))
         x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.4e+19) {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	} else if (x1 <= 1.6e+41) {
		tmp = x1 + fma((((3.0 * (x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), x1, ((x1 * x2) * fma(x2, 8.0, -12.0))));
	} else {
		tmp = x1 + ((x1 * (x1 * (x1 * x1))) * (6.0 + ((-3.0 + ((fma(x2, 8.0, -3.0) - (fma((2.0 * x2), -6.0, 18.0) / x1)) / x1)) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.4e+19)
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))));
	elseif (x1 <= 1.6e+41)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(3.0 * Float64(x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), x1, Float64(Float64(x1 * x2) * fma(x2, 8.0, -12.0)))));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * Float64(x1 * x1))) * Float64(6.0 + Float64(Float64(-3.0 + Float64(Float64(fma(x2, 8.0, -3.0) - Float64(fma(Float64(2.0 * x2), -6.0, 18.0) / x1)) / x1)) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.4e+19], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.6e+41], N[(x1 + N[(N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(x1 * x2), $MachinePrecision] * N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(6.0 + N[(N[(-3.0 + N[(N[(N[(x2 * 8.0 + -3.0), $MachinePrecision] - N[(N[(N[(2.0 * x2), $MachinePrecision] * -6.0 + 18.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.4e19

   1. Initial program 36.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 97.2%

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

   if -1.4e19 < x1 < 1.60000000000000005e41

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 83.1

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 94.1%

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

   if 1.60000000000000005e41 < x1

   1. Initial program 42.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3 + \frac{\mathsf{fma}\left(x2, 8, -3\right) - \frac{\mathsf{fma}\left(2 \cdot x2, -6, 18\right)}{x1}}{x1}}{x1}\right) + x1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(x1 \cdot x2\right) \cdot \mathsf{fma}\left(x2, 8, -12\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3 + \frac{\mathsf{fma}\left(x2, 8, -3\right) - \frac{\mathsf{fma}\left(2 \cdot x2, -6, 18\right)}{x1}}{x1}}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(x1 \cdot x2\right) \cdot \mathsf{fma}\left(x2, 8, -12\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3 + \frac{\mathsf{fma}\left(x2, 8, -3\right) - \frac{\mathsf{fma}\left(2 \cdot x2, -6, 18\right)}{x1}}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.4e+19)
   (+
    x1
    (*
     x1
     (fma
      x1
      (+ 9.0 (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (* 2.0 x2) -12.0)))
      (fma (* 2.0 x2) 6.0 -18.0))))
   (if (<= x1 1.6e+41)
     (+
      x1
      (fma
       (/ (- (* 3.0 (* x1 x1)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
       3.0
       (fma (fma x1 x1 1.0) x1 (* (* x1 x2) (fma x2 8.0 -12.0)))))
     (+
      x1
      (*
       (* x1 (* x1 (* x1 x1)))
       (+
        6.0
        (/
         (+
          -3.0
          (/ (- (fma x2 8.0 -3.0) (/ (fma (* 2.0 x2) -6.0 18.0) x1)) x1))
         x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.4e+19) {
		tmp = x1 + (x1 * fma(x1, (9.0 + fma(x1, fma(x1, 6.0, -3.0), fma(4.0, (2.0 * x2), -12.0))), fma((2.0 * x2), 6.0, -18.0)));
	} else if (x1 <= 1.6e+41) {
		tmp = x1 + fma((((3.0 * (x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), x1, ((x1 * x2) * fma(x2, 8.0, -12.0))));
	} else {
		tmp = x1 + ((x1 * (x1 * (x1 * x1))) * (6.0 + ((-3.0 + ((fma(x2, 8.0, -3.0) - (fma((2.0 * x2), -6.0, 18.0) / x1)) / x1)) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.4e+19)
		tmp = Float64(x1 + Float64(x1 * fma(x1, Float64(9.0 + fma(x1, fma(x1, 6.0, -3.0), fma(4.0, Float64(2.0 * x2), -12.0))), fma(Float64(2.0 * x2), 6.0, -18.0))));
	elseif (x1 <= 1.6e+41)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(3.0 * Float64(x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), x1, Float64(Float64(x1 * x2) * fma(x2, 8.0, -12.0)))));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * Float64(x1 * x1))) * Float64(6.0 + Float64(Float64(-3.0 + Float64(Float64(fma(x2, 8.0, -3.0) - Float64(fma(Float64(2.0 * x2), -6.0, 18.0) / x1)) / x1)) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.4e+19], N[(x1 + N[(x1 * N[(x1 * N[(9.0 + N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(2.0 * x2), $MachinePrecision] + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * x2), $MachinePrecision] * 6.0 + -18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.6e+41], N[(x1 + N[(N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(x1 * x2), $MachinePrecision] * N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(6.0 + N[(N[(-3.0 + N[(N[(N[(x2 * 8.0 + -3.0), $MachinePrecision] - N[(N[(N[(2.0 * x2), $MachinePrecision] * -6.0 + 18.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.4e19

