Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 69.9% → 99.0%

Time: 18.9s

Alternatives: 20

Speedup: 6.8×

• 47.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\\ t_4 := x1 + \left(t\_3 + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ t_5 := \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\\ t_6 := \frac{t\_5}{\mathsf{fma}\left(x1, x1, 1\right)}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{t\_0 - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(t\_6, 4, -6\right), \frac{\left(t\_6 + -3\right) \cdot \left(\left(x1 \cdot 2\right) \cdot t\_5\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot t\_6, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;x1 + \left(t\_3 + 3 \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \end{array} \end{array} \]

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
          (+
           (+
            (*
             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)))))
        (t_4 (+ x1 (+ t_3 (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1)))))
        (t_5 (fma 2.0 x2 (fma x1 (* x1 3.0) (- x1))))
        (t_6 (/ t_5 (fma x1 x1 1.0))))
   (if (<= t_4 5e+262)
     (+
      x1
      (fma
       (/ (- t_0 (fma 2.0 x2 x1)) (fma x1 x1 1.0))
       3.0
       (fma
        (fma x1 x1 1.0)
        (fma
         x1
         (* x1 (fma t_6 4.0 -6.0))
         (/ (* (+ t_6 -3.0) (* (* x1 2.0) t_5)) (fma x1 x1 1.0)))
        (fma x1 (* (* x1 3.0) t_6) (fma x1 (* x1 x1) x1)))))
     (if (<= t_4 INFINITY)
       (+ x1 (+ t_3 (* 3.0 3.0)))
       (* x1 (* x1 (* x1 (* x1 6.0))))))))

C

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

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(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))))
	t_4 = Float64(x1 + Float64(t_3 + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	t_5 = fma(2.0, x2, fma(x1, Float64(x1 * 3.0), Float64(-x1)))
	t_6 = Float64(t_5 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (t_4 <= 5e+262)
		tmp = Float64(x1 + fma(Float64(Float64(t_0 - fma(2.0, x2, x1)) / fma(x1, x1, 1.0)), 3.0, fma(fma(x1, x1, 1.0), fma(x1, Float64(x1 * fma(t_6, 4.0, -6.0)), Float64(Float64(Float64(t_6 + -3.0) * Float64(Float64(x1 * 2.0) * t_5)) / fma(x1, x1, 1.0))), fma(x1, Float64(Float64(x1 * 3.0) * t_6), fma(x1, Float64(x1 * x1), x1)))));
	elseif (t_4 <= Inf)
		tmp = Float64(x1 + Float64(t_3 + Float64(3.0 * 3.0)));
	else
		tmp = Float64(x1 * Float64(x1 * Float64(x1 * Float64(x1 * 6.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(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]}, Block[{t$95$4 = N[(x1 + N[(t$95$3 + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * x2 + N[(x1 * N[(x1 * 3.0), $MachinePrecision] + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e+262], N[(x1 + N[(N[(N[(t$95$0 - N[(2.0 * x2 + x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 * N[(x1 * N[(t$95$6 * 4.0 + -6.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$6 + -3.0), $MachinePrecision] * N[(N[(x1 * 2.0), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(N[(x1 * 3.0), $MachinePrecision] * t$95$6), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(x1 + N[(t$95$3 + N[(3.0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 5.00000000000000008e262

   1. Initial program 99.3%

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

   4. Applied rewrites 99.6%

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

   if 5.00000000000000008e262 < (+.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 inf

