Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.5% → 98.9%

Time: 20.0s

Alternatives: 18

Speedup: 9.0×

• 47.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ t_1 := \mathsf{fma}\left(x1, \mathsf{fma}\left(3, x1, -1\right), 2 \cdot x2\right)\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -1.3 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 + \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_1, \frac{4}{\mathsf{fma}\left(x1, x1, 1\right)}, -6\right), \left(-3 + t\_2\right) \cdot \frac{2 \cdot \left(x1 \cdot t\_1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, x1, \left(x1 \cdot 3\right) \cdot t\_2\right), x1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           (pow x1 4.0)
           (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))
        (t_1 (fma x1 (fma 3.0 x1 -1.0) (* 2.0 x2)))
        (t_2 (/ t_1 (fma x1 x1 1.0))))
   (if (<= x1 -1.3e+101)
     t_0
     (if (<= x1 4.5e+30)
       (fma
        (/ (- (* 3.0 (* x1 x1)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
        3.0
        (+
         x1
         (fma
          (fma x1 x1 1.0)
          (fma
           x1
           (* x1 (fma t_1 (/ 4.0 (fma x1 x1 1.0)) -6.0))
           (* (+ -3.0 t_2) (/ (* 2.0 (* x1 t_1)) (fma x1 x1 1.0))))
          (fma x1 (fma x1 x1 (* (* x1 3.0) t_2)) x1))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	double t_1 = fma(x1, fma(3.0, x1, -1.0), (2.0 * x2));
	double t_2 = t_1 / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -1.3e+101) {
		tmp = t_0;
	} else if (x1 <= 4.5e+30) {
		tmp = fma((((3.0 * (x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, (x1 + fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma(t_1, (4.0 / fma(x1, x1, 1.0)), -6.0)), ((-3.0 + t_2) * ((2.0 * (x1 * t_1)) / fma(x1, x1, 1.0)))), fma(x1, fma(x1, x1, ((x1 * 3.0) * t_2)), x1))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))))
	t_1 = fma(x1, fma(3.0, x1, -1.0), Float64(2.0 * x2))
	t_2 = Float64(t_1 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -1.3e+101)
		tmp = t_0;
	elseif (x1 <= 4.5e+30)
		tmp = fma(Float64(Float64(Float64(3.0 * Float64(x1 * x1)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, Float64(x1 + fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_1, Float64(4.0 / fma(x1, x1, 1.0)), -6.0)), Float64(Float64(-3.0 + t_2) * Float64(Float64(2.0 * Float64(x1 * t_1)) / fma(x1, x1, 1.0)))), fma(x1, fma(x1, x1, Float64(Float64(x1 * 3.0) * t_2)), x1))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(3.0 * x1 + -1.0), $MachinePrecision] + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.3e+101], t$95$0, If[LessEqual[x1, 4.5e+30], N[(N[(N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x1 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(t$95$1 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 + t$95$2), $MachinePrecision] * N[(N[(2.0 * N[(x1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1 + N[(N[(x1 * 3.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.3e101 or 4.49999999999999995e30 < x1

   1. Initial program 20.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if -1.3e101 < x1 < 4.49999999999999995e30

   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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.3 \cdot 10^{+101}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{3 \cdot \left(x1 \cdot x1\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1 + \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x1, \mathsf{fma}\left(3, x1, -1\right), 2 \cdot x2\right), \frac{4}{\mathsf{fma}\left(x1, x1, 1\right)}, -6\right), \left(-3 + \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(3, x1, -1\right), 2 \cdot x2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \frac{2 \cdot \left(x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(3, x1, -1\right), 2 \cdot x2\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, x1, \left(x1 \cdot 3\right) \cdot \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(3, x1, -1\right), 2 \cdot x2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -2:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-129}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x1 (* 8.0 (* x2 x2))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2e+224)
     t_1
     (if (<= t_4 -2.0)
       (fma x2 -6.0 x1)
       (if (<= t_4 -5e-129)
         (* x1 (fma x1 9.0 -1.0))
         (if (<= t_4 2e+228)
           (* x2 -6.0)
           (if (<= t_4 INFINITY) t_1 (fma x1 (* x1 9.0) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x1 * (8.0 * (x2 * x2));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2e+224) {
		tmp = t_1;
	} else if (t_4 <= -2.0) {
		tmp = fma(x2, -6.0, x1);
	} else if (t_4 <= -5e-129) {
		tmp = x1 * fma(x1, 9.0, -1.0);
	} else if (t_4 <= 2e+228) {
		tmp = x2 * -6.0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(x1, (x1 * 9.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x1 * Float64(8.0 * Float64(x2 * x2)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2e+224)
		tmp = t_1;
	elseif (t_4 <= -2.0)
		tmp = fma(x2, -6.0, x1);
	elseif (t_4 <= -5e-129)
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	elseif (t_4 <= 2e+228)
		tmp = Float64(x2 * -6.0);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = fma(x1, Float64(x1 * 9.0), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+224], t$95$1, If[LessEqual[t$95$4, -2.0], N[(x2 * -6.0 + x1), $MachinePrecision], If[LessEqual[t$95$4, -5e-129], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+228], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(x1 * 9.0), $MachinePrecision] + x1), $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)))))) < -1.99999999999999994e224 or 1.9999999999999998e228 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 66.6%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 64.9%

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

   9. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -1.99999999999999994e224 < (+.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

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -2 < (+.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)))))) < -5.00000000000000027e-129

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 99.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. associate-+l- N/A