   1. Initial program 36.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 97.2%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 97.2%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, x2 \cdot 2, -12\right)\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   if -1.4e19 < x1 < 1.60000000000000005e41

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 83.1

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 94.1%

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

   if 1.60000000000000005e41 < x1

   1. Initial program 42.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3 + \frac{\mathsf{fma}\left(x2, 8, -3\right) - \frac{\mathsf{fma}\left(2 \cdot x2, -6, 18\right)}{x1}}{x1}}{x1}\right) + x1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(x1 \cdot x2\right) \cdot \mathsf{fma}\left(x2, 8, -12\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3 + \frac{\mathsf{fma}\left(x2, 8, -3\right) - \frac{\mathsf{fma}\left(2 \cdot x2, -6, 18\right)}{x1}}{x1}}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(x1 \cdot x2\right) \cdot \mathsf{fma}\left(x2, 8, -12\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           x1
           (fma
            x1
            (+ 9.0 (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (* 2.0 x2) -12.0)))
            (fma (* 2.0 x2) 6.0 -18.0))))))
   (if (<= x1 -1.4e+19)
     t_0
     (if (<= x1 1.6e+41)
       (+
        x1
        (fma
         (/ (- (* 3.0 (* x1 x1)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
         3.0
         (fma (fma x1 x1 1.0) x1 (* (* x1 x2) (fma x2 8.0 -12.0)))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (x1 * fma(x1, (9.0 + fma(x1, fma(x1, 6.0, -3.0), fma(4.0, (2.0 * x2), -12.0))), fma((2.0 * x2), 6.0, -18.0)));
	double tmp;
	if (x1 <= -1.4e+19) {
		tmp = t_0;
	} else if (x1 <= 1.6e+41) {
		tmp = x1 + fma((((3.0 * (x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), x1, ((x1 * x2) * fma(x2, 8.0, -12.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(x1 * fma(x1, Float64(9.0 + fma(x1, fma(x1, 6.0, -3.0), fma(4.0, Float64(2.0 * x2), -12.0))), fma(Float64(2.0 * x2), 6.0, -18.0))))
	tmp = 0.0
	if (x1 <= -1.4e+19)
		tmp = t_0;
	elseif (x1 <= 1.6e+41)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(3.0 * Float64(x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), x1, Float64(Float64(x1 * x2) * fma(x2, 8.0, -12.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(x1 * N[(x1 * N[(9.0 + N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(2.0 * x2), $MachinePrecision] + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * x2), $MachinePrecision] * 6.0 + -18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.4e+19], t$95$0, If[LessEqual[x1, 1.6e+41], N[(x1 + N[(N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(x1 * x2), $MachinePrecision] * N[(x2 * 8.0 + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.4e19 or 1.60000000000000005e41 < x1

   1. Initial program 39.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 98.3%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 98.3%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, x2 \cdot 2, -12\right)\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   if -1.4e19 < x1 < 1.60000000000000005e41