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

   5. Applied rewrites 99.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+262}:\\ \;\;\;\;x1 + \mathsf{fma}\left(\frac{x1 \cdot \left(x1 \cdot 3\right) - \mathsf{fma}\left(2, x2, x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, x1 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 4, -6\right), \frac{\left(\frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)} + -3\right) \cdot \left(\left(x1 \cdot 2\right) \cdot \mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\right), \mathsf{fma}\left(x1, \left(x1 \cdot 3\right) \cdot \frac{\mathsf{fma}\left(2, x2, \mathsf{fma}\left(x1, x1 \cdot 3, -x1\right)\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, \mathsf{fma}\left(x1, x1 \cdot x1, x1\right)\right)\right)\right)\\ \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 + \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 3\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ t_3 := x1 \cdot x1 + 1\\ t_4 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_3}\\ t_5 := x1 + \left(\left(x1 + \left(\left(t\_3 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_4\right) \cdot \left(t\_4 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_4 \cdot 4 - 6\right)\right) + t\_1 \cdot t\_4\right) + t\_0\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_3}\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+36}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;t\_5 \leq 10^{+263}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot t\_0, 6, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1)))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (* x2 (fma x2 (* x1 8.0) -6.0)))
        (t_3 (+ (* x1 x1) 1.0))
        (t_4 (/ (- (+ t_1 (* 2.0 x2)) x1) t_3))
        (t_5
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_3
               (+
                (* (* (* x1 2.0) t_4) (- t_4 3.0))
                (* (* x1 x1) (- (* t_4 4.0) 6.0))))
              (* t_1 t_4))
             t_0))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_3))))))
   (if (<= t_5 -2e+36)
     t_2
     (if (<= t_5 5e+114)
       (fma -6.0 x2 (* x1 (fma x1 9.0 -1.0)))
       (if (<= t_5 1e+263) t_2 (fma (* x1 t_0) 6.0 x1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = x2 * fma(x2, (x1 * 8.0), -6.0);
	double t_3 = (x1 * x1) + 1.0;
	double t_4 = ((t_1 + (2.0 * x2)) - x1) / t_3;
	double t_5 = x1 + ((x1 + (((t_3 * ((((x1 * 2.0) * t_4) * (t_4 - 3.0)) + ((x1 * x1) * ((t_4 * 4.0) - 6.0)))) + (t_1 * t_4)) + t_0)) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_3)));
	double tmp;
	if (t_5 <= -2e+36) {
		tmp = t_2;
	} else if (t_5 <= 5e+114) {
		tmp = fma(-6.0, x2, (x1 * fma(x1, 9.0, -1.0)));
	} else if (t_5 <= 1e+263) {
		tmp = t_2;
	} else {
		tmp = fma((x1 * t_0), 6.0, x1);
	}
	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(x2 * fma(x2, Float64(x1 * 8.0), -6.0))
	t_3 = Float64(Float64(x1 * x1) + 1.0)
	t_4 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_3)
	t_5 = 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_1 * t_4)) + t_0)) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_3))))
	tmp = 0.0
	if (t_5 <= -2e+36)
		tmp = t_2;
	elseif (t_5 <= 5e+114)
		tmp = fma(-6.0, x2, Float64(x1 * fma(x1, 9.0, -1.0)));
	elseif (t_5 <= 1e+263)
		tmp = t_2;
	else
		tmp = fma(Float64(x1 * t_0), 6.0, x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = 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$1 * t$95$4), $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$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+36], t$95$2, If[LessEqual[t$95$5, 5e+114], N[(-6.0 * x2 + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1e+263], t$95$2, N[(N[(x1 * t$95$0), $MachinePrecision] * 6.0 + 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)))))) < -2.00000000000000008e36 or 5.0000000000000001e114 < (+.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)))))) < 1.00000000000000002e263

   1. Initial program 99.6%

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

   3. Taylor expanded in 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 54.3%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 86.6

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

   8. Applied rewrites 86.6%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 86.6%

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

   12. Taylor expanded in x2 around inf

       \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1 - 6 \cdot \frac{1}{x2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. rgt-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 86.4

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

   14. Applied rewrites 86.4%

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

   if -2.00000000000000008e36 < (+.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)))))) < 5.0000000000000001e114

   1. Initial program 99.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 90.8

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

   8. Applied rewrites 90.8%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 90.8%

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

   12. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-rgt-identity N/A

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

       7. distribute-lft-out N/A

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

       8. lower-*.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. associate-+l+ N/A

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

       12. *-commutative N/A

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 90.9

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

   14. Applied rewrites 90.9%

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

   if 1.00000000000000002e263 < (+.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 33.6%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 92.5

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

   10. Applied rewrites 92.5%

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

4. Final simplification 90.6%

   \[\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^{+36}:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -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 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, 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 10^{+263}:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right), 6, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;t\_4 \leq 10^{+263}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, \left(x1 \cdot x1\right) \cdot 6, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x2 (fma x2 (* x1 8.0) -6.0)))
        (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+36)
     t_1
     (if (<= t_4 5e+114)
       (fma -6.0 x2 (* x1 (fma x1 9.0 -1.0)))
       (if (<= t_4 1e+263) t_1 (fma (* x1 x1) (* (* x1 x1) 6.0) x1))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x2 * fma(x2, (x1 * 8.0), -6.0);
	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+36) {
		tmp = t_1;
	} else if (t_4 <= 5e+114) {
		tmp = fma(-6.0, x2, (x1 * fma(x1, 9.0, -1.0)));
	} else if (t_4 <= 1e+263) {
		tmp = t_1;
	} else {
		tmp = fma((x1 * x1), ((x1 * x1) * 6.0), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x2 * fma(x2, Float64(x1 * 8.0), -6.0))
	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+36)
		tmp = t_1;
	elseif (t_4 <= 5e+114)
		tmp = fma(-6.0, x2, Float64(x1 * fma(x1, 9.0, -1.0)));
	elseif (t_4 <= 1e+263)
		tmp = t_1;
	else
		tmp = fma(Float64(x1 * x1), Float64(Float64(x1 * x1) * 6.0), 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[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + -6.0), $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+36], t$95$1, If[LessEqual[t$95$4, 5e+114], N[(-6.0 * x2 + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+263], t$95$1, N[(N[(x1 * x1), $MachinePrecision] * N[(N[(x1 * x1), $MachinePrecision] * 6.0), $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)))))) < -2.00000000000000008e36 or 5.0000000000000001e114 < (+.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)))))) < 1.00000000000000002e263

