Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.1% → 98.2%

Time: 28.2s

Alternatives: 33

Speedup: 7.1×

• 47.1% 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: 70.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_1 := \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 1.25 \cdot 10^{+44}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2 \cdot -2 - x1, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_1, 4, -6\right), \frac{\left(-3 + t\_1\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_1 (/ t_0 (fma x1 x1 1.0))))
   (if (<= x1 -1.5e+55)
     (* 6.0 (pow x1 4.0))
     (if (<= x1 1.25e+44)
       (+
        x1
        (fma
         (- (* x2 -2.0) x1)
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (* x1 (fma t_1 4.0 -6.0))
           (/ (* (+ -3.0 t_1) (* (* x1 2.0) t_0)) (fma x1 x1 1.0)))
          (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))
       (*
        (pow x1 4.0)
        (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))))

C

double code(double x1, double x2) {
	double t_0 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double t_1 = t_0 / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -1.5e+55) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= 1.25e+44) {
		tmp = x1 + fma(((x2 * -2.0) - x1), 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma(t_1, 4.0, -6.0)), (((-3.0 + t_1) * ((x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	} else {
		tmp = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_1 = Float64(t_0 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -1.5e+55)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= 1.25e+44)
		tmp = Float64(x1 + fma(Float64(Float64(x2 * -2.0) - x1), 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_1, 4.0, -6.0)), Float64(Float64(Float64(-3.0 + t_1) * Float64(Float64(x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	else
		tmp = 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)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+55], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.25e+44], N[(x1 + N[(N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision] * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(t$95$1 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + t$95$1), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.50000000000000008e55

   1. Initial program 9.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.5

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.50000000000000008e55 < x1 < 1.2499999999999999e44

   1. Initial program 99.2%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 36.8%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 36.7%

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

   11. Taylor expanded in x1 around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 99.7

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

   13. Applied rewrites 99.7%

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

   if 1.2499999999999999e44 < x1

   1. Initial program 37.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 1.25 \cdot 10^{+44}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2 \cdot -2 - x1, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot \left(x1 \cdot 3\right)\\ t_3 := \left(t\_2 + 2 \cdot x2\right) - x1\\ t_4 := \frac{t\_3}{-1 - x1 \cdot x1}\\ t_5 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(t\_2 \cdot t\_4 + t\_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_4\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{t\_3}{t\_0}\right) \cdot \left(3 + t\_4\right)\right)\right)\right)\right) + 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq -0.001:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_5 \leq 0.0004:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot 9, x1, -x1\right)\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* 8.0 (* x1 (* x2 x2))))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (- (+ t_2 (* 2.0 x2)) x1))
        (t_4 (/ t_3 (- -1.0 (* x1 x1))))
        (t_5
         (+
          x1
          (+
           (+
            x1
            (-
             (* x1 (* x1 x1))
             (+
              (* t_2 t_4)
              (*
               t_0
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_4)))
                (* (* (* x1 2.0) (/ t_3 t_0)) (+ 3.0 t_4)))))))
           (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_5 -2e+292)
     t_1
     (if (<= t_5 -0.001)
       (* x2 -6.0)
       (if (<= t_5 0.0004)
         (fma (* x1 9.0) x1 (- x1))
         (if (<= t_5 5e+143)
           (+ x1 (* x2 -6.0))
           (if (<= t_5 INFINITY) t_1 (+ x1 (* (* x1 x1) 9.0)))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = 8.0 * (x1 * (x2 * x2));
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = (t_2 + (2.0 * x2)) - x1;
	double t_4 = t_3 / (-1.0 - (x1 * x1));
	double t_5 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_2 * t_4) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_4))) + (((x1 * 2.0) * (t_3 / t_0)) * (3.0 + t_4))))))) + (3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_5 <= -2e+292) {
		tmp = t_1;
	} else if (t_5 <= -0.001) {
		tmp = x2 * -6.0;
	} else if (t_5 <= 0.0004) {
		tmp = fma((x1 * 9.0), x1, -x1);
	} else if (t_5 <= 5e+143) {
		tmp = x1 + (x2 * -6.0);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(Float64(t_2 + Float64(2.0 * x2)) - x1)
	t_4 = Float64(t_3 / Float64(-1.0 - Float64(x1 * x1)))
	t_5 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) - Float64(Float64(t_2 * t_4) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_4))) + Float64(Float64(Float64(x1 * 2.0) * Float64(t_3 / t_0)) * Float64(3.0 + t_4))))))) + Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_5 <= -2e+292)
		tmp = t_1;
	elseif (t_5 <= -0.001)
		tmp = Float64(x2 * -6.0);
	elseif (t_5 <= 0.0004)
		tmp = fma(Float64(x1 * 9.0), x1, Float64(-x1));
	elseif (t_5 <= 5e+143)
		tmp = Float64(x1 + Float64(x2 * -6.0));
	elseif (t_5 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+292], t$95$1, If[LessEqual[t$95$5, -0.001], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$5, 0.0004], N[(N[(x1 * 9.0), $MachinePrecision] * x1 + (-x1)), $MachinePrecision], If[LessEqual[t$95$5, 5e+143], N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$1, N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e292 or 5.00000000000000012e143 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   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 5.7

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 52.1%

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

   if -2e292 < (+.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)))))) < -1e-3

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 67.2

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

   5. Applied rewrites 67.2%

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

   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 67.2

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

   8. Applied rewrites 67.2%

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

   if -1e-3 < (+.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)))))) < 4.00000000000000019e-4

   1. Initial program 98.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 25.0

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

   5. Applied rewrites 25.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 69.1%

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

   13. Applied rewrites 69.2%

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

   if 4.00000000000000019e-4 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 5.00000000000000012e143

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
   10. Step-by-step derivation

   11. Applied rewrites 80.5%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 -2 \cdot 10^{+292}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 -0.001:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 0.0004:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot 9, x1, -x1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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^{+143}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := \mathsf{fma}\left(x2, 8 \cdot \left(x1 \cdot x2\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ t_2 := x1 \cdot \left(x1 \cdot 3\right)\\ t_3 := \left(t\_2 + 2 \cdot x2\right) - x1\\ t_4 := \frac{t\_3}{-1 - x1 \cdot x1}\\ t_5 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(t\_2 \cdot t\_4 + t\_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_4\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{t\_3}{t\_0}\right) \cdot \left(3 + t\_4\right)\right)\right)\right)\right) + 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{+256}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (fma x2 (* 8.0 (* x1 x2)) (* x1 (fma x1 9.0 -1.0))))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (- (+ t_2 (* 2.0 x2)) x1))
        (t_4 (/ t_3 (- -1.0 (* x1 x1))))
        (t_5
         (+
          x1
          (+
           (+
            x1
            (-
             (* x1 (* x1 x1))
             (+
              (* t_2 t_4)
              (*
               t_0
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_4)))
                (* (* (* x1 2.0) (/ t_3 t_0)) (+ 3.0 t_4)))))))
           (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_5 -1e+256)
     t_1
     (if (<= t_5 5e+143)
       (fma x1 (fma x1 9.0 -1.0) (* x2 -6.0))
       (if (<= t_5 INFINITY) t_1 (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = fma(x2, (8.0 * (x1 * x2)), (x1 * fma(x1, 9.0, -1.0)));
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = (t_2 + (2.0 * x2)) - x1;
	double t_4 = t_3 / (-1.0 - (x1 * x1));
	double t_5 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_2 * t_4) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_4))) + (((x1 * 2.0) * (t_3 / t_0)) * (3.0 + t_4))))))) + (3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_5 <= -1e+256) {
		tmp = t_1;
	} else if (t_5 <= 5e+143) {
		tmp = fma(x1, fma(x1, 9.0, -1.0), (x2 * -6.0));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = fma(x2, Float64(8.0 * Float64(x1 * x2)), Float64(x1 * fma(x1, 9.0, -1.0)))
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(Float64(t_2 + Float64(2.0 * x2)) - x1)
	t_4 = Float64(t_3 / Float64(-1.0 - Float64(x1 * x1)))
	t_5 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) - Float64(Float64(t_2 * t_4) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_4))) + Float64(Float64(Float64(x1 * 2.0) * Float64(t_3 / t_0)) * Float64(3.0 + t_4))))))) + Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_5 <= -1e+256)
		tmp = t_1;
	elseif (t_5 <= 5e+143)
		tmp = fma(x1, fma(x1, 9.0, -1.0), Float64(x2 * -6.0));
	elseif (t_5 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x2 * N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -1e+256], t$95$1, If[LessEqual[t$95$5, 5e+143], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$1, N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1e256 or 5.00000000000000012e143 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   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 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 49.3%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.6%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x2 around inf

       \[\leadsto \mathsf{fma}\left(x2, 8 \cdot \left(x1 \cdot \color{blue}{x2}\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 57.7%

       \[\leadsto \mathsf{fma}\left(x2, 8 \cdot \left(x2 \cdot \color{blue}{x1}\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if -1e256 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 5.00000000000000012e143

   1. Initial program 99.0%

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

   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

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

   8. Applied rewrites 80.2%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 83.1%

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

   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 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
   10. Step-by-step derivation

   11. Applied rewrites 80.5%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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^{+256}:\\ \;\;\;\;\mathsf{fma}\left(x2, 8 \cdot \left(x1 \cdot x2\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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:\\ \;\;\;\;\mathsf{fma}\left(x2, 8 \cdot \left(x1 \cdot x2\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{-1 - x1 \cdot x1}\\ t_4 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(t\_1 \cdot t\_3 + t\_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_3\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{t\_2}{t\_0}\right) \cdot \left(3 + t\_3\right)\right)\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, 8 \cdot \left(x2 \cdot x2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 (- -1.0 (* x1 x1))))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (-
             (* x1 (* x1 x1))
             (+
              (* t_1 t_3)
              (*
               t_0
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_3)))
                (* (* (* x1 2.0) (/ t_2 t_0)) (+ 3.0 t_3)))))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_4 -2e+292)
     (* 8.0 (* x1 (* x2 x2)))
     (if (<= t_4 2e+41)
       (fma x1 (fma x1 9.0 -1.0) (* x2 -6.0))
       (if (<= t_4 INFINITY)
         (+ x1 (fma x1 (* 8.0 (* x2 x2)) (* x2 -6.0)))
         (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / (-1.0 - (x1 * x1));
	double t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_1 * t_3) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_3))) + (((x1 * 2.0) * (t_2 / t_0)) * (3.0 + t_3))))))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_4 <= -2e+292) {
		tmp = 8.0 * (x1 * (x2 * x2));
	} else if (t_4 <= 2e+41) {
		tmp = fma(x1, fma(x1, 9.0, -1.0), (x2 * -6.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = x1 + fma(x1, (8.0 * (x2 * x2)), (x2 * -6.0));
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / Float64(-1.0 - Float64(x1 * x1)))
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) - Float64(Float64(t_1 * t_3) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_3))) + Float64(Float64(Float64(x1 * 2.0) * Float64(t_2 / t_0)) * Float64(3.0 + t_3))))))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_4 <= -2e+292)
		tmp = Float64(8.0 * Float64(x1 * Float64(x2 * x2)));
	elseif (t_4 <= 2e+41)
		tmp = fma(x1, fma(x1, 9.0, -1.0), Float64(x2 * -6.0));
	elseif (t_4 <= Inf)
		tmp = Float64(x1 + fma(x1, Float64(8.0 * Float64(x2 * x2)), Float64(x2 * -6.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * t$95$3), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+292], N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+41], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(x1 + N[(x1 * N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e292