          \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       5. metadata-eval N/A

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

       6. lower-*.f64 N/A

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

       7. sub-neg N/A

          \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto x1 \cdot \left(\color{blue}{x1 \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

       9. metadata-eval N/A

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

       10. lower-fma.f64 81.9

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

   11. Applied rewrites 81.9%

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

   if -5.00000000000000027e-129 < (+.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)))))) < 1.9999999999999998e228

   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 52.7

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

   5. Applied rewrites 52.7%

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

   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 53.3

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

   8. Applied rewrites 53.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 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 56.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in x1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 88.2

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

   11. Applied rewrites 88.2%

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

   12. Taylor expanded in x1 around inf

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

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{9 \cdot {x1}^{2} + \frac{1}{x1} \cdot {x1}^{2}} \]

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \frac{1}{x1} \cdot \color{blue}{\left(x1 \cdot x1\right)} \]

       6. associate-*r* N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \color{blue}{\left(\frac{1}{x1} \cdot x1\right) \cdot x1} \]

       7. lft-mult-inverse N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \color{blue}{1} \cdot x1 \]

       8. *-lft-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 88.2

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

   14. Applied rewrites 88.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+224}:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{-129}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{+228}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+224}:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;x1 + 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_1
               (+
                (* (* (* x1 2.0) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* t_2 4.0) 6.0))))
              (* t_0 t_2))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -2e+224)
     (* x1 (* 8.0 (* x2 x2)))
     (if (<= t_3 1e+245)
       (fma x2 -6.0 (* x1 (fma x1 9.0 -1.0)))
       (if (<= t_3 INFINITY)
         (+ x1 (* 8.0 (* x1 (* x2 x2))))
         (fma x1 (* x1 9.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -2e+224) {
		tmp = x1 * (8.0 * (x2 * x2));
	} else if (t_3 <= 1e+245) {
		tmp = fma(x2, -6.0, (x1 * fma(x1, 9.0, -1.0)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = x1 + (8.0 * (x1 * (x2 * x2)));
	} else {
		tmp = fma(x1, (x1 * 9.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -2e+224)
		tmp = Float64(x1 * Float64(8.0 * Float64(x2 * x2)));
	elseif (t_3 <= 1e+245)
		tmp = fma(x2, -6.0, Float64(x1 * fma(x1, 9.0, -1.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(x1 + Float64(8.0 * Float64(x1 * Float64(x2 * x2))));
	else
		tmp = fma(x1, Float64(x1 * 9.0), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+224], N[(x1 * N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+245], N[(x2 * -6.0 + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(x1 + N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * 9.0), $MachinePrecision] + x1), $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)))))) < -1.99999999999999994e224

   1. Initial program 99.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 80.4%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 80.4

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

   11. Applied rewrites 80.4%

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

   if -1.99999999999999994e224 < (+.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)))))) < 1.00000000000000004e245

   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

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

   5. Applied rewrites 79.1%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-rgt-identity N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(x1 \cdot \left(9 \cdot x1 - 2\right) + \color{blue}{x1 \cdot 1}\right) \]

       5. distribute-lft-out N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{x1 \cdot \left(\left(9 \cdot x1 - 2\right) + 1\right)} \]

       6. associate-+l- N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 88.0%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 80.1%

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

   if 1.00000000000000004e245 < (+.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 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

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

   5. Applied rewrites 61.4%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 57.6

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

   8. Applied rewrites 57.6%

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

   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 56.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in x1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 88.2

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

   11. Applied rewrites 88.2%

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

   12. Taylor expanded in x1 around inf

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

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{9 \cdot {x1}^{2} + \frac{1}{x1} \cdot {x1}^{2}} \]

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \frac{1}{x1} \cdot \color{blue}{\left(x1 \cdot x1\right)} \]

       6. associate-*r* N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \color{blue}{\left(\frac{1}{x1} \cdot x1\right) \cdot x1} \]

       7. lft-mult-inverse N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \color{blue}{1} \cdot x1 \]

       8. *-lft-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 88.2

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

   14. Applied rewrites 88.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+224}:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 + 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x1 (* 8.0 (* x2 x2))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2e+224)
     t_1
     (if (<= t_4 1e+245)
       (fma x2 -6.0 (* x1 (fma x1 9.0 -1.0)))
       (if (<= t_4 INFINITY) t_1 (fma x1 (* x1 9.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x1 * (8.0 * (x2 * x2));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2e+224) {
		tmp = t_1;
	} else if (t_4 <= 1e+245) {
		tmp = fma(x2, -6.0, (x1 * fma(x1, 9.0, -1.0)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(x1, (x1 * 9.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x1 * Float64(8.0 * Float64(x2 * x2)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2e+224)
		tmp = t_1;
	elseif (t_4 <= 1e+245)
		tmp = fma(x2, -6.0, Float64(x1 * fma(x1, 9.0, -1.0)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = fma(x1, Float64(x1 * 9.0), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+224], t$95$1, If[LessEqual[t$95$4, 1e+245], N[(x2 * -6.0 + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(x1 * 9.0), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.99999999999999994e224 or 1.00000000000000004e245 < (+.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 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

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

   5. Applied rewrites 69.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 67.5%

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

   9. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 66.8

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

   11. Applied rewrites 66.8%

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

   if -1.99999999999999994e224 < (+.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)))))) < 1.00000000000000004e245

   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

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

   5. Applied rewrites 79.1%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-rgt-identity N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(x1 \cdot \left(9 \cdot x1 - 2\right) + \color{blue}{x1 \cdot 1}\right) \]