   1. Initial program 99.6%

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

   3. Taylor expanded in 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 54.3%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 86.6

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

   8. Applied rewrites 86.6%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 86.6%

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

   12. Taylor expanded in x2 around inf

       \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1 - 6 \cdot \frac{1}{x2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. rgt-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 86.4

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

   14. Applied rewrites 86.4%

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

   if -2.00000000000000008e36 < (+.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)))))) < 5.0000000000000001e114

   1. Initial program 99.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 90.8

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

   8. Applied rewrites 90.8%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 90.8%

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

   12. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-rgt-identity N/A

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

       7. distribute-lft-out N/A

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

       8. lower-*.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. associate-+l+ N/A

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

       12. *-commutative N/A

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 90.9

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

   14. Applied rewrites 90.9%

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

   if 1.00000000000000002e263 < (+.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 33.6%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 92.5

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

   5. Applied rewrites 92.5%

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]
   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. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 90.6%

   \[\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^{+36}:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -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 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, 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 10^{+263}:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, \left(x1 \cdot x1\right) \cdot 6, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\right)\\ \mathbf{elif}\;t\_4 \leq 10^{+263}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x2 (fma x2 (* x1 8.0) -6.0)))
        (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+36)
     t_1
     (if (<= t_4 5e+114)
       (fma -6.0 x2 (* x1 (fma x1 9.0 -1.0)))
       (if (<= t_4 1e+263) t_1 (* x1 (* x1 (* x1 (* x1 6.0)))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x2 * fma(x2, (x1 * 8.0), -6.0);
	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+36) {
		tmp = t_1;
	} else if (t_4 <= 5e+114) {
		tmp = fma(-6.0, x2, (x1 * fma(x1, 9.0, -1.0)));
	} else if (t_4 <= 1e+263) {
		tmp = t_1;
	} else {
		tmp = x1 * (x1 * (x1 * (x1 * 6.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x2 * fma(x2, Float64(x1 * 8.0), -6.0))
	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+36)
		tmp = t_1;
	elseif (t_4 <= 5e+114)
		tmp = fma(-6.0, x2, Float64(x1 * fma(x1, 9.0, -1.0)));
	elseif (t_4 <= 1e+263)
		tmp = t_1;
	else
		tmp = Float64(x1 * Float64(x1 * Float64(x1 * Float64(x1 * 6.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + -6.0), $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+36], t$95$1, If[LessEqual[t$95$4, 5e+114], N[(-6.0 * x2 + N[(x1 * N[(x1 * 9.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+263], t$95$1, N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2.00000000000000008e36 or 5.0000000000000001e114 < (+.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)))))) < 1.00000000000000002e263

   1. Initial program 99.6%

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

   3. Taylor expanded in 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 54.3%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 86.6

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

   8. Applied rewrites 86.6%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 86.6%

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

   12. Taylor expanded in x2 around inf

       \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1 - 6 \cdot \frac{1}{x2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. rgt-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 86.4

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

   14. Applied rewrites 86.4%

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

   if -2.00000000000000008e36 < (+.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)))))) < 5.0000000000000001e114

   1. Initial program 99.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 90.8

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

   8. Applied rewrites 90.8%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 90.8%

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

   12. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-rgt-identity N/A

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

       7. distribute-lft-out N/A

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

       8. lower-*.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. associate-+l+ N/A

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

       12. *-commutative N/A

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 90.9

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

   14. Applied rewrites 90.9%

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

   if 1.00000000000000002e263 < (+.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 33.6%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