   1. Initial program 100.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.7%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 82.7%

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

   if -2e292 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.00000000000000001e41

   1. Initial program 99.0%

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

   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 40.3

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

   5. Applied rewrites 40.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 84.5%

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

   if 2.00000000000000001e41 < (+.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 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 48.1%

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

   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 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
   10. Step-by-step derivation

   11. Applied rewrites 80.5%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 -2 \cdot 10^{+292}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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(x1, 8 \cdot \left(x2 \cdot x2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{-1 - x1 \cdot x1}\\ t_4 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(t\_1 \cdot t\_3 + t\_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_3\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{t\_2}{t\_0}\right) \cdot \left(3 + t\_3\right)\right)\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1, 8 \cdot \left(x2 \cdot x2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 (- -1.0 (* x1 x1))))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (-
             (* x1 (* x1 x1))
             (+
              (* t_1 t_3)
              (*
               t_0
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_3)))
                (* (* (* x1 2.0) (/ t_2 t_0)) (+ 3.0 t_3)))))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_4 -2e+292)
     (* 8.0 (* x1 (* x2 x2)))
     (if (<= t_4 2e+41)
       (fma x1 (fma x1 9.0 -1.0) (* x2 -6.0))
       (if (<= t_4 INFINITY)
         (fma x1 (* 8.0 (* x2 x2)) (* x2 -6.0))
         (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / (-1.0 - (x1 * x1));
	double t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_1 * t_3) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_3))) + (((x1 * 2.0) * (t_2 / t_0)) * (3.0 + t_3))))))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_4 <= -2e+292) {
		tmp = 8.0 * (x1 * (x2 * x2));
	} else if (t_4 <= 2e+41) {
		tmp = fma(x1, fma(x1, 9.0, -1.0), (x2 * -6.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = fma(x1, (8.0 * (x2 * x2)), (x2 * -6.0));
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / Float64(-1.0 - Float64(x1 * x1)))
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) - Float64(Float64(t_1 * t_3) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_3))) + Float64(Float64(Float64(x1 * 2.0) * Float64(t_2 / t_0)) * Float64(3.0 + t_3))))))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_4 <= -2e+292)
		tmp = Float64(8.0 * Float64(x1 * Float64(x2 * x2)));
	elseif (t_4 <= 2e+41)
		tmp = fma(x1, fma(x1, 9.0, -1.0), Float64(x2 * -6.0));
	elseif (t_4 <= Inf)
		tmp = fma(x1, Float64(8.0 * Float64(x2 * x2)), Float64(x2 * -6.0));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * t$95$3), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+292], N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+41], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(x1 * N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e292

   1. Initial program 100.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.7%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 82.7%

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

   if -2e292 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.00000000000000001e41

   1. Initial program 99.0%

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

   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 40.3

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

   5. Applied rewrites 40.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 84.5%

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

   if 2.00000000000000001e41 < (+.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 99.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 14.6

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

   5. Applied rewrites 14.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 48.9%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 47.8%

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

   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 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
   10. Step-by-step derivation

   11. Applied rewrites 80.5%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 -2 \cdot 10^{+292}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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:\\ \;\;\;\;\mathsf{fma}\left(x1, 8 \cdot \left(x2 \cdot x2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot \left(x1 \cdot 3\right)\\ t_3 := \left(t\_2 + 2 \cdot x2\right) - x1\\ t_4 := \frac{t\_3}{-1 - x1 \cdot x1}\\ t_5 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(t\_2 \cdot t\_4 + t\_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_4\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{t\_3}{t\_0}\right) \cdot \left(3 + t\_4\right)\right)\right)\right)\right) + 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;x1 + t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* 8.0 (* x1 (* x2 x2))))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (- (+ t_2 (* 2.0 x2)) x1))
        (t_4 (/ t_3 (- -1.0 (* x1 x1))))
        (t_5
         (+
          x1
          (+
           (+
            x1
            (-
             (* x1 (* x1 x1))
             (+
              (* t_2 t_4)
              (*
               t_0
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_4)))
                (* (* (* x1 2.0) (/ t_3 t_0)) (+ 3.0 t_4)))))))
           (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_5 -2e+292)
     t_1
     (if (<= t_5 5e+143)
       (fma x1 (fma x1 9.0 -1.0) (* x2 -6.0))
       (if (<= t_5 INFINITY) (+ x1 t_1) (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = 8.0 * (x1 * (x2 * x2));
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = (t_2 + (2.0 * x2)) - x1;
	double t_4 = t_3 / (-1.0 - (x1 * x1));
	double t_5 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_2 * t_4) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_4))) + (((x1 * 2.0) * (t_3 / t_0)) * (3.0 + t_4))))))) + (3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_5 <= -2e+292) {
		tmp = t_1;
	} else if (t_5 <= 5e+143) {
		tmp = fma(x1, fma(x1, 9.0, -1.0), (x2 * -6.0));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = x1 + t_1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(Float64(t_2 + Float64(2.0 * x2)) - x1)
	t_4 = Float64(t_3 / Float64(-1.0 - Float64(x1 * x1)))
	t_5 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) - Float64(Float64(t_2 * t_4) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_4))) + Float64(Float64(Float64(x1 * 2.0) * Float64(t_3 / t_0)) * Float64(3.0 + t_4))))))) + Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_5 <= -2e+292)
		tmp = t_1;
	elseif (t_5 <= 5e+143)
		tmp = fma(x1, fma(x1, 9.0, -1.0), Float64(x2 * -6.0));
	elseif (t_5 <= Inf)
		tmp = Float64(x1 + t_1);
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+292], t$95$1, If[LessEqual[t$95$5, 5e+143], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(x1 + t$95$1), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e292

   1. Initial program 100.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.7%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 82.7%

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

   if -2e292 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 5.00000000000000012e143

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 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 40.6

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 79.5%

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

   if 5.00000000000000012e143 < (+.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 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 45.6%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 45.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
   10. Step-by-step derivation

   11. Applied rewrites 80.5%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 -2 \cdot 10^{+292}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 + 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot \left(x1 \cdot 3\right)\\ t_3 := \left(t\_2 + 2 \cdot x2\right) - x1\\ t_4 := \frac{t\_3}{-1 - x1 \cdot x1}\\ t_5 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(t\_2 \cdot t\_4 + t\_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_4\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{t\_3}{t\_0}\right) \cdot \left(3 + t\_4\right)\right)\right)\right)\right) + 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* 8.0 (* x1 (* x2 x2))))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (- (+ t_2 (* 2.0 x2)) x1))
        (t_4 (/ t_3 (- -1.0 (* x1 x1))))
        (t_5
         (+
          x1
          (+
           (+
            x1
            (-
             (* x1 (* x1 x1))
             (+
              (* t_2 t_4)
              (*
               t_0
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_4)))
                (* (* (* x1 2.0) (/ t_3 t_0)) (+ 3.0 t_4)))))))
           (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_5 -2e+292)
     t_1
     (if (<= t_5 5e+143)
       (fma x1 (fma x1 9.0 -1.0) (* x2 -6.0))
       (if (<= t_5 INFINITY) t_1 (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = 8.0 * (x1 * (x2 * x2));
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = (t_2 + (2.0 * x2)) - x1;
	double t_4 = t_3 / (-1.0 - (x1 * x1));
	double t_5 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_2 * t_4) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_4))) + (((x1 * 2.0) * (t_3 / t_0)) * (3.0 + t_4))))))) + (3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_5 <= -2e+292) {
		tmp = t_1;
	} else if (t_5 <= 5e+143) {
		tmp = fma(x1, fma(x1, 9.0, -1.0), (x2 * -6.0));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(Float64(t_2 + Float64(2.0 * x2)) - x1)
	t_4 = Float64(t_3 / Float64(-1.0 - Float64(x1 * x1)))
	t_5 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) - Float64(Float64(t_2 * t_4) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_4))) + Float64(Float64(Float64(x1 * 2.0) * Float64(t_3 / t_0)) * Float64(3.0 + t_4))))))) + Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_5 <= -2e+292)
		tmp = t_1;
	elseif (t_5 <= 5e+143)
		tmp = fma(x1, fma(x1, 9.0, -1.0), Float64(x2 * -6.0));
	elseif (t_5 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+292], t$95$1, If[LessEqual[t$95$5, 5e+143], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$1, N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e292 or 5.00000000000000012e143 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   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 5.7

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 52.1%

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

   if -2e292 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 5.00000000000000012e143

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 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 40.6

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 79.5%

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

   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 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
   10. Step-by-step derivation