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 83.1

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 94.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(x1 \cdot x2\right) \cdot \mathsf{fma}\left(x2, 8, -12\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot -2 - x1, 3, x1 + \left(x1 + \mathsf{fma}\left(x1, x1 \cdot x1, 4 \cdot \left(\left(x1 \cdot x2\right) \cdot \mathsf{fma}\left(2, x2, -3\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           x1
           (fma
            x1
            (+ 9.0 (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (* 2.0 x2) -12.0)))
            (fma (* 2.0 x2) 6.0 -18.0))))))
   (if (<= x1 -1.4e+19)
     t_0
     (if (<= x1 1.6e+41)
       (fma
        (- (* x2 -2.0) x1)
        3.0
        (+
         x1
         (+ x1 (fma x1 (* x1 x1) (* 4.0 (* (* x1 x2) (fma 2.0 x2 -3.0)))))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (x1 * fma(x1, (9.0 + fma(x1, fma(x1, 6.0, -3.0), fma(4.0, (2.0 * x2), -12.0))), fma((2.0 * x2), 6.0, -18.0)));
	double tmp;
	if (x1 <= -1.4e+19) {
		tmp = t_0;
	} else if (x1 <= 1.6e+41) {
		tmp = fma(((x2 * -2.0) - x1), 3.0, (x1 + (x1 + fma(x1, (x1 * x1), (4.0 * ((x1 * x2) * fma(2.0, x2, -3.0)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(x1 * fma(x1, Float64(9.0 + fma(x1, fma(x1, 6.0, -3.0), fma(4.0, Float64(2.0 * x2), -12.0))), fma(Float64(2.0 * x2), 6.0, -18.0))))
	tmp = 0.0
	if (x1 <= -1.4e+19)
		tmp = t_0;
	elseif (x1 <= 1.6e+41)
		tmp = fma(Float64(Float64(x2 * -2.0) - x1), 3.0, Float64(x1 + Float64(x1 + fma(x1, Float64(x1 * x1), Float64(4.0 * Float64(Float64(x1 * x2) * fma(2.0, x2, -3.0)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(x1 * N[(x1 * N[(9.0 + N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(2.0 * x2), $MachinePrecision] + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * x2), $MachinePrecision] * 6.0 + -18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.4e+19], t$95$0, If[LessEqual[x1, 1.6e+41], N[(N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision] * 3.0 + N[(x1 + N[(x1 + N[(x1 * N[(x1 * x1), $MachinePrecision] + N[(4.0 * N[(N[(x1 * x2), $MachinePrecision] * N[(2.0 * x2 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.4e19 or 1.60000000000000005e41 < x1

   1. Initial program 39.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 98.3%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 98.3%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, x2 \cdot 2, -12\right)\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   if -1.4e19 < x1 < 1.60000000000000005e41

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.1

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

   8. Applied rewrites 83.1%

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

   10. Applied rewrites 94.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x2 \cdot -2 - x1, 3, x1 + \left(x1 + \mathsf{fma}\left(x1, x1 \cdot x1, 4 \cdot \left(\left(x1 \cdot x2\right) \cdot \mathsf{fma}\left(2, x2, -3\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           x1
           (fma
            x1
            (+ 9.0 (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (* 2.0 x2) -12.0)))
            (fma (* 2.0 x2) 6.0 -18.0))))))
   (if (<= x1 -1.4e+19)
     t_0
     (if (<= x1 1.6e+41)
       (fma x1 (fma 4.0 (* x2 (fma x2 2.0 -3.0)) -1.0) (* x2 -6.0))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (x1 * fma(x1, (9.0 + fma(x1, fma(x1, 6.0, -3.0), fma(4.0, (2.0 * x2), -12.0))), fma((2.0 * x2), 6.0, -18.0)));
	double tmp;
	if (x1 <= -1.4e+19) {
		tmp = t_0;
	} else if (x1 <= 1.6e+41) {
		tmp = fma(x1, fma(4.0, (x2 * fma(x2, 2.0, -3.0)), -1.0), (x2 * -6.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64(x1 * fma(x1, Float64(9.0 + fma(x1, fma(x1, 6.0, -3.0), fma(4.0, Float64(2.0 * x2), -12.0))), fma(Float64(2.0 * x2), 6.0, -18.0))))
	tmp = 0.0
	if (x1 <= -1.4e+19)
		tmp = t_0;
	elseif (x1 <= 1.6e+41)
		tmp = fma(x1, fma(4.0, Float64(x2 * fma(x2, 2.0, -3.0)), -1.0), Float64(x2 * -6.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(x1 * N[(x1 * N[(9.0 + N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(2.0 * x2), $MachinePrecision] + -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * x2), $MachinePrecision] * 6.0 + -18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.4e+19], t$95$0, If[LessEqual[x1, 1.6e+41], N[(x1 * N[(4.0 * N[(x2 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.4e19 or 1.60000000000000005e41 < x1

   1. Initial program 39.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 98.3%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 98.3%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, x2 \cdot 2, -12\right)\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   if -1.4e19 < x1 < 1.60000000000000005e41