   9. Taylor expanded in x1 around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 92.5

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

   11. Applied rewrites 92.5%

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

4. Final simplification 90.6%

   \[\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^{+36}:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -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 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, 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 10^{+263}:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ t_1 := x1 \cdot \left(x1 \cdot 3\right)\\ t_2 := x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ t_3 := x1 \cdot x1 + 1\\ t_4 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_3}\\ t_5 := x1 + \left(\left(x1 + \left(\left(t\_3 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_4\right) \cdot \left(t\_4 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_4 \cdot 4 - 6\right)\right) + t\_1 \cdot t\_4\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_3}\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{+36}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, t\_0\right)\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (fma x1 9.0 -1.0)))
        (t_1 (* x1 (* x1 3.0)))
        (t_2 (* x2 (fma x2 (* x1 8.0) -6.0)))
        (t_3 (+ (* x1 x1) 1.0))
        (t_4 (/ (- (+ t_1 (* 2.0 x2)) x1) t_3))
        (t_5
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_3
               (+
                (* (* (* x1 2.0) t_4) (- t_4 3.0))
                (* (* x1 x1) (- (* t_4 4.0) 6.0))))
              (* t_1 t_4))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_3))))))
   (if (<= t_5 -2e+36)
     t_2
     (if (<= t_5 5e+114) (fma -6.0 x2 t_0) (if (<= t_5 INFINITY) t_2 t_0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * fma(x1, 9.0, -1.0);
	double t_1 = x1 * (x1 * 3.0);
	double t_2 = x2 * fma(x2, (x1 * 8.0), -6.0);
	double t_3 = (x1 * x1) + 1.0;
	double t_4 = ((t_1 + (2.0 * x2)) - x1) / t_3;
	double t_5 = x1 + ((x1 + (((t_3 * ((((x1 * 2.0) * t_4) * (t_4 - 3.0)) + ((x1 * x1) * ((t_4 * 4.0) - 6.0)))) + (t_1 * t_4)) + (x1 * (x1 * x1)))) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_3)));
	double tmp;
	if (t_5 <= -2e+36) {
		tmp = t_2;
	} else if (t_5 <= 5e+114) {
		tmp = fma(-6.0, x2, t_0);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * fma(x1, 9.0, -1.0))
	t_1 = Float64(x1 * Float64(x1 * 3.0))
	t_2 = Float64(x2 * fma(x2, Float64(x1 * 8.0), -6.0))
	t_3 = Float64(Float64(x1 * x1) + 1.0)
	t_4 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_3)
	t_5 = 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_1 * t_4)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_3))))
	tmp = 0.0
	if (t_5 <= -2e+36)
		tmp = t_2;
	elseif (t_5 <= 5e+114)
		tmp = fma(-6.0, x2, t_0);
	elseif (t_5 <= Inf)
		tmp = t_2;
	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]}, Block[{t$95$1 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = 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$1 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e+36], t$95$2, If[LessEqual[t$95$5, 5e+114], N[(-6.0 * x2 + t$95$0), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$2, t$95$0]]]]]]]]]

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)))))) < -2.00000000000000008e36 or 5.0000000000000001e114 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 70.1

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

   8. Applied rewrites 70.1%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 70.2%

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

   12. Taylor expanded in x2 around inf

       \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1 - 6 \cdot \frac{1}{x2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. rgt-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 69.2

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

   14. Applied rewrites 69.2%

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

   if -2.00000000000000008e36 < (+.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)))))) < 5.0000000000000001e114

   1. Initial program 99.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 90.8

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

   8. Applied rewrites 90.8%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 90.8%

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

   12. Taylor expanded in x2 around 0

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

       1. +-commutative N/A

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

       2. associate-+l+ N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-rgt-identity N/A

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

       7. distribute-lft-out N/A

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

       8. lower-*.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. associate-+l+ N/A

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

       12. *-commutative N/A

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

       13. metadata-eval N/A

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

       14. lower-fma.f64 90.9

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

   14. Applied rewrites 90.9%

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

   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 72.4%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 41.3

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

   8. Applied rewrites 41.3%

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

   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. lower-*.f64 N/A

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

       5. associate-+l- N/A

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

       6. metadata-eval N/A

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

       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 85.9

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

   11. Applied rewrites 85.9%

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

4. Final simplification 81.3%

   \[\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^{+36}:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -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 5 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-6, x2, 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:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+100}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x2 (fma x2 (* x1 8.0) -6.0)))
        (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 -5e+185)
     t_1
     (if (<= t_4 1e+100)
       (- (* x2 -6.0) x1)
       (if (<= t_4 INFINITY) t_1 (* x1 (fma x1 9.0 -1.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x2 * fma(x2, (x1 * 8.0), -6.0);
	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 <= -5e+185) {
		tmp = t_1;
	} else if (t_4 <= 1e+100) {
		tmp = (x2 * -6.0) - x1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 * fma(x1, 9.0, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x2 * fma(x2, Float64(x1 * 8.0), -6.0))
	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 <= -5e+185)
		tmp = t_1;
	elseif (t_4 <= 1e+100)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + -6.0), $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, -5e+185], t$95$1, If[LessEqual[t$95$4, 1e+100], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(x1 * 9.0 + -1.0), $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)))))) < -4.9999999999999999e185 or 1.00000000000000002e100 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 65.2

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

   8. Applied rewrites 65.2%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 65.3%

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

   12. Taylor expanded in x2 around inf

       \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1 - 6 \cdot \frac{1}{x2}\right)} \]
   13. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. distribute-lft-in N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-neg-in N/A

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. rgt-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 64.1

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

   14. Applied rewrites 64.1%

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

   if -4.9999999999999999e185 < (+.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)))))) < 1.00000000000000002e100

   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 90.6%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 92.7

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

   8. Applied rewrites 92.7%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 91.6

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

   11. Applied rewrites 91.6%

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

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

   1. Initial program 0.0%

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

   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 72.4%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 41.3