   11. Applied rewrites 80.5%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 -2 \cdot 10^{+292}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot \left(x1 \cdot 3\right)\\ t_3 := \left(t\_2 + 2 \cdot x2\right) - x1\\ t_4 := \frac{t\_3}{-1 - x1 \cdot x1}\\ t_5 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(t\_2 \cdot t\_4 + t\_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_4\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{t\_3}{t\_0}\right) \cdot \left(3 + t\_4\right)\right)\right)\right)\right) + 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* 8.0 (* x1 (* x2 x2))))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (- (+ t_2 (* 2.0 x2)) x1))
        (t_4 (/ t_3 (- -1.0 (* x1 x1))))
        (t_5
         (+
          x1
          (+
           (+
            x1
            (-
             (* x1 (* x1 x1))
             (+
              (* t_2 t_4)
              (*
               t_0
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_4)))
                (* (* (* x1 2.0) (/ t_3 t_0)) (+ 3.0 t_4)))))))
           (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_5 -2e+292)
     t_1
     (if (<= t_5 5e+143)
       (fma x2 -6.0 (* x1 (fma x1 9.0 -1.0)))
       (if (<= t_5 INFINITY) t_1 (+ x1 (* (* x1 x1) 9.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = 8.0 * (x1 * (x2 * x2));
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = (t_2 + (2.0 * x2)) - x1;
	double t_4 = t_3 / (-1.0 - (x1 * x1));
	double t_5 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_2 * t_4) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_4))) + (((x1 * 2.0) * (t_3 / t_0)) * (3.0 + t_4))))))) + (3.0 * (((t_2 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_5 <= -2e+292) {
		tmp = t_1;
	} else if (t_5 <= 5e+143) {
		tmp = fma(x2, -6.0, (x1 * fma(x1, 9.0, -1.0)));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(8.0 * Float64(x1 * Float64(x2 * x2)))
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(Float64(t_2 + Float64(2.0 * x2)) - x1)
	t_4 = Float64(t_3 / Float64(-1.0 - Float64(x1 * x1)))
	t_5 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) - Float64(Float64(t_2 * t_4) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_4))) + Float64(Float64(Float64(x1 * 2.0) * Float64(t_3 / t_0)) * Float64(3.0 + t_4))))))) + Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_5 <= -2e+292)
		tmp = t_1;
	elseif (t_5 <= 5e+143)
		tmp = fma(x2, -6.0, Float64(x1 * fma(x1, 9.0, -1.0)));
	elseif (t_5 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * t$95$4), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+292], t$95$1, If[LessEqual[t$95$5, 5e+143], N[(x2 * -6.0 + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$1, N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2e292 or 5.00000000000000012e143 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   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 5.7

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 52.1%

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

   if -2e292 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 5.00000000000000012e143

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 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 40.6

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 88.5%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 79.5%

       \[\leadsto \mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
   10. Step-by-step derivation

   11. Applied rewrites 80.5%

       \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 -2 \cdot 10^{+292}:\\ \;\;\;\;8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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^{+143}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot x1 + 1\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := \left(t\_1 + 2 \cdot x2\right) - x1\\ t_3 := \frac{t\_2}{-1 - x1 \cdot x1}\\ t_4 := x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(t\_1 \cdot t\_3 + t\_0 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot t\_3\right) + \left(\left(x1 \cdot 2\right) \cdot \frac{t\_2}{t\_0}\right) \cdot \left(3 + t\_3\right)\right)\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_0}\right)\\ \mathbf{if}\;t\_4 \leq -0.001:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_4 \leq 0.0004:\\ \;\;\;\;-x1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+256}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (* x1 x1) 1.0))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (- (+ t_1 (* 2.0 x2)) x1))
        (t_3 (/ t_2 (- -1.0 (* x1 x1))))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (-
             (* x1 (* x1 x1))
             (+
              (* t_1 t_3)
              (*
               t_0
               (+
                (* (* x1 x1) (+ 6.0 (* 4.0 t_3)))
                (* (* (* x1 2.0) (/ t_2 t_0)) (+ 3.0 t_3)))))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_0))))))
   (if (<= t_4 -0.001)
     (* x2 -6.0)
     (if (<= t_4 0.0004)
       (- x1)
       (if (<= t_4 5e+256) (+ x1 (* x2 -6.0)) (* (* x1 x1) 9.0))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / (-1.0 - (x1 * x1));
	double t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_1 * t_3) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_3))) + (((x1 * 2.0) * (t_2 / t_0)) * (3.0 + t_3))))))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_4 <= -0.001) {
		tmp = x2 * -6.0;
	} else if (t_4 <= 0.0004) {
		tmp = -x1;
	} else if (t_4 <= 5e+256) {
		tmp = x1 + (x2 * -6.0);
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (x1 * x1) + 1.0d0
    t_1 = x1 * (x1 * 3.0d0)
    t_2 = (t_1 + (2.0d0 * x2)) - x1
    t_3 = t_2 / ((-1.0d0) - (x1 * x1))
    t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_1 * t_3) + (t_0 * (((x1 * x1) * (6.0d0 + (4.0d0 * t_3))) + (((x1 * 2.0d0) * (t_2 / t_0)) * (3.0d0 + t_3))))))) + (3.0d0 * (((t_1 - (2.0d0 * x2)) - x1) / t_0)))
    if (t_4 <= (-0.001d0)) then
        tmp = x2 * (-6.0d0)
    else if (t_4 <= 0.0004d0) then
        tmp = -x1
    else if (t_4 <= 5d+256) then
        tmp = x1 + (x2 * (-6.0d0))
    else
        tmp = (x1 * x1) * 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) + 1.0;
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (t_1 + (2.0 * x2)) - x1;
	double t_3 = t_2 / (-1.0 - (x1 * x1));
	double t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_1 * t_3) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_3))) + (((x1 * 2.0) * (t_2 / t_0)) * (3.0 + t_3))))))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	double tmp;
	if (t_4 <= -0.001) {
		tmp = x2 * -6.0;
	} else if (t_4 <= 0.0004) {
		tmp = -x1;
	} else if (t_4 <= 5e+256) {
		tmp = x1 + (x2 * -6.0);
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) + 1.0
	t_1 = x1 * (x1 * 3.0)
	t_2 = (t_1 + (2.0 * x2)) - x1
	t_3 = t_2 / (-1.0 - (x1 * x1))
	t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_1 * t_3) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_3))) + (((x1 * 2.0) * (t_2 / t_0)) * (3.0 + t_3))))))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)))
	tmp = 0
	if t_4 <= -0.001:
		tmp = x2 * -6.0
	elif t_4 <= 0.0004:
		tmp = -x1
	elif t_4 <= 5e+256:
		tmp = x1 + (x2 * -6.0)
	else:
		tmp = (x1 * x1) * 9.0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) + 1.0)
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(t_1 + Float64(2.0 * x2)) - x1)
	t_3 = Float64(t_2 / Float64(-1.0 - Float64(x1 * x1)))
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) - Float64(Float64(t_1 * t_3) + Float64(t_0 * Float64(Float64(Float64(x1 * x1) * Float64(6.0 + Float64(4.0 * t_3))) + Float64(Float64(Float64(x1 * 2.0) * Float64(t_2 / t_0)) * Float64(3.0 + t_3))))))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_0))))
	tmp = 0.0
	if (t_4 <= -0.001)
		tmp = Float64(x2 * -6.0);
	elseif (t_4 <= 0.0004)
		tmp = Float64(-x1);
	elseif (t_4 <= 5e+256)
		tmp = Float64(x1 + Float64(x2 * -6.0));
	else
		tmp = Float64(Float64(x1 * x1) * 9.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) + 1.0;
	t_1 = x1 * (x1 * 3.0);
	t_2 = (t_1 + (2.0 * x2)) - x1;
	t_3 = t_2 / (-1.0 - (x1 * x1));
	t_4 = x1 + ((x1 + ((x1 * (x1 * x1)) - ((t_1 * t_3) + (t_0 * (((x1 * x1) * (6.0 + (4.0 * t_3))) + (((x1 * 2.0) * (t_2 / t_0)) * (3.0 + t_3))))))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_0)));
	tmp = 0.0;
	if (t_4 <= -0.001)
		tmp = x2 * -6.0;
	elseif (t_4 <= 0.0004)
		tmp = -x1;
	elseif (t_4 <= 5e+256)
		tmp = x1 + (x2 * -6.0);
	else
		tmp = (x1 * x1) * 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * t$95$3), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(4.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x1 * 2.0), $MachinePrecision] * N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(3.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.001], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$4, 0.0004], (-x1), If[LessEqual[t$95$4, 5e+256], N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 99.8%

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

   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 50.5

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

   5. Applied rewrites 50.5%

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

   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 50.5

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

   8. Applied rewrites 50.5%

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

   if -1e-3 < (+.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)))))) < 4.00000000000000019e-4

   1. Initial program 98.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 25.0

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

   5. Applied rewrites 25.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 97.2%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 69.1%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around 0

       \[\leadsto -1 \cdot x1 \]
   13. Step-by-step derivation

   14. Applied rewrites 64.5%

       \[\leadsto -x1 \]

   if 4.00000000000000019e-4 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 5.00000000000000015e256

   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 32.7

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

   5. Applied rewrites 32.7%

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

   if 5.00000000000000015e256 < (+.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 25.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 66.5%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 60.8%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto 9 \cdot {x1}^{\color{blue}{2}} \]
   13. Step-by-step derivation

   14. Applied rewrites 60.9%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 -0.001:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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 0.0004:\\ \;\;\;\;-x1\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) - \left(\left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1} + \left(x1 \cdot x1 + 1\right) \cdot \left(\left(x1 \cdot x1\right) \cdot \left(6 + 4 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right) + \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(3 + \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{-1 - x1 \cdot x1}\right)\right)\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^{+256}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 99.3%

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

   4. Applied rewrites 95.4%

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

   6. Applied rewrites 99.6%

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

   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 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 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.7%

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

Alternative 11: 96.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.00125:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1)))))
   (if (<= x1 -1.5e+55)
     (* 6.0 (pow x1 4.0))
     (if (<= x1 -0.00125)
       (+
        x1
        (fma
         3.0
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (*
            x1
            (fma
             (/ (fma (* x1 x1) 3.0 (- (* 2.0 x2) x1)) (fma x1 x1 1.0))
             4.0
             -6.0))
           (/
            (* (+ -3.0 (/ t_0 (fma x1 x1 1.0))) (* (* x1 2.0) t_0))
            (fma x1 x1 1.0)))
          (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))
       (if (<= x1 650.0)
         (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double tmp;
	if (x1 <= -1.5e+55) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= -0.00125) {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma((fma((x1 * x1), 3.0, ((2.0 * x2) - x1)) / fma(x1, x1, 1.0)), 4.0, -6.0)), (((-3.0 + (t_0 / fma(x1, x1, 1.0))) * ((x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	} else if (x1 <= 650.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	tmp = 0.0
	if (x1 <= -1.5e+55)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= -0.00125)
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(Float64(fma(Float64(x1 * x1), 3.0, Float64(Float64(2.0 * x2) - x1)) / fma(x1, x1, 1.0)), 4.0, -6.0)), Float64(Float64(Float64(-3.0 + Float64(t_0 / fma(x1, x1, 1.0))) * Float64(Float64(x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	elseif (x1 <= 650.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = 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)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+55], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -0.00125], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(N[(N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(N[(2.0 * x2), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 650.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.50000000000000008e55