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 47.3

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

   5. Applied rewrites 47.3%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, 2 \cdot x2, -12\right)\right), \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x2 \cdot \mathsf{fma}\left(x2, 2, -3\right)\\ \mathbf{if}\;x1 \leq -7 \cdot 10^{+76}:\\ \;\;\;\;x1 + x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 8, 12\right), x1 \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -3, -3\right), -18\right)}{x2}\right)\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, t\_0, -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + 4 \cdot \left(x1 \cdot t\_0\right)\right)\right) + 3 \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x2 (fma x2 2.0 -3.0))))
   (if (<= x1 -7e+76)
     (+
      x1
      (*
       x2
       (fma
        x1
        (fma x1 8.0 12.0)
        (* x1 (/ (fma x1 (fma x1 -3.0 -3.0) -18.0) x2)))))
     (if (<= x1 1.4)
       (fma x1 (fma 4.0 t_0 -1.0) (* x2 -6.0))
       (+
        x1
        (+ (+ x1 (+ (* x1 (* x1 x1)) (* 4.0 (* x1 t_0)))) (* 3.0 3.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x2 * fma(x2, 2.0, -3.0);
	double tmp;
	if (x1 <= -7e+76) {
		tmp = x1 + (x2 * fma(x1, fma(x1, 8.0, 12.0), (x1 * (fma(x1, fma(x1, -3.0, -3.0), -18.0) / x2))));
	} else if (x1 <= 1.4) {
		tmp = fma(x1, fma(4.0, t_0, -1.0), (x2 * -6.0));
	} else {
		tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + (4.0 * (x1 * t_0)))) + (3.0 * 3.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x2 * fma(x2, 2.0, -3.0))
	tmp = 0.0
	if (x1 <= -7e+76)
		tmp = Float64(x1 + Float64(x2 * fma(x1, fma(x1, 8.0, 12.0), Float64(x1 * Float64(fma(x1, fma(x1, -3.0, -3.0), -18.0) / x2)))));
	elseif (x1 <= 1.4)
		tmp = fma(x1, fma(4.0, t_0, -1.0), Float64(x2 * -6.0));
	else
		tmp = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(4.0 * Float64(x1 * t_0)))) + Float64(3.0 * 3.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x2 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -7e+76], N[(x1 + N[(x2 * N[(x1 * N[(x1 * 8.0 + 12.0), $MachinePrecision] + N[(x1 * N[(N[(x1 * N[(x1 * -3.0 + -3.0), $MachinePrecision] + -18.0), $MachinePrecision] / x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.4], N[(x1 * N[(4.0 * t$95$0 + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(x1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.00000000000000001e76

   1. Initial program 14.5%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + x2 \cdot \left(x1 \cdot \left(12 + 8 \cdot x1\right) + \color{blue}{\frac{x1 \cdot \left(x1 \cdot \left(-3 \cdot x1 - 3\right) - 18\right)}{x2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto x1 + x2 \cdot \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x1, 8, 12\right)}, x1 \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -3, -3\right), -18\right)}{x2}\right) \]