       5. distribute-lft-out N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{x1 \cdot \left(\left(9 \cdot x1 - 2\right) + 1\right)} \]

       6. associate-+l- N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 88.0%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 80.1%

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

   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 56.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in x1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 88.2

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

   11. Applied rewrites 88.2%

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

   12. Taylor expanded in x1 around inf

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

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{9 \cdot {x1}^{2} + \frac{1}{x1} \cdot {x1}^{2}} \]

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \frac{1}{x1} \cdot \color{blue}{\left(x1 \cdot x1\right)} \]

       6. associate-*r* N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \color{blue}{\left(\frac{1}{x1} \cdot x1\right) \cdot x1} \]

       7. lft-mult-inverse N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \color{blue}{1} \cdot x1 \]

       8. *-lft-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 88.2

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

   14. Applied rewrites 88.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+224}:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x1 (* 8.0 (* x2 x2))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2e+224)
     t_1
     (if (<= t_4 1e+245)
       (fma x1 (fma x1 9.0 -1.0) (* x2 -6.0))
       (if (<= t_4 INFINITY) t_1 (fma x1 (* x1 9.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x1 * (8.0 * (x2 * x2));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2e+224) {
		tmp = t_1;
	} else if (t_4 <= 1e+245) {
		tmp = fma(x1, fma(x1, 9.0, -1.0), (x2 * -6.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(x1, (x1 * 9.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x1 * Float64(8.0 * Float64(x2 * x2)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2e+224)
		tmp = t_1;
	elseif (t_4 <= 1e+245)
		tmp = fma(x1, fma(x1, 9.0, -1.0), Float64(x2 * -6.0));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = fma(x1, Float64(x1 * 9.0), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+224], t$95$1, If[LessEqual[t$95$4, 1e+245], N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(x1 * 9.0), $MachinePrecision] + x1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.99999999999999994e224 or 1.00000000000000004e245 < (+.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 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

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

   5. Applied rewrites 69.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 67.5%

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

   9. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 66.8

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

   11. Applied rewrites 66.8%

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

   if -1.99999999999999994e224 < (+.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)))))) < 1.00000000000000004e245

   1. Initial program 99.1%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right) - 6\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 80.1

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

   10. Applied rewrites 80.1%

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

   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 56.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in x1 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 88.2

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

   11. Applied rewrites 88.2%

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

   12. Taylor expanded in x1 around inf

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

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{9 \cdot {x1}^{2} + \frac{1}{x1} \cdot {x1}^{2}} \]

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \frac{1}{x1} \cdot \color{blue}{\left(x1 \cdot x1\right)} \]

       6. associate-*r* N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \color{blue}{\left(\frac{1}{x1} \cdot x1\right) \cdot x1} \]

       7. lft-mult-inverse N/A

          \[\leadsto x1 \cdot \left(9 \cdot x1\right) + \color{blue}{1} \cdot x1 \]

       8. *-lft-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 88.2

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

   14. Applied rewrites 88.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+224}:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 9, -1\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1, x1 \cdot 9, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_1 \cdot t\_3\right) + t\_0\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+224}:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x1 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1)))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_1 t_3))
             t_0))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -2e+224)
     (* x1 (* 8.0 (* x2 x2)))
     (if (<= t_4 2e+208)
       (fma x2 -6.0 (* x1 (fma x1 9.0 -1.0)))
       (* 6.0 (* x1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_1 * t_3)) + t_0)) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -2e+224) {
		tmp = x1 * (8.0 * (x2 * x2));
	} else if (t_4 <= 2e+208) {
		tmp = fma(x2, -6.0, (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = 6.0 * (x1 * t_0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_1 * t_3)) + t_0)) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -2e+224)
		tmp = Float64(x1 * Float64(8.0 * Float64(x2 * x2)));
	elseif (t_4 <= 2e+208)
		tmp = fma(x2, -6.0, Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(6.0 * Float64(x1 * t_0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+224], N[(x1 * N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+208], N[(x2 * -6.0 + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(x1 * t$95$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)))))) < -1.99999999999999994e224

   1. Initial program 99.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 80.4%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.4%

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

   9. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 80.4

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

   11. Applied rewrites 80.4%

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

   if -1.99999999999999994e224 < (+.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)))))) < 2e208

   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

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

   5. Applied rewrites 84.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-rgt-identity N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(x1 \cdot \left(9 \cdot x1 - 2\right) + \color{blue}{x1 \cdot 1}\right) \]

       5. distribute-lft-out N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{x1 \cdot \left(\left(9 \cdot x1 - 2\right) + 1\right)} \]

       6. associate-+l- N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 93.4%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 84.9%

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

   if 2e208 < (+.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 30.9%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 89.0

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x1}^{\color{blue}{\left(2 \cdot 2\right)}}, 6, x1\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x1}^{2} \cdot {x1}^{2}}, 6, x1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}, 6, x1\right) \]

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot x1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(\color{blue}{\left(x1 \cdot {x1}^{2}\right)} \cdot x1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f64 89.0