   1. Initial program 9.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.5

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.50000000000000008e55 < x1 < -0.00125000000000000003

   1. Initial program 99.4%

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 94.7%

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

       1. lift-fma.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-neg.f64 N/A

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

       5. lift-neg.f64 N/A

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

       6. associate-+r+ N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. +-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*r* N/A

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

       15. lift-neg.f64 N/A

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

       16. sub-neg N/A

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

       17. associate--l+ N/A

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

       18. associate-*r* N/A

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

       19. lift-*.f64 N/A

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

       20. *-commutative N/A

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

       21. lower-fma.f64 N/A

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

       22. lower--.f64 N/A

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

   12. Applied rewrites 94.8%

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

   if -0.00125000000000000003 < x1 < 650

   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 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 650 < 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 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 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 95.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.00125:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x1 \cdot x1, 3, 2 \cdot x2 - x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_1 := \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.00125:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_1, 4, -6\right), \frac{\left(-3 + t\_1\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_1 (/ t_0 (fma x1 x1 1.0))))
   (if (<= x1 -1.5e+55)
     (* 6.0 (pow x1 4.0))
     (if (<= x1 -0.00125)
       (+
        x1
        (fma
         3.0
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (* x1 (fma t_1 4.0 -6.0))
           (/ (* (+ -3.0 t_1) (* (* x1 2.0) t_0)) (fma x1 x1 1.0)))
          (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))
       (if (<= x1 650.0)
         (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double t_1 = t_0 / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -1.5e+55) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= -0.00125) {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma(t_1, 4.0, -6.0)), (((-3.0 + t_1) * ((x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	} else if (x1 <= 650.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_1 = Float64(t_0 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -1.5e+55)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= -0.00125)
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_1, 4.0, -6.0)), Float64(Float64(Float64(-3.0 + t_1) * Float64(Float64(x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	elseif (x1 <= 650.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = 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)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+55], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -0.00125], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(t$95$1 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + t$95$1), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 650.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.50000000000000008e55

   1. Initial program 9.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.5

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.50000000000000008e55 < x1 < -0.00125000000000000003

   1. Initial program 99.4%

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 94.7%

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

   if -0.00125000000000000003 < x1 < 650

   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 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 650 < 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 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 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 95.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.00125:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.95:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(\left(x1 \cdot 2\right) \cdot t\_0\right) \cdot \frac{-1 + \frac{\mathsf{fma}\left(x2, 2, -3\right)}{x1}}{x1}}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1)))))
   (if (<= x1 -1.5e+55)
     (* 6.0 (pow x1 4.0))
     (if (<= x1 -0.95)
       (+
        x1
        (fma
         3.0
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (* x1 (fma (/ t_0 (fma x1 x1 1.0)) 4.0 -6.0))
           (/
            (* (* (* x1 2.0) t_0) (/ (+ -1.0 (/ (fma x2 2.0 -3.0) x1)) x1))
            (fma x1 x1 1.0)))
          (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))
       (if (<= x1 650.0)
         (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double tmp;
	if (x1 <= -1.5e+55) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= -0.95) {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma((t_0 / fma(x1, x1, 1.0)), 4.0, -6.0)), ((((x1 * 2.0) * t_0) * ((-1.0 + (fma(x2, 2.0, -3.0) / x1)) / x1)) / fma(x1, x1, 1.0))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	} else if (x1 <= 650.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	tmp = 0.0
	if (x1 <= -1.5e+55)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= -0.95)
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(Float64(t_0 / fma(x1, x1, 1.0)), 4.0, -6.0)), Float64(Float64(Float64(Float64(x1 * 2.0) * t_0) * Float64(Float64(-1.0 + Float64(fma(x2, 2.0, -3.0) / x1)) / x1)) / fma(x1, x1, 1.0))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	elseif (x1 <= 650.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = 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)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+55], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -0.95], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(-1.0 + N[(N[(x2 * 2.0 + -3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 650.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.50000000000000008e55

   1. Initial program 9.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.5

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.50000000000000008e55 < x1 < -0.94999999999999996

   1. Initial program 99.4%

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 94.7%

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

   11. Taylor expanded in x1 around -inf

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

   13. Applied rewrites 94.6%

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

   if -0.94999999999999996 < x1 < 650

   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 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 650 < 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 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 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 95.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.95:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right) \cdot \frac{-1 + \frac{\mathsf{fma}\left(x2, 2, -3\right)}{x1}}{x1}}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_1 := \frac{t\_0}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.00125:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_1, 4, -6\right), \frac{\left(-3 + t\_1\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_1 (/ t_0 (fma x1 x1 1.0))))
   (if (<= x1 -1.5e+55)
     (* 6.0 (pow x1 4.0))
     (if (<= x1 -0.00125)
       (+
        x1
        (fma
         3.0
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (* x1 (fma t_1 4.0 -6.0))
           (/ (* (+ -3.0 t_1) (* (* x1 2.0) t_0)) (fma x1 x1 1.0)))
          (* x1 (* x1 x1)))))
       (if (<= x1 650.0)
         (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double t_1 = t_0 / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -1.5e+55) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= -0.00125) {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma(t_1, 4.0, -6.0)), (((-3.0 + t_1) * ((x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), (x1 * (x1 * x1))));
	} else if (x1 <= 650.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_1 = Float64(t_0 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -1.5e+55)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= -0.00125)
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_1, 4.0, -6.0)), Float64(Float64(Float64(-3.0 + t_1) * Float64(Float64(x1 * 2.0) * t_0)) / fma(x1, x1, 1.0))), Float64(x1 * Float64(x1 * x1)))));
	elseif (x1 <= 650.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = 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)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+55], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -0.00125], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(t$95$1 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + t$95$1), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 650.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.50000000000000008e55

   1. Initial program 9.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.5

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.50000000000000008e55 < x1 < -0.00125000000000000003

   1. Initial program 99.4%

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in x1 around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 94.0

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

   10. Applied rewrites 94.0%

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

   if -0.00125000000000000003 < x1 < 650

   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 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 650 < 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 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 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 95.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.00125:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 96.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{if}\;x1 \leq -6.5 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -0.0013:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right) \cdot \mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (pow x1 4.0)
          (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))
   (if (<= x1 -6.5e+46)
     t_0
     (if (<= x1 -0.0013)
       (+
        x1
        (fma
         3.0
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (*
            x1
            (fma
             (/ (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))) (fma x1 x1 1.0))
             4.0
             -6.0))
           (/ (* 8.0 (* x1 (* x2 x2))) (* (fma x1 x1 1.0) (fma x1 x1 1.0))))
          (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))
       (if (<= x1 650.0)
         (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1));
	double tmp;
	if (x1 <= -6.5e+46) {
		tmp = t_0;
	} else if (x1 <= -0.0013) {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma((fma(2.0, x2, fma(x1, (x1 * 3.0), -x1)) / fma(x1, x1, 1.0)), 4.0, -6.0)), ((8.0 * (x1 * (x2 * x2))) / (fma(x1, x1, 1.0) * fma(x1, x1, 1.0)))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	} else if (x1 <= 650.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = 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)))
	tmp = 0.0
	if (x1 <= -6.5e+46)
		tmp = t_0;
	elseif (x1 <= -0.0013)
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(Float64(fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1))) / fma(x1, x1, 1.0)), 4.0, -6.0)), Float64(Float64(8.0 * Float64(x1 * Float64(x2 * x2))) / Float64(fma(x1, x1, 1.0) * fma(x1, x1, 1.0)))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	elseif (x1 <= 650.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = 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]}, If[LessEqual[x1, -6.5e+46], t$95$0, If[LessEqual[x1, -0.0013], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(N[(N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 650.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -6.50000000000000008e46 or 650 < x1

   1. Initial program 26.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 97.5%

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

   if -6.50000000000000008e46 < x1 < -0.0012999999999999999

   1. Initial program 99.4%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 94.3%

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

   11. Taylor expanded in x2 around inf

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. lower-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 N/A

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

       15. +-commutative N/A

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

       16. unpow2 N/A

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

       17. lower-fma.f64 94.2

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

   13. Applied rewrites 94.2%

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

   if -0.0012999999999999999 < x1 < 650

   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 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -6.5 \cdot 10^{+46}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq -0.0013:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right) \cdot \mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 96.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.0013:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right) \cdot \mathsf{fma}\left(x1, x1, 1\right)}\right), t\_0\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right), t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1))))
   (if (<= x1 -1.5e+55)
     (* 6.0 (pow x1 4.0))
     (if (<= x1 -0.0013)
       (+
        x1
        (fma
         3.0
         3.0
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (*
            x1
            (fma
             (/ (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))) (fma x1 x1 1.0))
             4.0
             -6.0))
           (/ (* 8.0 (* x1 (* x2 x2))) (* (fma x1 x1 1.0) (fma x1 x1 1.0))))
          t_0)))
       (if (<= x1 650.0)
         (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
         (+
          x1
          (fma
           3.0
           3.0
           (fma
            (fma x1 x1 1.0)
            (*
             (* x1 x1)
             (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) -6.0) x1) 4.0) x1)))
            t_0))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1));
	double tmp;
	if (x1 <= -1.5e+55) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= -0.0013) {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma((fma(2.0, x2, fma(x1, (x1 * 3.0), -x1)) / fma(x1, x1, 1.0)), 4.0, -6.0)), ((8.0 * (x1 * (x2 * x2))) / (fma(x1, x1, 1.0) * fma(x1, x1, 1.0)))), t_0));
	} else if (x1 <= 650.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), ((x1 * x1) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))), t_0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1))
	tmp = 0.0
	if (x1 <= -1.5e+55)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= -0.0013)
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(Float64(fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1))) / fma(x1, x1, 1.0)), 4.0, -6.0)), Float64(Float64(8.0 * Float64(x1 * Float64(x2 * x2))) / Float64(fma(x1, x1, 1.0) * fma(x1, x1, 1.0)))), t_0)));
	elseif (x1 <= 650.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))), t_0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+55], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -0.0013], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(N[(N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 650.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + -6.0), $MachinePrecision] / x1), $MachinePrecision] - 4.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.50000000000000008e55

   1. Initial program 9.6%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.5

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -1.50000000000000008e55 < x1 < -0.0012999999999999999

   1. Initial program 99.4%

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 94.7%

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

   11. Taylor expanded in x2 around inf

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. lower-/.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. unpow2 N/A