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

   8. Applied rewrites 41.3%

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

   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. lower-*.f64 N/A

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

       5. associate-+l- N/A

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

       6. metadata-eval N/A

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

       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 85.9

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

   11. Applied rewrites 85.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+185}:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -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 10^{+100}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \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:\\ \;\;\;\;x2 \cdot \mathsf{fma}\left(x2, x1 \cdot 8, -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+263}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x2 (* 8.0 (* x1 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 -5e+263)
     t_1
     (if (<= t_4 5e+186)
       (- (* x2 -6.0) x1)
       (if (<= t_4 INFINITY) t_1 (* x1 (fma x1 9.0 -1.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x2 * (8.0 * (x1 * 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 <= -5e+263) {
		tmp = t_1;
	} else if (t_4 <= 5e+186) {
		tmp = (x2 * -6.0) - x1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 * fma(x1, 9.0, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x2 * Float64(8.0 * Float64(x1 * 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 <= -5e+263)
		tmp = t_1;
	elseif (t_4 <= 5e+186)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x2 * N[(8.0 * N[(x1 * 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, -5e+263], t$95$1, If[LessEqual[t$95$4, 5e+186], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(x1 * 9.0 + -1.0), $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)))))) < -5.00000000000000022e263 or 4.99999999999999954e186 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 47.7%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 61.2

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

   8. Applied rewrites 61.2%

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

   9. Taylor expanded in x2 around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 60.0

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

   11. Applied rewrites 60.0%

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

   if -5.00000000000000022e263 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999954e186

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 84.5

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 83.6

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

   11. Applied rewrites 83.6%

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

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

   1. Initial program 0.0%

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

   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 72.4%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 41.3

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

   8. Applied rewrites 41.3%

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

   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. lower-*.f64 N/A

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

       5. associate-+l- N/A

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

       6. metadata-eval N/A

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

       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 85.9

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

   11. Applied rewrites 85.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+263}:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \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:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\\ t_4 := x1 + \left(t\_3 + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_4 \leq 10^{+22}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;x1 + \left(t\_3 + 3 \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \end{array} \end{array} \]

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
          (+
           (+
            (*
             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)))))
        (t_4 (+ x1 (+ t_3 (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_4 1e+22)
     (+ x1 (fma x2 (fma x2 (* x1 8.0) -6.0) (* x1 (fma 9.0 x1 -2.0))))
     (if (<= t_4 INFINITY)
       (+ x1 (+ t_3 (* 3.0 3.0)))
       (* x1 (* x1 (* x1 (* x1 6.0))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((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)));
	double t_4 = x1 + (t_3 + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_4 <= 1e+22) {
		tmp = x1 + fma(x2, fma(x2, (x1 * 8.0), -6.0), (x1 * fma(9.0, x1, -2.0)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = x1 + (t_3 + (3.0 * 3.0));
	} else {
		tmp = x1 * (x1 * (x1 * (x1 * 6.0)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(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))))
	t_4 = Float64(x1 + Float64(t_3 + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_4 <= 1e+22)
		tmp = Float64(x1 + fma(x2, fma(x2, Float64(x1 * 8.0), -6.0), Float64(x1 * fma(9.0, x1, -2.0))));
	elseif (t_4 <= Inf)
		tmp = Float64(x1 + Float64(t_3 + Float64(3.0 * 3.0)));
	else
		tmp = Float64(x1 * Float64(x1 * Float64(x1 * Float64(x1 * 6.0))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(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]}, Block[{t$95$4 = N[(x1 + N[(t$95$3 + 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$4, 1e+22], N[(x1 + N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(x1 + N[(t$95$3 + N[(3.0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1e22

   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

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 99.5%

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

   if 1e22 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 88.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around inf

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

       1. metadata-eval N/A

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

       2. pow-plus N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. associate-*l* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+22}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\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 + \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 3\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;x2 \cdot \left(\left(x1 \cdot x1\right) \cdot 8\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \end{array} \]

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 5e+186)
     (- (* x2 -6.0) x1)
     (if (<= t_3 INFINITY)
       (* x2 (* (* x1 x1) 8.0))
       (* x1 (fma x1 9.0 -1.0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= 5e+186) {
		tmp = (x2 * -6.0) - x1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = x2 * ((x1 * x1) * 8.0);
	} else {
		tmp = x1 * fma(x1, 9.0, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= 5e+186)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (t_3 <= Inf)
		tmp = Float64(x2 * Float64(Float64(x1 * x1) * 8.0));
	else
		tmp = Float64(x1 * fma(x1, 9.0, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e+186], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(x2 * N[(N[(x1 * x1), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(x1 * 9.0 + -1.0), $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)))))) < 4.99999999999999954e186

   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

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 75.6

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 74.7

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

   11. Applied rewrites 74.7%

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

   if 4.99999999999999954e186 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in x2 around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 19.5

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

   8. Applied rewrites 19.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in x2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f64 41.3