   if -7.00000000000000001e76 < x1 < 1.3999999999999999

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 43.7

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

   5. Applied rewrites 43.7%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.9

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

   8. Applied rewrites 75.9%

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

   if 1.3999999999999999 < x1

   1. Initial program 50.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 27.2

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

   5. Applied rewrites 27.2%

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

   6. Taylor expanded in x1 around inf

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

   8. Applied rewrites 76.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -7 \cdot 10^{+76}:\\ \;\;\;\;x1 + x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 8, 12\right), x1 \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -3, -3\right), -18\right)}{x2}\right)\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + 4 \cdot \left(x1 \cdot \left(x2 \cdot \mathsf{fma}\left(x2, 2, -3\right)\right)\right)\right)\right) + 3 \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -7 \cdot 10^{+76}:\\ \;\;\;\;x1 + x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 8, 12\right), x1 \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -3, -3\right), -18\right)}{x2}\right)\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + x2 \cdot 8, \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -7e+76)
   (+
    x1
    (*
     x2
     (fma
      x1
      (fma x1 8.0 12.0)
      (* x1 (/ (fma x1 (fma x1 -3.0 -3.0) -18.0) x2)))))
   (if (<= x1 2.7e+42)
     (fma x1 (fma 4.0 (* x2 (fma x2 2.0 -3.0)) -1.0) (* x2 -6.0))
     (+ x1 (* x1 (fma x1 (+ 9.0 (* x2 8.0)) (fma (* 2.0 x2) 6.0 -18.0)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -7e+76) {
		tmp = x1 + (x2 * fma(x1, fma(x1, 8.0, 12.0), (x1 * (fma(x1, fma(x1, -3.0, -3.0), -18.0) / x2))));
	} else if (x1 <= 2.7e+42) {
		tmp = fma(x1, fma(4.0, (x2 * fma(x2, 2.0, -3.0)), -1.0), (x2 * -6.0));
	} else {
		tmp = x1 + (x1 * fma(x1, (9.0 + (x2 * 8.0)), fma((2.0 * x2), 6.0, -18.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -7e+76)
		tmp = Float64(x1 + Float64(x2 * fma(x1, fma(x1, 8.0, 12.0), Float64(x1 * Float64(fma(x1, fma(x1, -3.0, -3.0), -18.0) / x2)))));
	elseif (x1 <= 2.7e+42)
		tmp = fma(x1, fma(4.0, Float64(x2 * fma(x2, 2.0, -3.0)), -1.0), Float64(x2 * -6.0));
	else
		tmp = Float64(x1 + Float64(x1 * fma(x1, Float64(9.0 + Float64(x2 * 8.0)), fma(Float64(2.0 * x2), 6.0, -18.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -7e+76], N[(x1 + N[(x2 * N[(x1 * N[(x1 * 8.0 + 12.0), $MachinePrecision] + N[(x1 * N[(N[(x1 * N[(x1 * -3.0 + -3.0), $MachinePrecision] + -18.0), $MachinePrecision] / x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.7e+42], N[(x1 * N[(4.0 * N[(x2 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(x1 * N[(x1 * N[(9.0 + N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * x2), $MachinePrecision] * 6.0 + -18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.00000000000000001e76

   1. Initial program 14.5%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + x2 \cdot \left(x1 \cdot \left(12 + 8 \cdot x1\right) + \color{blue}{\frac{x1 \cdot \left(x1 \cdot \left(-3 \cdot x1 - 3\right) - 18\right)}{x2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto x1 + x2 \cdot \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x1, 8, 12\right)}, x1 \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -3, -3\right), -18\right)}{x2}\right) \]

   if -7.00000000000000001e76 < x1 < 2.7000000000000001e42

   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. *-commutative N/A

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

      2. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.2

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

   8. Applied rewrites 75.2%

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

   if 2.7000000000000001e42 < x1

   1. Initial program 42.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 15.7%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + 8 \cdot x2, \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 60.8%

       \[\leadsto x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + x2 \cdot 8, \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -7 \cdot 10^{+76}:\\ \;\;\;\;x1 + x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 8, 12\right), x1 \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -3, -3\right), -18\right)}{x2}\right)\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + x2 \cdot 8, \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot -3\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + x2 \cdot 8, \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.12e+83)
   (+ x1 (* (* x1 (* x1 x1)) -3.0))
   (if (<= x1 2.7e+42)
     (fma x1 (fma 4.0 (* x2 (fma x2 2.0 -3.0)) -1.0) (* x2 -6.0))
     (+ x1 (* x1 (fma x1 (+ 9.0 (* x2 8.0)) (fma (* 2.0 x2) 6.0 -18.0)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.12e+83) {
		tmp = x1 + ((x1 * (x1 * x1)) * -3.0);
	} else if (x1 <= 2.7e+42) {
		tmp = fma(x1, fma(4.0, (x2 * fma(x2, 2.0, -3.0)), -1.0), (x2 * -6.0));
	} else {
		tmp = x1 + (x1 * fma(x1, (9.0 + (x2 * 8.0)), fma((2.0 * x2), 6.0, -18.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.12e+83)
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) * -3.0));
	elseif (x1 <= 2.7e+42)
		tmp = fma(x1, fma(4.0, Float64(x2 * fma(x2, 2.0, -3.0)), -1.0), Float64(x2 * -6.0));
	else
		tmp = Float64(x1 + Float64(x1 * fma(x1, Float64(9.0 + Float64(x2 * 8.0)), fma(Float64(2.0 * x2), 6.0, -18.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.12e+83], N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.7e+42], N[(x1 * N[(4.0 * N[(x2 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(x1 * N[(x1 * N[(9.0 + N[(x2 * 8.0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * x2), $MachinePrecision] * 6.0 + -18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.12e83