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

       14. lower-fma.f64 N/A

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

       15. +-commutative N/A

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

       16. unpow2 N/A

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

       17. lower-fma.f64 88.0

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

   13. Applied rewrites 88.0%

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

   if -0.0012999999999999999 < x1 < 650

   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 38.3

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 650 < 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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 32.5%

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

   11. Taylor expanded in x1 around -inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. lower--.f64 N/A

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

       7. lower-/.f64 N/A

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

   13. Applied rewrites 95.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq -0.0013:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right) \cdot \mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 94.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -10500000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(x2, 2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 + \mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x2, 6, x1 \cdot -2\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4.5e+153)
   (* (* x1 x1) 9.0)
   (if (<= x1 -10500000000.0)
     (+
      x1
      (fma
       (/ (- (* 3.0 (* x1 x1)) (fma x2 2.0 x1)) (fma x1 x1 1.0))
       3.0
       (+
        x1
        (fma
         (* (* x1 x1) 6.0)
         (fma x1 x1 1.0)
         (* (* x1 x1) (fma x2 6.0 (* x1 -2.0)))))))
     (if (<= x1 650.0)
       (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
       (+
        x1
        (fma
         3.0
         3.0
         (fma
          (fma x1 x1 1.0)
          (*
           (* x1 x1)
           (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) -6.0) x1) 4.0) x1)))
          (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4.5e+153) {
		tmp = (x1 * x1) * 9.0;
	} else if (x1 <= -10500000000.0) {
		tmp = x1 + fma((((3.0 * (x1 * x1)) - fma(x2, 2.0, x1)) / fma(x1, x1, 1.0)), 3.0, (x1 + fma(((x1 * x1) * 6.0), fma(x1, x1, 1.0), ((x1 * x1) * fma(x2, 6.0, (x1 * -2.0))))));
	} else if (x1 <= 650.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), ((x1 * x1) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4.5e+153)
		tmp = Float64(Float64(x1 * x1) * 9.0);
	elseif (x1 <= -10500000000.0)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(3.0 * Float64(x1 * x1)) - fma(x2, 2.0, x1)) / fma(x1, x1, 1.0)), 3.0, Float64(x1 + fma(Float64(Float64(x1 * x1) * 6.0), fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * fma(x2, 6.0, Float64(x1 * -2.0)))))));
	elseif (x1 <= 650.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4.5e+153], N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision], If[LessEqual[x1, -10500000000.0], N[(x1 + N[(N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(x2 * 2.0 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x1 + N[(N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(x2 * 6.0 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 650.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + -6.0), $MachinePrecision] / x1), $MachinePrecision] - 4.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -4.5000000000000001e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.4

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto 9 \cdot {x1}^{\color{blue}{2}} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto 9 \cdot \left(x1 \cdot \color{blue}{x1}\right) \]

   if -4.5000000000000001e153 < x1 < -1.05e10

   1. Initial program 45.4%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.8

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

   8. Applied rewrites 30.8%

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

   10. Applied rewrites 30.7%

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

   11. Taylor expanded in x1 around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 82.1

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

   13. Applied rewrites 82.1%

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

   if -1.05e10 < x1 < 650

   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 650 < 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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 32.5%

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

   11. Taylor expanded in x1 around -inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. lower--.f64 N/A

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

       7. lower-/.f64 N/A

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

   13. Applied rewrites 95.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -10500000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(x2, 2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 + \mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x2, 6, x1 \cdot -2\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 94.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -13000000000:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -13000000000.0)
   (* 6.0 (pow x1 4.0))
   (if (<= x1 650.0)
     (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
     (+
      x1
      (fma
       3.0
       3.0
       (fma
        (fma x1 x1 1.0)
        (*
         (* x1 x1)
         (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) -6.0) x1) 4.0) x1)))
        (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1))))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -13000000000.0) {
		tmp = 6.0 * pow(x1, 4.0);
	} else if (x1 <= 650.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), ((x1 * x1) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -13000000000.0)
		tmp = Float64(6.0 * (x1 ^ 4.0));
	elseif (x1 <= 650.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1) - 4.0) / x1))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -13000000000.0], N[(6.0 * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 650.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + -6.0), $MachinePrecision] / x1), $MachinePrecision] - 4.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.3e10

   1. Initial program 22.1%

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

   3. Taylor expanded in x1 around 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 0.9

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

   5. Applied rewrites 0.9%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 92.4

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

   8. Applied rewrites 92.4%

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

   if -1.3e10 < x1 < 650

   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 650 < 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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 32.5%

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

   11. Taylor expanded in x1 around -inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. lower--.f64 N/A

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

       7. lower-/.f64 N/A

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

   13. Applied rewrites 95.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -13000000000:\\ \;\;\;\;6 \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1} - 4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 93.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -10500000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(x2, 2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 + \mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x2, 6, x1 \cdot -2\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 46000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{-4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4.5e+153)
   (* (* x1 x1) 9.0)
   (if (<= x1 -10500000000.0)
     (+
      x1
      (fma
       (/ (- (* 3.0 (* x1 x1)) (fma x2 2.0 x1)) (fma x1 x1 1.0))
       3.0
       (+
        x1
        (fma
         (* (* x1 x1) 6.0)
         (fma x1 x1 1.0)
         (* (* x1 x1) (fma x2 6.0 (* x1 -2.0)))))))
     (if (<= x1 46000.0)
       (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
       (+
        x1
        (fma
         3.0
         3.0
         (fma
          (fma x1 x1 1.0)
          (* (* x1 x1) (+ 6.0 (/ -4.0 x1)))
          (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4.5e+153) {
		tmp = (x1 * x1) * 9.0;
	} else if (x1 <= -10500000000.0) {
		tmp = x1 + fma((((3.0 * (x1 * x1)) - fma(x2, 2.0, x1)) / fma(x1, x1, 1.0)), 3.0, (x1 + fma(((x1 * x1) * 6.0), fma(x1, x1, 1.0), ((x1 * x1) * fma(x2, 6.0, (x1 * -2.0))))));
	} else if (x1 <= 46000.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), ((x1 * x1) * (6.0 + (-4.0 / x1))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4.5e+153)
		tmp = Float64(Float64(x1 * x1) * 9.0);
	elseif (x1 <= -10500000000.0)
		tmp = Float64(x1 + fma(Float64(Float64(Float64(3.0 * Float64(x1 * x1)) - fma(x2, 2.0, x1)) / fma(x1, x1, 1.0)), 3.0, Float64(x1 + fma(Float64(Float64(x1 * x1) * 6.0), fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * fma(x2, 6.0, Float64(x1 * -2.0)))))));
	elseif (x1 <= 46000.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * Float64(6.0 + Float64(-4.0 / x1))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4.5e+153], N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision], If[LessEqual[x1, -10500000000.0], N[(x1 + N[(N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(x2 * 2.0 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x1 + N[(N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(x2 * 6.0 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 46000.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(-4.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -4.5000000000000001e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.4

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto 9 \cdot {x1}^{\color{blue}{2}} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto 9 \cdot \left(x1 \cdot \color{blue}{x1}\right) \]

   if -4.5000000000000001e153 < x1 < -1.05e10

   1. Initial program 45.4%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 30.1

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.8

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

   8. Applied rewrites 30.8%

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

   10. Applied rewrites 30.7%

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

   11. Taylor expanded in x1 around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 82.1

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

   13. Applied rewrites 82.1%

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

   if -1.05e10 < x1 < 46000

   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 46000 < 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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 32.5%

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

   11. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. lower-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. distribute-neg-frac N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f64 93.2

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

   13. Applied rewrites 93.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -10500000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(x2, 2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 + \mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(x2, 6, x1 \cdot -2\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 46000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{-4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 93.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(t\_0, 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{elif}\;x1 \leq -13000000000:\\ \;\;\;\;x1 + \left(\left(x1 + \left(t\_0 + \left(\left(x1 \cdot x1 + 1\right) \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \left(2 \cdot x2\right)\right)\right)\right) + 3 \cdot 3\right)\\ \mathbf{elif}\;x1 \leq 46000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{-4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1))))
   (if (<= x1 -1.5e+153)
     (* x1 (* x1 9.0))
     (if (<= x1 -5.5e+102)
       (* x1 (/ (fma t_0 729.0 -1.0) (fma (* x1 x1) 81.0 (+ (* x1 9.0) 1.0))))
       (if (<= x1 -13000000000.0)
         (+
          x1
          (+
           (+
            x1
            (+
             t_0
             (+
              (* (+ (* x1 x1) 1.0) (* x1 (* x1 6.0)))
              (* (* x1 (* x1 3.0)) (* 2.0 x2)))))
           (* 3.0 3.0)))
         (if (<= x1 46000.0)
           (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
           (+
            x1
            (fma
             3.0
             3.0
             (fma
              (fma x1 x1 1.0)
              (* (* x1 x1) (+ 6.0 (/ -4.0 x1)))
              (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double tmp;
	if (x1 <= -1.5e+153) {
		tmp = x1 * (x1 * 9.0);
	} else if (x1 <= -5.5e+102) {
		tmp = x1 * (fma(t_0, 729.0, -1.0) / fma((x1 * x1), 81.0, ((x1 * 9.0) + 1.0)));
	} else if (x1 <= -13000000000.0) {
		tmp = x1 + ((x1 + (t_0 + ((((x1 * x1) + 1.0) * (x1 * (x1 * 6.0))) + ((x1 * (x1 * 3.0)) * (2.0 * x2))))) + (3.0 * 3.0));
	} else if (x1 <= 46000.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), ((x1 * x1) * (6.0 + (-4.0 / x1))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -1.5e+153)
		tmp = Float64(x1 * Float64(x1 * 9.0));
	elseif (x1 <= -5.5e+102)
		tmp = Float64(x1 * Float64(fma(t_0, 729.0, -1.0) / fma(Float64(x1 * x1), 81.0, Float64(Float64(x1 * 9.0) + 1.0))));
	elseif (x1 <= -13000000000.0)
		tmp = Float64(x1 + Float64(Float64(x1 + Float64(t_0 + Float64(Float64(Float64(Float64(x1 * x1) + 1.0) * Float64(x1 * Float64(x1 * 6.0))) + Float64(Float64(x1 * Float64(x1 * 3.0)) * Float64(2.0 * x2))))) + Float64(3.0 * 3.0)));
	elseif (x1 <= 46000.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * Float64(6.0 + Float64(-4.0 / x1))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+153], N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -5.5e+102], N[(x1 * N[(N[(t$95$0 * 729.0 + -1.0), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] * 81.0 + N[(N[(x1 * 9.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -13000000000.0], N[(x1 + N[(N[(x1 + N[(t$95$0 + N[(N[(N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 46000.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(-4.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -1.50000000000000009e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.4