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

   8. Applied rewrites 41.3%

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

   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. lower-*.f64 N/A

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

       5. associate-+l- N/A

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

       6. metadata-eval N/A

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

       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 85.9

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

   11. Applied rewrites 85.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x2 \cdot \left(\left(x1 \cdot x1\right) \cdot 8\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1, 9, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.2% accurate, 0.5× speedup

Math

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

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 INFINITY) t_3 (* x1 (* x1 (* x1 (* x1 6.0)))))))

C

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

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double 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 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = x1 * (x1 * (x1 * (x1 * 6.0)));
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 3.0)
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	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)))
	tmp = 0
	if t_3 <= math.inf:
		tmp = t_3
	else:
		tmp = x1 * (x1 * (x1 * (x1 * 6.0)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 3.0);
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	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)));
	tmp = 0.0;
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = x1 * (x1 * (x1 * (x1 * 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, 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, Infinity], t$95$3, N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0), $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 99.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{x1 \cdot \left(1 + 6 \cdot {x1}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \color{blue}{\left(6 \cdot {x1}^{3} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(6 \cdot {x1}^{3}\right) \cdot x1 + 1 \cdot x1} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({x1}^{3} \cdot 6\right)} \cdot x1 + 1 \cdot x1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x1}^{3} \cdot \left(6 \cdot x1\right)} + 1 \cdot x1 \]

      5. *-lft-identity N/A

         \[\leadsto {x1}^{3} \cdot \left(6 \cdot x1\right) + \color{blue}{x1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x1}^{3}, 6 \cdot x1, x1\right)} \]

      7. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{{x1}^{2}}, 6 \cdot x1, x1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot {x1}^{2}}, 6 \cdot x1, x1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), \color{blue}{x1 \cdot 6}, x1\right) \]

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), \color{blue}{x1 \cdot 6}, x1\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)} \]

   9. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto 6 \cdot {x1}^{\color{blue}{\left(3 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto 6 \cdot \color{blue}{\left({x1}^{3} \cdot x1\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(6 \cdot {x1}^{3}\right) \cdot x1} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{x1 \cdot \left(6 \cdot {x1}^{3}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(6 \cdot {x1}^{3}\right)} \]

       6. *-commutative N/A

          \[\leadsto x1 \cdot \color{blue}{\left({x1}^{3} \cdot 6\right)} \]

       7. cube-mult N/A

          \[\leadsto x1 \cdot \left(\color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)} \cdot 6\right) \]

       8. unpow2 N/A

          \[\leadsto x1 \cdot \left(\left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot 6\right) \]

       9. associate-*l* N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left({x1}^{2} \cdot 6\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left({x1}^{2} \cdot 6\right)\right)} \]

       11. unpow2 N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot 6\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot 6\right)\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(6 \cdot x1\right)}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot \left(6 \cdot x1\right)\right)}\right) \]

       15. *-commutative N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot 6\right)}\right)\right) \]

       16. lower-*.f64 100.0

           \[\leadsto x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot 6\right)}\right)\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_1
              (+
               (* (* (* x1 2.0) t_2) (- t_2 3.0))
               (* (* x1 x1) (- (* t_2 4.0) 6.0))))
             (* t_0 t_2))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))
        5e+186)
     (- (* x2 -6.0) x1)
     (+ x1 (* (* x1 x1) 9.0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+186) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x1 * (x1 * 3.0d0)
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0d0) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((t_2 * 4.0d0) - 6.0d0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))) <= 5d+186) then
        tmp = (x2 * (-6.0d0)) - x1
    else
        tmp = x1 + ((x1 * x1) * 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+186) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 3.0)
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	tmp = 0
	if (x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+186:
		tmp = (x2 * -6.0) - x1
	else:
		tmp = x1 + ((x1 * x1) * 9.0)
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= 5e+186)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 3.0);
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = 0.0;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+186)
		tmp = (x2 * -6.0) - x1;
	else
		tmp = x1 + ((x1 * x1) * 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+186], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999954e186

   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

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 79.1%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 - 2}, x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)}, x2 \cdot -6\right) \]

      2. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, 9 \cdot x1 + \color{blue}{-2}, x2 \cdot -6\right) \]

      3. lower-fma.f64 75.6

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

   8. Applied rewrites 75.6%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto -6 \cdot x2 + \color{blue}{\left(\mathsf{neg}\left(x1\right)\right)} \]

       2. unsub-neg N/A

          \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]

       3. lower--.f64 N/A

          \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{x2 \cdot -6} - x1 \]

       5. lower-*.f64 74.7

          \[\leadsto \color{blue}{x2 \cdot -6} - x1 \]

   11. Applied rewrites 74.7%

       \[\leadsto \color{blue}{x2 \cdot -6 - x1} \]

   if 4.99999999999999954e186 < (+.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 37.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 60.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 - 2}, x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)}, x2 \cdot -6\right) \]