   1. Initial program 14.5%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.5%

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

   if -1.12e83 < x1 < 2.7000000000000001e42

   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. *-commutative N/A

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

      2. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.2

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

   8. Applied rewrites 75.2%

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

   if 2.7000000000000001e42 < x1

   1. Initial program 42.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 15.7%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + 8 \cdot x2, \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 60.8%

       \[\leadsto x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + x2 \cdot 8, \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot -3\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \mathsf{fma}\left(x1, 9 + x2 \cdot 8, \mathsf{fma}\left(2 \cdot x2, 6, -18\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot -3\\ \mathbf{elif}\;x1 \leq -1.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{8 \cdot \left(x2 \cdot x2\right)}{x1}\\ \mathbf{elif}\;x1 \leq 2.1 \cdot 10^{-96}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 \leq 4.6 \cdot 10^{+42}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(x2 \cdot \mathsf{fma}\left(x1, 8, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.12e+83)
   (+ x1 (* (* x1 (* x1 x1)) -3.0))
   (if (<= x1 -1.3e-52)
     (/ (* 8.0 (* x2 x2)) x1)
     (if (<= x1 2.1e-96)
       (* x2 -6.0)
       (if (<= x1 4.6e+42)
         (* 8.0 (* x1 (* x2 x2)))
         (+ x1 (* x1 (* x2 (fma x1 8.0 12.0)))))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.12e+83) {
		tmp = x1 + ((x1 * (x1 * x1)) * -3.0);
	} else if (x1 <= -1.3e-52) {
		tmp = (8.0 * (x2 * x2)) / x1;
	} else if (x1 <= 2.1e-96) {
		tmp = x2 * -6.0;
	} else if (x1 <= 4.6e+42) {
		tmp = 8.0 * (x1 * (x2 * x2));
	} else {
		tmp = x1 + (x1 * (x2 * fma(x1, 8.0, 12.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.12e+83)
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) * -3.0));
	elseif (x1 <= -1.3e-52)
		tmp = Float64(Float64(8.0 * Float64(x2 * x2)) / x1);
	elseif (x1 <= 2.1e-96)
		tmp = Float64(x2 * -6.0);
	elseif (x1 <= 4.6e+42)
		tmp = Float64(8.0 * Float64(x1 * Float64(x2 * x2)));
	else
		tmp = Float64(x1 + Float64(x1 * Float64(x2 * fma(x1, 8.0, 12.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.12e+83], N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.3e-52], N[(N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision], If[LessEqual[x1, 2.1e-96], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[x1, 4.6e+42], N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(x1 * N[(x2 * N[(x1 * 8.0 + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -1.12e83

   1. Initial program 14.5%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.5%

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

   if -1.12e83 < x1 < -1.2999999999999999e-52

   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. *-commutative N/A

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

      2. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 42.7

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

   10. Applied rewrites 42.7%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 34.4%

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

   if -1.2999999999999999e-52 < x1 < 2.10000000000000001e-96

   1. Initial program 98.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   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 62.7

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

   8. Applied rewrites 62.7%

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

   if 2.10000000000000001e-96 < x1 < 4.6e42

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 16.1

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

   5. Applied rewrites 16.1%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 49.7

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

   10. Applied rewrites 49.7%

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

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 49.8%

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

   if 4.6e42 < x1

   1. Initial program 42.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 15.7%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 41.0%

       \[\leadsto x1 + x1 \cdot \left(x2 \cdot \mathsf{fma}\left(x1, \color{blue}{8}, 12\right)\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 16: 53.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{if}\;x1 \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot -3\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.1 \cdot 10^{-96}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 \leq 4.6 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(x2 \cdot \mathsf{fma}\left(x1, 8, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 8.0 (* x1 (* x2 x2)))))
   (if (<= x1 -1.12e+83)
     (+ x1 (* (* x1 (* x1 x1)) -3.0))
     (if (<= x1 -1.5e-53)
       t_0
       (if (<= x1 2.1e-96)
         (* x2 -6.0)
         (if (<= x1 4.6e+42) t_0 (+ x1 (* x1 (* x2 (fma x1 8.0 12.0))))))))))