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 98.1%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto x1 \cdot \left(9 \cdot x1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

   if -1.50000000000000009e153 < x1 < -5.49999999999999981e102

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 21.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 7.7%

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

   13. Applied rewrites 100.0%

       \[\leadsto x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, \color{blue}{81}, 1 - \left(x1 \cdot 9\right) \cdot -1\right)} \]

   if -5.49999999999999981e102 < x1 < -1.3e10

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

   9. Taylor expanded in x1 around inf

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

   11. Applied rewrites 67.3%

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

   if -1.3e10 < x1 < 46000

   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 46000 < 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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 32.5%

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

   11. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. lower-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. distribute-neg-frac N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f64 93.2

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

   13. Applied rewrites 93.2%

       \[\leadsto x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \color{blue}{\left(x1 \cdot x1\right) \cdot \left(6 + \frac{-4}{x1}\right)}, \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{elif}\;x1 \leq -13000000000:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(x1 \cdot x1 + 1\right) \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \left(2 \cdot x2\right)\right)\right)\right) + 3 \cdot 3\right)\\ \mathbf{elif}\;x1 \leq 46000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{-4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 94.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(t\_0, 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{elif}\;x1 \leq -13000000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, x1 + \mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(3 \cdot \left(x1 \cdot x1\right), 2 \cdot x2, t\_0\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 46000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{-4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1))))
   (if (<= x1 -1.5e+153)
     (* x1 (* x1 9.0))
     (if (<= x1 -5.5e+102)
       (* x1 (/ (fma t_0 729.0 -1.0) (fma (* x1 x1) 81.0 (+ (* x1 9.0) 1.0))))
       (if (<= x1 -13000000000.0)
         (+
          x1
          (fma
           3.0
           3.0
           (+
            x1
            (fma
             (* (* x1 x1) 6.0)
             (fma x1 x1 1.0)
             (fma (* 3.0 (* x1 x1)) (* 2.0 x2) t_0)))))
         (if (<= x1 46000.0)
           (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
           (+
            x1
            (fma
             3.0
             3.0
             (fma
              (fma x1 x1 1.0)
              (* (* x1 x1) (+ 6.0 (/ -4.0 x1)))
              (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double tmp;
	if (x1 <= -1.5e+153) {
		tmp = x1 * (x1 * 9.0);
	} else if (x1 <= -5.5e+102) {
		tmp = x1 * (fma(t_0, 729.0, -1.0) / fma((x1 * x1), 81.0, ((x1 * 9.0) + 1.0)));
	} else if (x1 <= -13000000000.0) {
		tmp = x1 + fma(3.0, 3.0, (x1 + fma(((x1 * x1) * 6.0), fma(x1, x1, 1.0), fma((3.0 * (x1 * x1)), (2.0 * x2), t_0))));
	} else if (x1 <= 46000.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), ((x1 * x1) * (6.0 + (-4.0 / x1))), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -1.5e+153)
		tmp = Float64(x1 * Float64(x1 * 9.0));
	elseif (x1 <= -5.5e+102)
		tmp = Float64(x1 * Float64(fma(t_0, 729.0, -1.0) / fma(Float64(x1 * x1), 81.0, Float64(Float64(x1 * 9.0) + 1.0))));
	elseif (x1 <= -13000000000.0)
		tmp = Float64(x1 + fma(3.0, 3.0, Float64(x1 + fma(Float64(Float64(x1 * x1) * 6.0), fma(x1, x1, 1.0), fma(Float64(3.0 * Float64(x1 * x1)), Float64(2.0 * x2), t_0)))));
	elseif (x1 <= 46000.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * Float64(6.0 + Float64(-4.0 / x1))), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+153], N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -5.5e+102], N[(x1 * N[(N[(t$95$0 * 729.0 + -1.0), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] * 81.0 + N[(N[(x1 * 9.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -13000000000.0], N[(x1 + N[(3.0 * 3.0 + N[(x1 + N[(N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * x2), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 46000.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * N[(6.0 + N[(-4.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -1.50000000000000009e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.4

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 98.1%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto x1 \cdot \left(9 \cdot x1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

   if -1.50000000000000009e153 < x1 < -5.49999999999999981e102

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 21.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 7.7%

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

   13. Applied rewrites 100.0%

       \[\leadsto x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, \color{blue}{81}, 1 - \left(x1 \cdot 9\right) \cdot -1\right)} \]

   if -5.49999999999999981e102 < x1 < -1.3e10

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

   10. Applied rewrites 67.1%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 67.1%

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

   if -1.3e10 < x1 < 46000

   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 46000 < 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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 32.5%

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

   11. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. lower-+.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. distribute-neg-frac N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f64 93.2

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

   13. Applied rewrites 93.2%

       \[\leadsto x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \color{blue}{\left(x1 \cdot x1\right) \cdot \left(6 + \frac{-4}{x1}\right)}, \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{elif}\;x1 \leq -13000000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, x1 + \mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(3 \cdot \left(x1 \cdot x1\right), 2 \cdot x2, x1 \cdot \left(x1 \cdot x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 46000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot \left(6 + \frac{-4}{x1}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 93.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ t_1 := \left(x1 \cdot x1\right) \cdot 6\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(t\_0, 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{elif}\;x1 \leq -13000000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, x1 + \mathsf{fma}\left(t\_1, \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(3 \cdot \left(x1 \cdot x1\right), 2 \cdot x2, t\_0\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 18000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), t\_1, \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1))) (t_1 (* (* x1 x1) 6.0)))
   (if (<= x1 -1.5e+153)
     (* x1 (* x1 9.0))
     (if (<= x1 -5.5e+102)
       (* x1 (/ (fma t_0 729.0 -1.0) (fma (* x1 x1) 81.0 (+ (* x1 9.0) 1.0))))
       (if (<= x1 -13000000000.0)
         (+
          x1
          (fma
           3.0
           3.0
           (+
            x1
            (fma t_1 (fma x1 x1 1.0) (fma (* 3.0 (* x1 x1)) (* 2.0 x2) t_0)))))
         (if (<= x1 18000000.0)
           (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
           (+
            x1
            (fma
             3.0
             3.0
             (fma
              (fma x1 x1 1.0)
              t_1
              (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double t_1 = (x1 * x1) * 6.0;
	double tmp;
	if (x1 <= -1.5e+153) {
		tmp = x1 * (x1 * 9.0);
	} else if (x1 <= -5.5e+102) {
		tmp = x1 * (fma(t_0, 729.0, -1.0) / fma((x1 * x1), 81.0, ((x1 * 9.0) + 1.0)));
	} else if (x1 <= -13000000000.0) {
		tmp = x1 + fma(3.0, 3.0, (x1 + fma(t_1, fma(x1, x1, 1.0), fma((3.0 * (x1 * x1)), (2.0 * x2), t_0))));
	} else if (x1 <= 18000000.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), t_1, fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	t_1 = Float64(Float64(x1 * x1) * 6.0)
	tmp = 0.0
	if (x1 <= -1.5e+153)
		tmp = Float64(x1 * Float64(x1 * 9.0));
	elseif (x1 <= -5.5e+102)
		tmp = Float64(x1 * Float64(fma(t_0, 729.0, -1.0) / fma(Float64(x1 * x1), 81.0, Float64(Float64(x1 * 9.0) + 1.0))));
	elseif (x1 <= -13000000000.0)
		tmp = Float64(x1 + fma(3.0, 3.0, Float64(x1 + fma(t_1, fma(x1, x1, 1.0), fma(Float64(3.0 * Float64(x1 * x1)), Float64(2.0 * x2), t_0)))));
	elseif (x1 <= 18000000.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), t_1, fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[x1, -1.5e+153], N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -5.5e+102], N[(x1 * N[(N[(t$95$0 * 729.0 + -1.0), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] * 81.0 + N[(N[(x1 * 9.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -13000000000.0], N[(x1 + N[(3.0 * 3.0 + N[(x1 + N[(t$95$1 * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * x2), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 18000000.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * t$95$1 + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -1.50000000000000009e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.4

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 98.1%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto x1 \cdot \left(9 \cdot x1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

   if -1.50000000000000009e153 < x1 < -5.49999999999999981e102

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 21.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 7.7%

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

   13. Applied rewrites 100.0%

       \[\leadsto x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, \color{blue}{81}, 1 - \left(x1 \cdot 9\right) \cdot -1\right)} \]

   if -5.49999999999999981e102 < x1 < -1.3e10

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

   10. Applied rewrites 67.1%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 67.1%

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

   if -1.3e10 < x1 < 1.8e7

   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   if 1.8e7 < 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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 32.5%

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

   11. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 92.1

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

   13. Applied rewrites 92.1%

       \[\leadsto x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \color{blue}{6 \cdot \left(x1 \cdot x1\right)}, \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{elif}\;x1 \leq -13000000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, x1 + \mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(3 \cdot \left(x1 \cdot x1\right), 2 \cdot x2, x1 \cdot \left(x1 \cdot x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 18000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 93.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{elif}\;x1 \leq -13000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 18000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (fma
           3.0
           3.0
           (fma
            (fma x1 x1 1.0)
            (* (* x1 x1) 6.0)
            (fma x1 (* (* x1 3.0) 3.0) (fma x1 (* x1 x1) x1)))))))
   (if (<= x1 -1.5e+153)
     (* x1 (* x1 9.0))
     (if (<= x1 -5.5e+102)
       (*
        x1
        (/
         (fma (* x1 (* x1 x1)) 729.0 -1.0)
         (fma (* x1 x1) 81.0 (+ (* x1 9.0) 1.0))))
       (if (<= x1 -13000000000.0)
         t_0
         (if (<= x1 18000000.0)
           (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))
           t_0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), ((x1 * x1) * 6.0), fma(x1, ((x1 * 3.0) * 3.0), fma(x1, (x1 * x1), x1))));
	double tmp;
	if (x1 <= -1.5e+153) {
		tmp = x1 * (x1 * 9.0);
	} else if (x1 <= -5.5e+102) {
		tmp = x1 * (fma((x1 * (x1 * x1)), 729.0, -1.0) / fma((x1 * x1), 81.0, ((x1 * 9.0) + 1.0)));
	} else if (x1 <= -13000000000.0) {
		tmp = t_0;
	} else if (x1 <= 18000000.0) {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + fma(3.0, 3.0, fma(fma(x1, x1, 1.0), Float64(Float64(x1 * x1) * 6.0), fma(x1, Float64(Float64(x1 * 3.0) * 3.0), fma(x1, Float64(x1 * x1), x1)))))
	tmp = 0.0
	if (x1 <= -1.5e+153)
		tmp = Float64(x1 * Float64(x1 * 9.0));
	elseif (x1 <= -5.5e+102)
		tmp = Float64(x1 * Float64(fma(Float64(x1 * Float64(x1 * x1)), 729.0, -1.0) / fma(Float64(x1 * x1), 81.0, Float64(Float64(x1 * 9.0) + 1.0))));
	elseif (x1 <= -13000000000.0)
		tmp = t_0;
	elseif (x1 <= 18000000.0)
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(3.0 * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.5e+153], N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -5.5e+102], N[(x1 * N[(N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * 729.0 + -1.0), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] * 81.0 + N[(N[(x1 * 9.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -13000000000.0], t$95$0, If[LessEqual[x1, 18000000.0], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.50000000000000009e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.4