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

   10. Applied rewrites 89.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -2 \cdot 10^{+224}:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma x1 (fma 3.0 x1 -1.0) (* 2.0 x2)))
        (t_1 (/ t_0 (fma x1 x1 1.0)))
        (t_2 (* x1 (* x1 3.0)))
        (t_3 (+ (* x1 x1) 1.0))
        (t_4 (/ (- (+ t_2 (* 2.0 x2)) x1) t_3)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_3
              (+
               (* (* (* x1 2.0) t_4) (- t_4 3.0))
               (* (* x1 x1) (- (* t_4 4.0) 6.0))))
             (* t_2 t_4))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_3))))
        INFINITY)
     (+
      x1
      (+
       x1
       (fma
        3.0
        (/ (- t_2 (fma 2.0 x2 x1)) (fma x1 x1 1.0))
        (fma
         (fma x1 x1 1.0)
         (fma
          (* (* x1 2.0) (+ -3.0 t_1))
          t_1
          (* x1 (* x1 (fma t_0 (/ 4.0 (fma x1 x1 1.0)) -6.0))))
         (fma
          x1
          (* x1 x1)
          (*
           t_2
           (/ (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))) (fma x1 x1 1.0))))))))
     (+
      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(x1, fma(3.0, x1, -1.0), (2.0 * x2));
	double t_1 = t_0 / fma(x1, x1, 1.0);
	double t_2 = x1 * (x1 * 3.0);
	double t_3 = (x1 * x1) + 1.0;
	double t_4 = ((t_2 + (2.0 * x2)) - x1) / t_3;
	double tmp;
	if ((x1 + ((x1 + (((t_3 * ((((x1 * 2.0) * t_4) * (t_4 - 3.0)) + ((x1 * x1) * ((t_4 * 4.0) - 6.0)))) + (t_2 * t_4)) + (x1 * (x1 * x1)))) + (3.0 * (((t_2 - (2.0 * x2)) - x1) / t_3)))) <= ((double) INFINITY)) {
		tmp = x1 + (x1 + fma(3.0, ((t_2 - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), fma(fma(x1, x1, 1.0), fma(((x1 * 2.0) * (-3.0 + t_1)), t_1, (x1 * (x1 * fma(t_0, (4.0 / fma(x1, x1, 1.0)), -6.0)))), fma(x1, (x1 * x1), (t_2 * (fma(2.0, x2, fma(x1, (x1 * 3.0), -x1)) / fma(x1, x1, 1.0)))))));
	} else {
		tmp = x1 + (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(x1, fma(3.0, x1, -1.0), Float64(2.0 * x2))
	t_1 = Float64(t_0 / fma(x1, x1, 1.0))
	t_2 = Float64(x1 * Float64(x1 * 3.0))
	t_3 = Float64(Float64(x1 * x1) + 1.0)
	t_4 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_3)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_3 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_4) * Float64(t_4 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_4 * 4.0) - 6.0)))) + Float64(t_2 * t_4)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_3)))) <= Inf)
		tmp = Float64(x1 + Float64(x1 + fma(3.0, Float64(Float64(t_2 - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), fma(fma(x1, x1, 1.0), fma(Float64(Float64(x1 * 2.0) * Float64(-3.0 + t_1)), t_1, Float64(x1 * Float64(x1 * fma(t_0, Float64(4.0 / fma(x1, x1, 1.0)), -6.0)))), fma(x1, Float64(x1 * x1), Float64(t_2 * Float64(fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1))) / fma(x1, x1, 1.0))))))));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))));
	end
	return tmp
end

Wolfram

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

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 95.5%

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

   6. Applied rewrites 99.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.3%

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

Alternative 8: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ t_1 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;x1 \leq -1.3 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;x1 + \left(x1 + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_2, 4, -6\right), \frac{\left(-3 + t\_2\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot x1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           (pow x1 4.0)
           (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1)))))
        (t_1 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_2 (/ t_1 (fma x1 x1 1.0))))
   (if (<= x1 -1.3e+101)
     t_0
     (if (<= x1 4.5e+30)
       (+
        x1
        (+
         x1
         (fma
          3.0
          (/ (- (* x1 (* x1 3.0)) (fma 2.0 x2 x1)) (fma x1 x1 1.0))
          (fma
           (fma x1 x1 1.0)
           (fma
            x1
            (* x1 (fma t_2 4.0 -6.0))
            (/ (* (+ -3.0 t_2) (* (* x1 2.0) t_1)) (fma x1 x1 1.0)))
           (* x1 (* x1 x1))))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	double t_1 = fma(2.0, x2, fma(x1, (x1 * 3.0), -x1));
	double t_2 = t_1 / fma(x1, x1, 1.0);
	double tmp;
	if (x1 <= -1.3e+101) {
		tmp = t_0;
	} else if (x1 <= 4.5e+30) {
		tmp = x1 + (x1 + fma(3.0, (((x1 * (x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), fma(fma(x1, x1, 1.0), fma(x1, (x1 * fma(t_2, 4.0, -6.0)), (((-3.0 + t_2) * ((x1 * 2.0) * t_1)) / fma(x1, x1, 1.0))), (x1 * (x1 * x1)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))))
	t_1 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_2 = Float64(t_1 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (x1 <= -1.3e+101)
		tmp = t_0;
	elseif (x1 <= 4.5e+30)
		tmp = Float64(x1 + Float64(x1 + fma(3.0, Float64(Float64(Float64(x1 * Float64(x1 * 3.0)) - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_2, 4.0, -6.0)), Float64(Float64(Float64(-3.0 + t_2) * Float64(Float64(x1 * 2.0) * t_1)) / fma(x1, x1, 1.0))), Float64(x1 * Float64(x1 * x1))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.3e+101], t$95$0, If[LessEqual[x1, 4.5e+30], N[(x1 + N[(x1 + N[(3.0 * N[(N[(N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(t$95$2 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-3.0 + t$95$2), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.3e101 or 4.49999999999999995e30 < x1