C

double code(double x1, double x2) {
	double t_0 = 8.0 * (x1 * (x2 * x2));
	double tmp;
	if (x1 <= -1.12e+83) {
		tmp = x1 + ((x1 * (x1 * x1)) * -3.0);
	} else if (x1 <= -1.5e-53) {
		tmp = t_0;
	} else if (x1 <= 2.1e-96) {
		tmp = x2 * -6.0;
	} else if (x1 <= 4.6e+42) {
		tmp = t_0;
	} else {
		tmp = x1 + (x1 * (x2 * fma(x1, 8.0, 12.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	tmp = 0.0
	if (x1 <= -1.12e+83)
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) * -3.0));
	elseif (x1 <= -1.5e-53)
		tmp = t_0;
	elseif (x1 <= 2.1e-96)
		tmp = Float64(x2 * -6.0);
	elseif (x1 <= 4.6e+42)
		tmp = t_0;
	else
		tmp = Float64(x1 + Float64(x1 * Float64(x2 * fma(x1, 8.0, 12.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.12e+83], N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.5e-53], t$95$0, If[LessEqual[x1, 2.1e-96], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[x1, 4.6e+42], t$95$0, N[(x1 + N[(x1 * N[(x2 * N[(x1 * 8.0 + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.12e83

   1. Initial program 14.5%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.5%

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

   if -1.12e83 < x1 < -1.5000000000000001e-53 or 2.10000000000000001e-96 < x1 < 4.6e42

   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. *-commutative N/A

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

      2. lower-*.f64 9.1

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

   5. Applied rewrites 9.1%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 30.1%

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

   8. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 46.0

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

   10. Applied rewrites 46.0%

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

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 41.4%

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

   if -1.5000000000000001e-53 < x1 < 2.10000000000000001e-96

   1. Initial program 98.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   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 62.7

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

   8. Applied rewrites 62.7%

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

   if 4.6e42 < x1

   1. Initial program 42.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 15.7%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 41.0%

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

Alternative 17: 69.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot -3\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + x1 \cdot \left(x2 \cdot \mathsf{fma}\left(x1, 8, 12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.12e+83)
   (+ x1 (* (* x1 (* x1 x1)) -3.0))
   (if (<= x1 2.7e+42)
     (fma x1 (fma 4.0 (* x2 (fma x2 2.0 -3.0)) -1.0) (* x2 -6.0))
     (+ x1 (* x1 (* x2 (fma x1 8.0 12.0)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.12e+83) {
		tmp = x1 + ((x1 * (x1 * x1)) * -3.0);
	} else if (x1 <= 2.7e+42) {
		tmp = fma(x1, fma(4.0, (x2 * fma(x2, 2.0, -3.0)), -1.0), (x2 * -6.0));
	} else {
		tmp = x1 + (x1 * (x2 * fma(x1, 8.0, 12.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.12e+83)
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) * -3.0));
	elseif (x1 <= 2.7e+42)
		tmp = fma(x1, fma(4.0, Float64(x2 * fma(x2, 2.0, -3.0)), -1.0), Float64(x2 * -6.0));
	else
		tmp = Float64(x1 + Float64(x1 * Float64(x2 * fma(x1, 8.0, 12.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.12e+83], N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 2.7e+42], N[(x1 * N[(4.0 * N[(x2 * N[(x2 * 2.0 + -3.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(x1 * N[(x2 * N[(x1 * 8.0 + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.12e83

   1. Initial program 14.5%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.5%

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

   if -1.12e83 < x1 < 2.7000000000000001e42

   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. *-commutative N/A

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

      2. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.2

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

   8. Applied rewrites 75.2%

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

   if 2.7000000000000001e42 < x1

   1. Initial program 42.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 99.9%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 15.7%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 41.0%

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

Alternative 18: 55.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{if}\;x1 \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot -3\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.1 \cdot 10^{-96}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 8.0 (* x1 (* x2 x2)))))
   (if (<= x1 -1.12e+83)
     (+ x1 (* (* x1 (* x1 x1)) -3.0))
     (if (<= x1 -1.5e-53) t_0 (if (<= x1 2.1e-96) (* x2 -6.0) t_0)))))