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 98.1%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto x1 \cdot \left(9 \cdot x1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

   if -1.50000000000000009e153 < x1 < -5.49999999999999981e102

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.6

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

   5. Applied rewrites 0.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 21.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 7.7%

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

   13. Applied rewrites 100.0%

       \[\leadsto x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, \color{blue}{81}, 1 - \left(x1 \cdot 9\right) \cdot -1\right)} \]

   if -5.49999999999999981e102 < x1 < -1.3e10 or 1.8e7 < x1

   1. Initial program 55.1%

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in x1 around inf

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in x1 around inf

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

   10. Applied rewrites 45.6%

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

   11. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 86.4

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

   13. Applied rewrites 86.4%

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

   if -1.3e10 < x1 < 1.8e7

   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 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 96.6%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{elif}\;x1 \leq -13000000000:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 18000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \mathsf{fma}\left(3, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(x1 \cdot x1\right) \cdot 6, \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot 3, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 83.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.5e+153)
   (* x1 (* x1 9.0))
   (if (<= x1 -1.5e+55)
     (*
      x1
      (/
       (fma (* x1 (* x1 x1)) 729.0 -1.0)
       (fma (* x1 x1) 81.0 (+ (* x1 9.0) 1.0))))
     (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.5e+153) {
		tmp = x1 * (x1 * 9.0);
	} else if (x1 <= -1.5e+55) {
		tmp = x1 * (fma((x1 * (x1 * x1)), 729.0, -1.0) / fma((x1 * x1), 81.0, ((x1 * 9.0) + 1.0)));
	} else {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.5e+153)
		tmp = Float64(x1 * Float64(x1 * 9.0));
	elseif (x1 <= -1.5e+55)
		tmp = Float64(x1 * Float64(fma(Float64(x1 * Float64(x1 * x1)), 729.0, -1.0) / fma(Float64(x1 * x1), 81.0, Float64(Float64(x1 * 9.0) + 1.0))));
	else
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.5e+153], N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.5e+55], N[(x1 * N[(N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * 729.0 + -1.0), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] * 81.0 + N[(N[(x1 * 9.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.50000000000000009e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.4

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 98.1%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto x1 \cdot \left(9 \cdot x1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

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

   if -1.50000000000000009e153 < x1 < -1.50000000000000008e55

   1. Initial program 24.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 16.9%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 7.1%

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

   13. Applied rewrites 80.3%

       \[\leadsto x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, \color{blue}{81}, 1 - \left(x1 \cdot 9\right) \cdot -1\right)} \]

   if -1.50000000000000008e55 < x1

   1. Initial program 82.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 25.9

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

   5. Applied rewrites 25.9%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.2%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 84.8%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{+55}:\\ \;\;\;\;x1 \cdot \frac{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), 729, -1\right)}{\mathsf{fma}\left(x1 \cdot x1, 81, x1 \cdot 9 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 83.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;x1 + x2 \cdot \mathsf{fma}\left(12, x1 \cdot x1, 9 \cdot \frac{x1 \cdot x1}{x2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4.5e+153)
   (* (* x1 x1) 9.0)
   (if (<= x1 -1.1e+55)
     (+ x1 (* x2 (fma 12.0 (* x1 x1) (* 9.0 (/ (* x1 x1) x2)))))
     (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4.5e+153) {
		tmp = (x1 * x1) * 9.0;
	} else if (x1 <= -1.1e+55) {
		tmp = x1 + (x2 * fma(12.0, (x1 * x1), (9.0 * ((x1 * x1) / x2))));
	} else {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4.5e+153)
		tmp = Float64(Float64(x1 * x1) * 9.0);
	elseif (x1 <= -1.1e+55)
		tmp = Float64(x1 + Float64(x2 * fma(12.0, Float64(x1 * x1), Float64(9.0 * Float64(Float64(x1 * x1) / x2)))));
	else
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4.5e+153], N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision], If[LessEqual[x1, -1.1e+55], N[(x1 + N[(x2 * N[(12.0 * N[(x1 * x1), $MachinePrecision] + N[(9.0 * N[(N[(x1 * x1), $MachinePrecision] / x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.5000000000000001e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 0.4

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

   5. Applied rewrites 0.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto 9 \cdot {x1}^{\color{blue}{2}} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto 9 \cdot \left(x1 \cdot \color{blue}{x1}\right) \]

   if -4.5000000000000001e153 < x1 < -1.10000000000000005e55

   1. Initial program 26.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 16.9%

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

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 28.9%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 74.6%

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

   if -1.10000000000000005e55 < x1

   1. Initial program 82.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 26.1

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.5%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 85.2%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;x1 + x2 \cdot \mathsf{fma}\left(12, x1 \cdot x1, 9 \cdot \frac{x1 \cdot x1}{x2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 81.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.3 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -1.25 \cdot 10^{+110}:\\ \;\;\;\;\left(x2 \cdot x2\right) \cdot \mathsf{fma}\left(8, x1, \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)}{x2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -3.3e+153)
   (* x1 (* x1 9.0))
   (if (<= x1 -1.25e+110)
     (* (* x2 x2) (fma 8.0 x1 (/ (fma x1 (fma 12.0 x1 -12.0) -6.0) x2)))
     (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.3e+153) {
		tmp = x1 * (x1 * 9.0);
	} else if (x1 <= -1.25e+110) {
		tmp = (x2 * x2) * fma(8.0, x1, (fma(x1, fma(12.0, x1, -12.0), -6.0) / x2));
	} else {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3.3e+153)
		tmp = Float64(x1 * Float64(x1 * 9.0));
	elseif (x1 <= -1.25e+110)
		tmp = Float64(Float64(x2 * x2) * fma(8.0, x1, Float64(fma(x1, fma(12.0, x1, -12.0), -6.0) / x2)));
	else
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3.3e+153], N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.25e+110], N[(N[(x2 * x2), $MachinePrecision] * N[(8.0 * x1 + N[(N[(x1 * N[(12.0 * x1 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] / x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.29999999999999994e153

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. lower-*.f64 0.4

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 0.4%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.1%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto x1 \cdot \left(9 \cdot x1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.1%

       \[\leadsto x1 \cdot \left(9 \cdot x1\right) \]

   if -3.29999999999999994e153 < x1 < -1.24999999999999995e110

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. lower-*.f64 0.6

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 0.6%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.9%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x2 around inf

       \[\leadsto {x2}^{2} \cdot \color{blue}{\left(\left(8 \cdot x1 + \frac{x1 \cdot \left(12 \cdot x1 - 12\right)}{x2}\right) - 6 \cdot \frac{1}{x2}\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 43.8%

       \[\leadsto \left(x2 \cdot x2\right) \cdot \color{blue}{\mathsf{fma}\left(8, x1, \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)}{x2}\right)} \]

   if -1.24999999999999995e110 < x1

   1. Initial program 82.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 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 24.9

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 24.9%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.1%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 81.6%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3.3 \cdot 10^{+153}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \mathbf{elif}\;x1 \leq -1.25 \cdot 10^{+110}:\\ \;\;\;\;\left(x2 \cdot x2\right) \cdot \mathsf{fma}\left(8, x1, \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)}{x2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 81.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot 9\\ \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;x1 + \mathsf{fma}\left(12, x2 \cdot \left(x1 \cdot x1\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) 9.0)))
   (if (<= x1 -4.5e+153)
     t_0
     (if (<= x1 -1.1e+55)
       (+ x1 (fma 12.0 (* x2 (* x1 x1)) t_0))
       (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0)))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * 9.0;
	double tmp;
	if (x1 <= -4.5e+153) {
		tmp = t_0;
	} else if (x1 <= -1.1e+55) {
		tmp = x1 + fma(12.0, (x2 * (x1 * x1)), t_0);
	} else {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * 9.0)
	tmp = 0.0
	if (x1 <= -4.5e+153)
		tmp = t_0;
	elseif (x1 <= -1.1e+55)
		tmp = Float64(x1 + fma(12.0, Float64(x2 * Float64(x1 * x1)), t_0));
	else
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]}, If[LessEqual[x1, -4.5e+153], t$95$0, If[LessEqual[x1, -1.1e+55], N[(x1 + N[(12.0 * N[(x2 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.5000000000000001e153