   1. Initial program 20.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if -1.3e101 < x1 < 4.49999999999999995e30

   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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x1 around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 97.9

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

   7. Applied rewrites 97.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.3 \cdot 10^{+101}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;x1 + \left(x1 + \mathsf{fma}\left(3, \frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(-3 + \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), x1 \cdot \left(x1 \cdot x1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{if}\;x1 \leq -2100000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 215000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           (pow x1 4.0)
           (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))
   (if (<= x1 -2100000.0)
     t_0
     (if (<= x1 215000000.0)
       (fma x2 (fma x1 (fma 8.0 x2 -12.0) -6.0) (* x1 (fma x1 9.0 -1.0)))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (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 <= -2100000.0) {
		tmp = t_0;
	} else if (x1 <= 215000000.0) {
		tmp = fma(x2, fma(x1, fma(8.0, x2, -12.0), -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))))
	tmp = 0.0
	if (x1 <= -2100000.0)
		tmp = t_0;
	elseif (x1 <= 215000000.0)
		tmp = fma(x2, fma(x1, fma(8.0, x2, -12.0), -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[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -2100000.0], t$95$0, If[LessEqual[x1, 215000000.0], N[(x2 * N[(x1 * N[(8.0 * x2 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.1e6 or 2.15e8 < x1

   1. Initial program 31.1%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 95.8%

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

   if -2.1e6 < x1 < 2.15e8

   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

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

   5. Applied rewrites 86.0%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-rgt-identity N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(x1 \cdot \left(9 \cdot x1 - 2\right) + \color{blue}{x1 \cdot 1}\right) \]

       5. distribute-lft-out N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{x1 \cdot \left(\left(9 \cdot x1 - 2\right) + 1\right)} \]

       6. associate-+l- N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 98.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2100000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 215000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\\ \mathbf{if}\;x1 \leq -145000000000:\\ \;\;\;\;6 \cdot t\_0\\ \mathbf{elif}\;x1 \leq 215000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + t\_0 \cdot \left(6 + \frac{-3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 (* x1 x1)))))
   (if (<= x1 -145000000000.0)
     (* 6.0 t_0)
     (if (<= x1 215000000.0)
       (fma x2 (fma x1 (fma 8.0 x2 -12.0) -6.0) (* x1 (fma x1 9.0 -1.0)))
       (+ x1 (* t_0 (+ 6.0 (/ -3.0 x1))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * (x1 * x1));
	double tmp;
	if (x1 <= -145000000000.0) {
		tmp = 6.0 * t_0;
	} else if (x1 <= 215000000.0) {
		tmp = fma(x2, fma(x1, fma(8.0, x2, -12.0), -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = x1 + (t_0 * (6.0 + (-3.0 / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * Float64(x1 * x1)))
	tmp = 0.0
	if (x1 <= -145000000000.0)
		tmp = Float64(6.0 * t_0);
	elseif (x1 <= 215000000.0)
		tmp = fma(x2, fma(x1, fma(8.0, x2, -12.0), -6.0), Float64(x1 * fma(x1, 9.0, -1.0)));
	else
		tmp = Float64(x1 + Float64(t_0 * Float64(6.0 + Float64(-3.0 / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -145000000000.0], N[(6.0 * t$95$0), $MachinePrecision], If[LessEqual[x1, 215000000.0], N[(x2 * N[(x1 * N[(8.0 * x2 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(t$95$0 * N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.45e11

   1. Initial program 19.2%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 94.7

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

   5. Applied rewrites 94.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 94.7

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x1}^{\color{blue}{\left(2 \cdot 2\right)}}, 6, x1\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x1}^{2} \cdot {x1}^{2}}, 6, x1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}, 6, x1\right) \]

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot x1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(\color{blue}{\left(x1 \cdot {x1}^{2}\right)} \cdot x1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f64 94.7

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

   10. Applied rewrites 94.7%

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

   if -1.45e11 < x1 < 2.15e8

   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

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

   5. Applied rewrites 86.0%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-rgt-identity N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(x1 \cdot \left(9 \cdot x1 - 2\right) + \color{blue}{x1 \cdot 1}\right) \]

       5. distribute-lft-out N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{x1 \cdot \left(\left(9 \cdot x1 - 2\right) + 1\right)} \]

       6. associate-+l- N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 98.2%

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

   if 2.15e8 < x1

   1. Initial program 42.8%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 36.4

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

   8. Applied rewrites 36.4%

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

   9. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-plus N/A

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

       4. lower-*.f64 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. sub-neg N/A

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

       11. lower-+.f64 N/A

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

       12. associate-*r/ N/A

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

       13. metadata-eval N/A

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

       14. distribute-neg-frac N/A

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

       15. metadata-eval N/A

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

       16. lower-/.f64 91.7

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

   11. Applied rewrites 91.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -145000000000:\\ \;\;\;\;6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 215000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right) \cdot \left(6 + \frac{-3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{if}\;x1 \leq -145000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1260000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x1 (* x1 (* x1 x1))))))
   (if (<= x1 -145000000000.0)
     t_0
     (if (<= x1 1260000000.0)
       (fma x2 (fma x1 (fma 8.0 x2 -12.0) -6.0) (* x1 (fma x1 9.0 -1.0)))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = 6.0 * (x1 * (x1 * (x1 * x1)));
	double tmp;
	if (x1 <= -145000000000.0) {
		tmp = t_0;
	} else if (x1 <= 1260000000.0) {
		tmp = fma(x2, fma(x1, fma(8.0, x2, -12.0), -6.0), (x1 * fma(x1, 9.0, -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(6.0 * Float64(x1 * Float64(x1 * Float64(x1 * x1))))
	tmp = 0.0
	if (x1 <= -145000000000.0)
		tmp = t_0;
	elseif (x1 <= 1260000000.0)
		tmp = fma(x2, fma(x1, fma(8.0, x2, -12.0), -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[(6.0 * N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -145000000000.0], t$95$0, If[LessEqual[x1, 1260000000.0], N[(x2 * N[(x1 * N[(8.0 * x2 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.45e11 or 1.26e9 < x1