C

double code(double x1, double x2) {
	double t_0 = 8.0 * (x1 * (x2 * x2));
	double tmp;
	if (x1 <= -1.12e+83) {
		tmp = x1 + ((x1 * (x1 * x1)) * -3.0);
	} else if (x1 <= -1.5e-53) {
		tmp = t_0;
	} else if (x1 <= 2.1e-96) {
		tmp = x2 * -6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 8.0d0 * (x1 * (x2 * x2))
    if (x1 <= (-1.12d+83)) then
        tmp = x1 + ((x1 * (x1 * x1)) * (-3.0d0))
    else if (x1 <= (-1.5d-53)) then
        tmp = t_0
    else if (x1 <= 2.1d-96) then
        tmp = x2 * (-6.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = 8.0 * (x1 * (x2 * x2));
	double tmp;
	if (x1 <= -1.12e+83) {
		tmp = x1 + ((x1 * (x1 * x1)) * -3.0);
	} else if (x1 <= -1.5e-53) {
		tmp = t_0;
	} else if (x1 <= 2.1e-96) {
		tmp = x2 * -6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = 8.0 * (x1 * (x2 * x2))
	tmp = 0
	if x1 <= -1.12e+83:
		tmp = x1 + ((x1 * (x1 * x1)) * -3.0)
	elif x1 <= -1.5e-53:
		tmp = t_0
	elif x1 <= 2.1e-96:
		tmp = x2 * -6.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	tmp = 0.0
	if (x1 <= -1.12e+83)
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) * -3.0));
	elseif (x1 <= -1.5e-53)
		tmp = t_0;
	elseif (x1 <= 2.1e-96)
		tmp = Float64(x2 * -6.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = 8.0 * (x1 * (x2 * x2));
	tmp = 0.0;
	if (x1 <= -1.12e+83)
		tmp = x1 + ((x1 * (x1 * x1)) * -3.0);
	elseif (x1 <= -1.5e-53)
		tmp = t_0;
	elseif (x1 <= 2.1e-96)
		tmp = x2 * -6.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.12e+83], N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.5e-53], t$95$0, If[LessEqual[x1, 2.1e-96], N[(x2 * -6.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.12e83

   1. Initial program 14.5%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto x1 + x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), x1 \cdot -3\right), \mathsf{fma}\left(x2 \cdot 2, 6, -18\right)\right)} \]

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 86.5%

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

   if -1.12e83 < x1 < -1.5000000000000001e-53 or 2.10000000000000001e-96 < x1

   1. Initial program 72.9%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 7.4

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

   5. Applied rewrites 7.4%

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

      1. lift-+.f64 N/A

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

   7. Applied rewrites 40.5%

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

   8. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 37.0%

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

   if -1.5000000000000001e-53 < x1 < 2.10000000000000001e-96

   1. Initial program 98.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   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 62.7

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

   8. Applied rewrites 62.7%

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

Alternative 19: 27.3% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.65 \cdot 10^{+19}:\\ \;\;\;\;x1 + x1 \cdot -18\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.65e+19) (+ x1 (* x1 -18.0)) (+ x1 (* x2 -6.0))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.65e+19) {
		tmp = x1 + (x1 * -18.0);
	} else {
		tmp = x1 + (x2 * -6.0);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if (x1 <= (-1.65d+19)) then
        tmp = x1 + (x1 * (-18.0d0))
    else
        tmp = x1 + (x2 * (-6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.65e+19) {
		tmp = x1 + (x1 * -18.0);
	} else {
		tmp = x1 + (x2 * -6.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x1 <= -1.65e+19:
		tmp = x1 + (x1 * -18.0)
	else:
		tmp = x1 + (x2 * -6.0)
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.65e+19)
		tmp = Float64(x1 + Float64(x1 * -18.0));
	else
		tmp = Float64(x1 + Float64(x2 * -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x1 <= -1.65e+19)
		tmp = x1 + (x1 * -18.0);
	else
		tmp = x1 + (x2 * -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.65e+19], N[(x1 + N[(x1 * -18.0), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.65e19

   1. Initial program 36.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 97.2%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 17.0%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + -18 \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 6.5%

       \[\leadsto x1 + x1 \cdot -18 \]

   if -1.65e19 < x1

   1. Initial program 83.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 35.6

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

   5. Applied rewrites 35.6%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 25.9% accurate, 49.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x1_, x2_] := N[(x2 * -6.0), $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. *-commutative N/A

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

   2. lower-*.f64 25.5

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

5. Applied rewrites 25.5%

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

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 25.4

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

8. Applied rewrites 25.4%

   \[\leadsto \color{blue}{x2 \cdot -6} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024238 
(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))))))