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. lower-*.f64 0.4

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 0.4%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto 9 \cdot {x1}^{\color{blue}{2}} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto 9 \cdot \left(x1 \cdot \color{blue}{x1}\right) \]

   if -4.5000000000000001e153 < x1 < -1.10000000000000005e55

   1. Initial program 26.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 16.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x1 around inf

      \[\leadsto x1 + {x1}^{2} \cdot \color{blue}{\left(\left(-4 \cdot x2 + \left(2 \cdot \left(3 + -2 \cdot x2\right) + \left(3 \cdot \left(3 + 2 \cdot x2\right) + 14 \cdot x2\right)\right)\right) - 6\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 28.9%

      \[\leadsto x1 + \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(3, x2 \cdot 2, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, -6\right)\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(9 \cdot {x1}^{2} + 12 \cdot \color{blue}{\left({x1}^{2} \cdot x2\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 28.9%

       \[\leadsto x1 + \mathsf{fma}\left(12, x2 \cdot \color{blue}{\left(x1 \cdot x1\right)}, \left(x1 \cdot x1\right) \cdot 9\right) \]

   if -1.10000000000000005e55 < x1

   1. Initial program 82.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 26.1

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 26.1%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.5%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 85.2%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;x1 + \mathsf{fma}\left(12, x2 \cdot \left(x1 \cdot x1\right), \left(x1 \cdot x1\right) \cdot 9\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 81.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -9 \cdot 10^{+146}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(12, x2, 9\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -9e+146)
   (* (* x1 x1) 9.0)
   (if (<= x1 -1.1e+55)
     (* (* x1 x1) (fma 12.0 x2 9.0))
     (fma x2 (fma (fma x2 8.0 -12.0) x1 -6.0) (* x1 (fma x1 9.0 -1.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -9e+146) {
		tmp = (x1 * x1) * 9.0;
	} else if (x1 <= -1.1e+55) {
		tmp = (x1 * x1) * fma(12.0, x2, 9.0);
	} else {
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), (x1 * fma(x1, 9.0, -1.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -9e+146)
		tmp = Float64(Float64(x1 * x1) * 9.0);
	elseif (x1 <= -1.1e+55)
		tmp = Float64(Float64(x1 * x1) * fma(12.0, x2, 9.0));
	else
		tmp = fma(x2, fma(fma(x2, 8.0, -12.0), x1, -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -9e+146], N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision], If[LessEqual[x1, -1.1e+55], N[(N[(x1 * x1), $MachinePrecision] * N[(12.0 * x2 + 9.0), $MachinePrecision]), $MachinePrecision], N[(x2 * N[(N[(x2 * 8.0 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -9.00000000000000051e146

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. lower-*.f64 0.4

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 0.4%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.7%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around inf

       \[\leadsto 9 \cdot {x1}^{\color{blue}{2}} \]
   13. Step-by-step derivation

   14. Applied rewrites 93.7%

       \[\leadsto 9 \cdot \left(x1 \cdot \color{blue}{x1}\right) \]

   if -9.00000000000000051e146 < x1 < -1.10000000000000005e55

   1. Initial program 30.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 0.7

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 0.7%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 17.1%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around inf

       \[\leadsto {x1}^{2} \cdot \left(9 + \color{blue}{12 \cdot x2}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 30.6%

       \[\leadsto \mathsf{fma}\left(12, x2, 9\right) \cdot \left(x1 \cdot \color{blue}{x1}\right) \]

   if -1.10000000000000005e55 < x1

   1. Initial program 82.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 26.1

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 26.1%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \left(9 \cdot x1 - 1\right) + \color{blue}{x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.5%

       \[\leadsto \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), \mathsf{fma}\left(8, x2 \cdot x1, -6\right)\right)}, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]

   12. Taylor expanded in x1 around 0

       \[\leadsto \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 85.2%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -9 \cdot 10^{+146}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \mathbf{elif}\;x1 \leq -1.1 \cdot 10^{+55}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot \mathsf{fma}\left(12, x2, 9\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(\mathsf{fma}\left(x2, 8, -12\right), x1, -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 30.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-129}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 5 \cdot 10^{-34}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;x1 + x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -5e-129)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 5e-34) (- x1) (+ x1 (* x2 -6.0)))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-129) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 5e-34) {
		tmp = -x1;
	} 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 ((2.0d0 * x2) <= (-5d-129)) then
        tmp = x2 * (-6.0d0)
    else if ((2.0d0 * x2) <= 5d-34) then
        tmp = -x1
    else
        tmp = x1 + (x2 * (-6.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-129) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 5e-34) {
		tmp = -x1;
	} else {
		tmp = x1 + (x2 * -6.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (2.0 * x2) <= -5e-129:
		tmp = x2 * -6.0
	elif (2.0 * x2) <= 5e-34:
		tmp = -x1
	else:
		tmp = x1 + (x2 * -6.0)
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -5e-129)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 5e-34)
		tmp = Float64(-x1);
	else
		tmp = Float64(x1 + Float64(x2 * -6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((2.0 * x2) <= -5e-129)
		tmp = x2 * -6.0;
	elseif ((2.0 * x2) <= 5e-34)
		tmp = -x1;
	else
		tmp = x1 + (x2 * -6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -5e-129], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 5e-34], (-x1), N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -5.00000000000000027e-129

   1. Initial program 71.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 24.5

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 24.5%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   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 25.1

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Applied rewrites 25.1%

      \[\leadsto \color{blue}{x2 \cdot -6} \]

   if -5.00000000000000027e-129 < (*.f64 #s(literal 2 binary64) x2) < 5.0000000000000003e-34

   1. Initial program 58.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 8.3

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 8.3%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.9%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around 0

       \[\leadsto -1 \cdot x1 \]
   13. Step-by-step derivation

   14. Applied rewrites 34.3%

       \[\leadsto -x1 \]

   if 5.0000000000000003e-34 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 66.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 31.3

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 31.3%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 29.9% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -5 \cdot 10^{-129}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 5 \cdot 10^{-34}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;x2 \cdot -6\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -5e-129)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 5e-34) (- x1) (* x2 -6.0))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-129) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 5e-34) {
		tmp = -x1;
	} else {
		tmp = 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 ((2.0d0 * x2) <= (-5d-129)) then
        tmp = x2 * (-6.0d0)
    else if ((2.0d0 * x2) <= 5d-34) then
        tmp = -x1
    else
        tmp = x2 * (-6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -5e-129) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 5e-34) {
		tmp = -x1;
	} else {
		tmp = x2 * -6.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (2.0 * x2) <= -5e-129:
		tmp = x2 * -6.0
	elif (2.0 * x2) <= 5e-34:
		tmp = -x1
	else:
		tmp = x2 * -6.0
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -5e-129)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 5e-34)
		tmp = Float64(-x1);
	else
		tmp = Float64(x2 * -6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((2.0 * x2) <= -5e-129)
		tmp = x2 * -6.0;
	elseif ((2.0 * x2) <= 5e-34)
		tmp = -x1;
	else
		tmp = x2 * -6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -5e-129], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 5e-34], (-x1), N[(x2 * -6.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -5.00000000000000027e-129 or 5.0000000000000003e-34 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 69.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 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 27.9

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 27.9%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   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 27.8

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Applied rewrites 27.8%

      \[\leadsto \color{blue}{x2 \cdot -6} \]

   if -5.00000000000000027e-129 < (*.f64 #s(literal 2 binary64) x2) < 5.0000000000000003e-34

   1. Initial program 58.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 8.3

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 8.3%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.9%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   12. Taylor expanded in x1 around 0

       \[\leadsto -1 \cdot x1 \]
   13. Step-by-step derivation

   14. Applied rewrites 34.3%

       \[\leadsto -x1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 54.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.7 \cdot 10^{-157}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \mathbf{elif}\;x1 \leq 8.8 \cdot 10^{-139}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot 9, x1, -x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.7e-157)
   (* x1 (fma x1 9.0 -1.0))
   (if (<= x1 8.8e-139) (* x2 -6.0) (fma (* x1 9.0) x1 (- x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.7e-157) {
		tmp = x1 * fma(x1, 9.0, -1.0);
	} else if (x1 <= 8.8e-139) {
		tmp = x2 * -6.0;
	} else {
		tmp = fma((x1 * 9.0), x1, -x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.7e-157)
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	elseif (x1 <= 8.8e-139)
		tmp = Float64(x2 * -6.0);
	else
		tmp = fma(Float64(x1 * 9.0), x1, Float64(-x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.7e-157], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8.8e-139], N[(x2 * -6.0), $MachinePrecision], N[(N[(x1 * 9.0), $MachinePrecision] * x1 + (-x1)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.69999999999999989e-157

   1. Initial program 47.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 5.8

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 5.8%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.5%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   if -1.69999999999999989e-157 < x1 < 8.80000000000000041e-139

   1. Initial program 99.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. lower-*.f64 65.8

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 65.8%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   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 66.1

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Applied rewrites 66.1%

      \[\leadsto \color{blue}{x2 \cdot -6} \]

   if 8.80000000000000041e-139 < x1

   1. Initial program 64.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 8.2

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 8.2%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 54.6%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 54.7%

       \[\leadsto \mathsf{fma}\left(x1 \cdot 9, x1, -x1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 32: 54.7% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \mathbf{if}\;x1 \leq -1.7 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 8.8 \cdot 10^{-139}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma x1 9.0 -1.0))))
   (if (<= x1 -1.7e-157) t_0 (if (<= x1 8.8e-139) (* x2 -6.0) t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma(x1, 9.0, -1.0);
	double tmp;
	if (x1 <= -1.7e-157) {
		tmp = t_0;
	} else if (x1 <= 8.8e-139) {
		tmp = x2 * -6.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(x1, 9.0, -1.0))
	tmp = 0.0
	if (x1 <= -1.7e-157)
		tmp = t_0;
	elseif (x1 <= 8.8e-139)
		tmp = Float64(x2 * -6.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.7e-157], t$95$0, If[LessEqual[x1, 8.8e-139], N[(x2 * -6.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.69999999999999989e-157 or 8.80000000000000041e-139 < x1

   1. Initial program 55.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 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 6.9

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 6.9%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

   8. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   9. Taylor expanded in x2 around 0

      \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.5%

       \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

   if -1.69999999999999989e-157 < x1 < 8.80000000000000041e-139

   1. Initial program 99.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

      2. lower-*.f64 65.8

         \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   5. Applied rewrites 65.8%

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   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 66.1

         \[\leadsto \color{blue}{x2 \cdot -6} \]

   8. Applied rewrites 66.1%

      \[\leadsto \color{blue}{x2 \cdot -6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 14.0% accurate, 99.3× speedup

Math

\[\begin{array}{l} \\ -x1 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (- x1))

C

double code(double x1, double x2) {
	return -x1;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = -x1
end function

Java

public static double code(double x1, double x2) {
	return -x1;
}

Python

def code(x1, x2):
	return -x1

Julia

function code(x1, x2)
	return Float64(-x1)
end

MATLAB

function tmp = code(x1, x2)
	tmp = -x1;
end

Wolfram

code[x1_, x2_] := (-x1)

Derivation

1. Initial program 64.8%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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 19.8

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Applied rewrites 19.8%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

6. Taylor expanded in x1 around 0

   \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 1\right)} \]

8. Applied rewrites 69.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, \mathsf{fma}\left(x2, -2, 3\right)\right), x2 \cdot 14\right)\right) + -6, \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

9. Taylor expanded in x2 around 0

   \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 45.4%

    \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9, -1\right)} \]

12. Taylor expanded in x1 around 0

    \[\leadsto -1 \cdot x1 \]
13. Step-by-step derivation

14. Applied rewrites 17.0%

    \[\leadsto -x1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))))