   1. Initial program 31.1%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 92.9

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

   5. Applied rewrites 92.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 92.9

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x1}^{\color{blue}{\left(2 \cdot 2\right)}}, 6, x1\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x1}^{2} \cdot {x1}^{2}}, 6, x1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}, 6, x1\right) \]

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot x1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(\color{blue}{\left(x1 \cdot {x1}^{2}\right)} \cdot x1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f64 93.0

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

   10. Applied rewrites 93.0%

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

   if -1.45e11 < x1 < 1.26e9

   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

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

   5. Applied rewrites 86.0%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-rgt-identity N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(x1 \cdot \left(9 \cdot x1 - 2\right) + \color{blue}{x1 \cdot 1}\right) \]

       5. distribute-lft-out N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{x1 \cdot \left(\left(9 \cdot x1 - 2\right) + 1\right)} \]

       6. associate-+l- N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 98.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -145000000000:\\ \;\;\;\;6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 1260000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, 9\right), -1\right)\\ \mathbf{if}\;x1 \leq -8.6 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 9 \cdot 10^{-132}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 \leq 0.0095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.7 \cdot 10^{+142}:\\ \;\;\;\;x1 \cdot \left(8 \cdot \left(x2 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot 9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma x1 (fma x1 -19.0 9.0) -1.0))))
   (if (<= x1 -8.6e-88)
     t_0
     (if (<= x1 9e-132)
       (* x2 -6.0)
       (if (<= x1 0.0095)
         t_0
         (if (<= x1 2.7e+142) (* x1 (* 8.0 (* x2 x2))) (* x1 (* x1 9.0))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma(x1, fma(x1, -19.0, 9.0), -1.0);
	double tmp;
	if (x1 <= -8.6e-88) {
		tmp = t_0;
	} else if (x1 <= 9e-132) {
		tmp = x2 * -6.0;
	} else if (x1 <= 0.0095) {
		tmp = t_0;
	} else if (x1 <= 2.7e+142) {
		tmp = x1 * (8.0 * (x2 * x2));
	} else {
		tmp = x1 * (x1 * 9.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(x1, fma(x1, -19.0, 9.0), -1.0))
	tmp = 0.0
	if (x1 <= -8.6e-88)
		tmp = t_0;
	elseif (x1 <= 9e-132)
		tmp = Float64(x2 * -6.0);
	elseif (x1 <= 0.0095)
		tmp = t_0;
	elseif (x1 <= 2.7e+142)
		tmp = Float64(x1 * Float64(8.0 * Float64(x2 * x2)));
	else
		tmp = Float64(x1 * Float64(x1 * 9.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * N[(x1 * -19.0 + 9.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -8.6e-88], t$95$0, If[LessEqual[x1, 9e-132], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[x1, 0.0095], t$95$0, If[LessEqual[x1, 2.7e+142], N[(x1 * N[(8.0 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -8.5999999999999995e-88 or 8.9999999999999999e-132 < x1 < 0.00949999999999999976

   1. Initial program 55.2%

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in x1 around 0

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

   7. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x1, \mathsf{fma}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(x2, -4, 6\right), x2, \mathsf{fma}\left(x2, 6, -9\right)\right) + \left(1 + \mathsf{fma}\left(x2, -4, 6\right) \cdot x2\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -3\right)\right), \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \left(\mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right) + -6\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -1\right)\right), x2 \cdot -6\right)} \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + -19 \cdot x1\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto x1 \cdot \left(x1 \cdot \left(9 + -19 \cdot x1\right) + \color{blue}{-1}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x1 \cdot \color{blue}{\mathsf{fma}\left(x1, 9 + -19 \cdot x1, -1\right)} \]

      5. +-commutative N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(x1, \color{blue}{-19 \cdot x1 + 9}, -1\right) \]

      6. *-commutative N/A

         \[\leadsto x1 \cdot \mathsf{fma}\left(x1, \color{blue}{x1 \cdot -19} + 9, -1\right) \]

      7. lower-fma.f64 69.8

         \[\leadsto x1 \cdot \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(x1, -19, 9\right)}, -1\right) \]

   10. Applied rewrites 69.8%

       \[\leadsto \color{blue}{x1 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, -19, 9\right), -1\right)} \]

   if -8.5999999999999995e-88 < x1 < 8.9999999999999999e-132

   1. Initial program 98.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 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 67.5

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

   8. Applied rewrites 67.5%

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

   if 0.00949999999999999976 < x1 < 2.69999999999999983e142

   1. Initial program 95.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 41.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 42.1%

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

   9. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 40.9

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

   11. Applied rewrites 40.9%

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

   if 2.69999999999999983e142 < x1

   1. Initial program 5.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 79.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 94.7%

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

   9. Taylor expanded in x1 around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x1}^{2} \cdot 9} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 94.7