      2. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, 9 \cdot x1 + \color{blue}{-2}, x2 \cdot -6\right) \]

      3. lower-fma.f64 55.4

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

   8. Applied rewrites 55.4%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

   9. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{9 \cdot {x1}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto x1 + \color{blue}{{x1}^{2} \cdot 9} \]

       2. lower-*.f64 N/A

          \[\leadsto x1 + \color{blue}{{x1}^{2} \cdot 9} \]

       3. unpow2 N/A

          \[\leadsto x1 + \color{blue}{\left(x1 \cdot x1\right)} \cdot 9 \]

       4. lower-*.f64 55.7

          \[\leadsto x1 + \color{blue}{\left(x1 \cdot x1\right)} \cdot 9 \]

   11. Applied rewrites 55.7%

       \[\leadsto x1 + \color{blue}{\left(x1 \cdot x1\right) \cdot 9} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_1
              (+
               (* (* (* x1 2.0) t_2) (- t_2 3.0))
               (* (* x1 x1) (- (* t_2 4.0) 6.0))))
             (* t_0 t_2))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))
        5e+186)
     (- (* x2 -6.0) x1)
     (* (* x1 x1) 9.0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+186) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x1 * (x1 * 3.0d0)
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0d0) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((t_2 * 4.0d0) - 6.0d0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))) <= 5d+186) then
        tmp = (x2 * (-6.0d0)) - x1
    else
        tmp = (x1 * x1) * 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+186) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 3.0)
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	tmp = 0
	if (x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+186:
		tmp = (x2 * -6.0) - x1
	else:
		tmp = (x1 * x1) * 9.0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= 5e+186)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = Float64(Float64(x1 * x1) * 9.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 3.0);
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = 0.0;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 5e+186)
		tmp = (x2 * -6.0) - x1;
	else
		tmp = (x1 * x1) * 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+186], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.99999999999999954e186

   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

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 79.1%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 - 2}, x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)}, x2 \cdot -6\right) \]

      2. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, 9 \cdot x1 + \color{blue}{-2}, x2 \cdot -6\right) \]

      3. lower-fma.f64 75.6

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

   8. Applied rewrites 75.6%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

   9. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto -6 \cdot x2 + \color{blue}{\left(\mathsf{neg}\left(x1\right)\right)} \]

       2. unsub-neg N/A

          \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]

       3. lower--.f64 N/A

          \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{x2 \cdot -6} - x1 \]

       5. lower-*.f64 74.7

          \[\leadsto \color{blue}{x2 \cdot -6} - x1 \]

   11. Applied rewrites 74.7%

       \[\leadsto \color{blue}{x2 \cdot -6 - x1} \]

   if 4.99999999999999954e186 < (+.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 37.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 60.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 - 2}, x2 \cdot -6\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)}, x2 \cdot -6\right) \]

      2. metadata-eval N/A

         \[\leadsto x1 + \mathsf{fma}\left(x1, 9 \cdot x1 + \color{blue}{-2}, x2 \cdot -6\right) \]

      3. lower-fma.f64 55.4

         \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

   8. Applied rewrites 55.4%

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x1}^{2} \cdot 9} \]

       3. unpow2 N/A

          \[\leadsto \color{blue}{\left(x1 \cdot x1\right)} \cdot 9 \]

       4. lower-*.f64 55.7

          \[\leadsto \color{blue}{\left(x1 \cdot x1\right)} \cdot 9 \]

   11. Applied rewrites 55.7%

       \[\leadsto \color{blue}{\left(x1 \cdot x1\right) \cdot 9} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), \frac{4}{x1 \cdot x1}, \frac{9}{x1 \cdot x1}\right) + \left(6 + \frac{-3}{x1}\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -31000.0)
   (+
    x1
    (*
     (pow x1 4.0)
     (+
      (fma (fma x2 2.0 -3.0) (/ 4.0 (* x1 x1)) (/ 9.0 (* x1 x1)))
      (+ 6.0 (/ -3.0 x1)))))
   (if (<= x1 4.8e+21)
     (+
      x1
      (fma
       x2
       (fma x2 (* x1 8.0) (fma x1 (fma 12.0 x1 -12.0) -6.0))
       (* x1 (fma 9.0 x1 -2.0))))
     (+
      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 tmp;
	if (x1 <= -31000.0) {
		tmp = x1 + (pow(x1, 4.0) * (fma(fma(x2, 2.0, -3.0), (4.0 / (x1 * x1)), (9.0 / (x1 * x1))) + (6.0 + (-3.0 / x1))));
	} else if (x1 <= 4.8e+21) {
		tmp = x1 + fma(x2, fma(x2, (x1 * 8.0), fma(x1, fma(12.0, x1, -12.0), -6.0)), (x1 * fma(9.0, x1, -2.0)));
	} 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)
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(fma(fma(x2, 2.0, -3.0), Float64(4.0 / Float64(x1 * x1)), Float64(9.0 / Float64(x1 * x1))) + Float64(6.0 + Float64(-3.0 / x1)))));
	elseif (x1 <= 4.8e+21)
		tmp = Float64(x1 + fma(x2, fma(x2, Float64(x1 * 8.0), fma(x1, fma(12.0, x1, -12.0), -6.0)), Float64(x1 * fma(9.0, x1, -2.0))));
	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_] := If[LessEqual[x1, -31000.0], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(N[(N[(x2 * 2.0 + -3.0), $MachinePrecision] * N[(4.0 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(9.0 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e+21], N[(x1 + N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + N[(x1 * N[(12.0 * x1 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $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 3 regimes