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

   11. Applied rewrites 94.7%

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

Alternative 13: 54.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 9\right)\\ t_1 := x1 + x1 \cdot -2\\ \mathbf{if}\;x1 \leq -1450000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -8.6 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 9 \cdot 10^{-132}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;x1 \leq 0.04:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 9.0))) (t_1 (+ x1 (* x1 -2.0))))
   (if (<= x1 -1450000.0)
     t_0
     (if (<= x1 -8.6e-88)
       t_1
       (if (<= x1 9e-132) (* x2 -6.0) (if (<= x1 0.04) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 9.0);
	double t_1 = x1 + (x1 * -2.0);
	double tmp;
	if (x1 <= -1450000.0) {
		tmp = t_0;
	} else if (x1 <= -8.6e-88) {
		tmp = t_1;
	} else if (x1 <= 9e-132) {
		tmp = x2 * -6.0;
	} else if (x1 <= 0.04) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x1 * (x1 * 9.0d0)
    t_1 = x1 + (x1 * (-2.0d0))
    if (x1 <= (-1450000.0d0)) then
        tmp = t_0
    else if (x1 <= (-8.6d-88)) then
        tmp = t_1
    else if (x1 <= 9d-132) then
        tmp = x2 * (-6.0d0)
    else if (x1 <= 0.04d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 9.0);
	double t_1 = x1 + (x1 * -2.0);
	double tmp;
	if (x1 <= -1450000.0) {
		tmp = t_0;
	} else if (x1 <= -8.6e-88) {
		tmp = t_1;
	} else if (x1 <= 9e-132) {
		tmp = x2 * -6.0;
	} else if (x1 <= 0.04) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 9.0)
	t_1 = x1 + (x1 * -2.0)
	tmp = 0
	if x1 <= -1450000.0:
		tmp = t_0
	elif x1 <= -8.6e-88:
		tmp = t_1
	elif x1 <= 9e-132:
		tmp = x2 * -6.0
	elif x1 <= 0.04:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 9.0))
	t_1 = Float64(x1 + Float64(x1 * -2.0))
	tmp = 0.0
	if (x1 <= -1450000.0)
		tmp = t_0;
	elseif (x1 <= -8.6e-88)
		tmp = t_1;
	elseif (x1 <= 9e-132)
		tmp = Float64(x2 * -6.0);
	elseif (x1 <= 0.04)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 9.0);
	t_1 = x1 + (x1 * -2.0);
	tmp = 0.0;
	if (x1 <= -1450000.0)
		tmp = t_0;
	elseif (x1 <= -8.6e-88)
		tmp = t_1;
	elseif (x1 <= 9e-132)
		tmp = x2 * -6.0;
	elseif (x1 <= 0.04)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1450000.0], t$95$0, If[LessEqual[x1, -8.6e-88], t$95$1, If[LessEqual[x1, 9e-132], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[x1, 0.04], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.45e6 or 0.0400000000000000008 < x1

   1. Initial program 32.9%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 51.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 59.7%

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

   9. Taylor expanded in x1 around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x1}^{2} \cdot 9} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 61.6

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

   11. Applied rewrites 61.6%

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

   if -1.45e6 < x1 < -8.5999999999999995e-88 or 8.9999999999999999e-132 < x1 < 0.0400000000000000008

   1. Initial program 98.9%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-2 \cdot x1} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x1 \cdot -2} \]

      2. lower-*.f64 51.7

         \[\leadsto x1 + \color{blue}{x1 \cdot -2} \]

   8. Applied rewrites 51.7%

      \[\leadsto x1 + \color{blue}{x1 \cdot -2} \]

   if -8.5999999999999995e-88 < x1 < 8.9999999999999999e-132

   1. Initial program 98.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 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 67.5

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

   8. Applied rewrites 67.5%

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

Alternative 14: 93.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{if}\;x1 \leq -145000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1200000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 6.0 (* x1 (* x1 (* x1 x1))))))
   (if (<= x1 -145000000000.0)
     t_0
     (if (<= x1 1200000000.0)
       (fma x2 (fma x1 (fma 8.0 x2 -12.0) -6.0) (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = 6.0 * (x1 * (x1 * (x1 * x1)));
	double tmp;
	if (x1 <= -145000000000.0) {
		tmp = t_0;
	} else if (x1 <= 1200000000.0) {
		tmp = fma(x2, fma(x1, fma(8.0, x2, -12.0), -6.0), -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(6.0 * Float64(x1 * Float64(x1 * Float64(x1 * x1))))
	tmp = 0.0
	if (x1 <= -145000000000.0)
		tmp = t_0;
	elseif (x1 <= 1200000000.0)
		tmp = fma(x2, fma(x1, fma(8.0, x2, -12.0), -6.0), Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(6.0 * N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -145000000000.0], t$95$0, If[LessEqual[x1, 1200000000.0], N[(x2 * N[(x1 * N[(8.0 * x2 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision] + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.45e11 or 1.2e9 < x1

   1. Initial program 31.1%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 92.9

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

   5. Applied rewrites 92.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 92.9

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x1}^{\color{blue}{\left(2 \cdot 2\right)}}, 6, x1\right) \]

      9. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x1}^{2} \cdot {x1}^{2}}, 6, x1\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot x1\right)} \cdot {x1}^{2}, 6, x1\right) \]

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

   8. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot x1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left(\color{blue}{\left(x1 \cdot {x1}^{2}\right)} \cdot x1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f64 93.0

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

   10. Applied rewrites 93.0%

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

   if -1.45e11 < x1 < 1.2e9

   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

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

   5. Applied rewrites 86.0%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 86.0%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-rgt-identity N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \left(x1 \cdot \left(9 \cdot x1 - 2\right) + \color{blue}{x1 \cdot 1}\right) \]

       5. distribute-lft-out N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + \color{blue}{x1 \cdot \left(\left(9 \cdot x1 - 2\right) + 1\right)} \]

       6. associate-+l- N/A

          \[\leadsto x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right) + x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

   11. Applied rewrites 98.2%

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

   12. Taylor expanded in x1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 97.4

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

   14. Applied rewrites 97.4%

       \[\leadsto \mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), \color{blue}{-x1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -145000000000:\\ \;\;\;\;6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 1200000000:\\ \;\;\;\;\mathsf{fma}\left(x2, \mathsf{fma}\left(x1, \mathsf{fma}\left(8, x2, -12\right), -6\right), -x1\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot x2 \leq -2 \cdot 10^{-174}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{elif}\;2 \cdot x2 \leq 10^{-190}:\\ \;\;\;\;x1 + x1 \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* 2.0 x2) -2e-174)
   (* x2 -6.0)
   (if (<= (* 2.0 x2) 1e-190) (+ x1 (* x1 -2.0)) (fma x2 -6.0 x1))))

C

double code(double x1, double x2) {
	double tmp;
	if ((2.0 * x2) <= -2e-174) {
		tmp = x2 * -6.0;
	} else if ((2.0 * x2) <= 1e-190) {
		tmp = x1 + (x1 * -2.0);
	} else {
		tmp = fma(x2, -6.0, x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(2.0 * x2) <= -2e-174)
		tmp = Float64(x2 * -6.0);
	elseif (Float64(2.0 * x2) <= 1e-190)
		tmp = Float64(x1 + Float64(x1 * -2.0));
	else
		tmp = fma(x2, -6.0, x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(2.0 * x2), $MachinePrecision], -2e-174], N[(x2 * -6.0), $MachinePrecision], If[LessEqual[N[(2.0 * x2), $MachinePrecision], 1e-190], N[(x1 + N[(x1 * -2.0), $MachinePrecision]), $MachinePrecision], N[(x2 * -6.0 + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -2e-174

   1. Initial program 68.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 36.5

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

   5. Applied rewrites 36.5%

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

   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 36.9

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

   8. Applied rewrites 36.9%

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

   if -2e-174 < (*.f64 #s(literal 2 binary64) x2) < 1e-190

   1. Initial program 74.5%

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

   3. Taylor expanded in x2 around 0

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

   5. Applied rewrites 63.2%

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

   6. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-2 \cdot x1} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x1 + \color{blue}{x1 \cdot -2} \]

      2. lower-*.f64 49.9

         \[\leadsto x1 + \color{blue}{x1 \cdot -2} \]

   8. Applied rewrites 49.9%

      \[\leadsto x1 + \color{blue}{x1 \cdot -2} \]

   if 1e-190 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 64.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 28.0

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

   5. Applied rewrites 28.0%

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

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 28.1

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

   7. Applied rewrites 28.1%

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

Alternative 16: 54.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \mathbf{if}\;x1 \leq -8.6 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 9 \cdot 10^{-132}:\\ \;\;\;\;x2 \cdot -6\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma x1 9.0 -1.0))))
   (if (<= x1 -8.6e-88) t_0 (if (<= x1 9e-132) (* 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 <= -8.6e-88) {
		tmp = t_0;
	} else if (x1 <= 9e-132) {
		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 <= -8.6e-88)
		tmp = t_0;
	elseif (x1 <= 9e-132)
		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, -8.6e-88], t$95$0, If[LessEqual[x1, 9e-132], N[(x2 * -6.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -8.5999999999999995e-88 or 8.9999999999999999e-132 < x1

   1. Initial program 51.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 63.6%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \color{blue}{9}, \mathsf{fma}\left(x2, \mathsf{fma}\left(4, x2 \cdot 2, -12\right), -2\right)\right), x2 \cdot -6\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.1%

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

   9. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. *-rgt-identity N/A

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

       3. distribute-lft-out N/A

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

       4. associate-+l- N/A

          \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 - \left(2 - 1\right)\right)} \]

       5. metadata-eval N/A

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

       6. lower-*.f64 N/A

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

       7. sub-neg N/A

          \[\leadsto x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto x1 \cdot \left(\color{blue}{x1 \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

       9. metadata-eval N/A

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

       10. lower-fma.f64 59.5

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

   11. Applied rewrites 59.5%

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

   if -8.5999999999999995e-88 < x1 < 8.9999999999999999e-132

   1. Initial program 98.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 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 67.5

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

   8. Applied rewrites 67.5%

      \[\leadsto \color{blue}{x2 \cdot -6} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 26.1% accurate, 42.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x2, -6, x1\right) \end{array} \]

FPCore

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

C

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

Julia

function code(x1, x2)
	return fma(x2, -6.0, x1)
end

Wolfram

code[x1_, x2_] := N[(x2 * -6.0 + x1), $MachinePrecision]

Derivation

1. Initial program 68.3%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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 27.9

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

5. Applied rewrites 27.9%

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

   1. lift-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 27.9

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

7. Applied rewrites 27.9%

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

Alternative 18: 26.0% accurate, 49.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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 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 27.7

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

8. Applied rewrites 27.7%

   \[\leadsto \color{blue}{x2 \cdot -6} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))))