2. if x1 < -31000

   1. Initial program 37.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto x1 + \color{blue}{{x1}^{4}} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right) \]

      3. +-commutative N/A

         \[\leadsto x1 + {x1}^{4} \cdot \left(\color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + 6\right)} - 3 \cdot \frac{1}{x1}\right) \]

      4. associate--l+ N/A

         \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + \left(6 - 3 \cdot \frac{1}{x1}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + \left(6 - 3 \cdot \frac{1}{x1}\right)\right)} \]

   5. Applied rewrites 93.6%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), \frac{4}{x1 \cdot x1}, \frac{9}{x1 \cdot x1}\right) + \left(6 + \frac{-3}{x1}\right)\right)} \]

   if -31000 < x1 < 4.8e21

   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

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(9 \cdot x1 - 2\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right) + x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6, x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      3. associate--l+ N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{8 \cdot \left(x1 \cdot x2\right) + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x1 \cdot x2\right) \cdot 8} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x2 \cdot x1\right)} \cdot 8 + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{x2 \cdot \left(x1 \cdot 8\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, x2 \cdot \color{blue}{\left(8 \cdot x1\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, \color{blue}{8 \cdot x1}, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      10. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{x1 \cdot \left(12 \cdot x1 - 12\right) + \left(\mathsf{neg}\left(6\right)\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) + \color{blue}{-6}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{\mathsf{fma}\left(x1, 12 \cdot x1 - 12, -6\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      13. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{12 \cdot x1 + \left(\mathsf{neg}\left(12\right)\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, 12 \cdot x1 + \color{blue}{-12}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(12, x1, -12\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \left(9 \cdot x1 + \color{blue}{-2}\right)\right) \]

      19. lower-fma.f64 97.5

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}\right) \]

   8. Applied rewrites 97.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)} \]

   if 4.8e21 < x1

   1. Initial program 41.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 95.6%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1}}{x1}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), \frac{4}{x1 \cdot x1}, \frac{9}{x1 \cdot x1}\right) + \left(6 + \frac{-3}{x1}\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 95.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -31000.0)
     t_0
     (if (<= x1 4.8e+21)
       (+
        x1
        (fma
         x2
         (fma x2 (* x1 8.0) (fma x1 (fma 12.0 x1 -12.0) -6.0))
         (* x1 (fma 9.0 x1 -2.0))))
       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 <= -31000.0) {
		tmp = t_0;
	} else if (x1 <= 4.8e+21) {
		tmp = x1 + fma(x2, fma(x2, (x1 * 8.0), fma(x1, fma(12.0, x1, -12.0), -6.0)), (x1 * fma(9.0, x1, -2.0)));
	} 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 <= -31000.0)
		tmp = t_0;
	elseif (x1 <= 4.8e+21)
		tmp = Float64(x1 + fma(x2, fma(x2, Float64(x1 * 8.0), fma(x1, fma(12.0, x1, -12.0), -6.0)), Float64(x1 * fma(9.0, x1, -2.0))));
	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, -31000.0], t$95$0, If[LessEqual[x1, 4.8e+21], N[(x1 + N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + N[(x1 * N[(12.0 * x1 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -31000 or 4.8e21 < x1

   1. Initial program 39.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   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 94.6%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1}}{x1}\right)} \]

   if -31000 < x1 < 4.8e21

   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

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(9 \cdot x1 - 2\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right) + x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6, x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      3. associate--l+ N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{8 \cdot \left(x1 \cdot x2\right) + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x1 \cdot x2\right) \cdot 8} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x2 \cdot x1\right)} \cdot 8 + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{x2 \cdot \left(x1 \cdot 8\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, x2 \cdot \color{blue}{\left(8 \cdot x1\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, \color{blue}{8 \cdot x1}, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      10. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{x1 \cdot \left(12 \cdot x1 - 12\right) + \left(\mathsf{neg}\left(6\right)\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) + \color{blue}{-6}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{\mathsf{fma}\left(x1, 12 \cdot x1 - 12, -6\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      13. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{12 \cdot x1 + \left(\mathsf{neg}\left(12\right)\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, 12 \cdot x1 + \color{blue}{-12}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(12, x1, -12\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \left(9 \cdot x1 + \color{blue}{-2}\right)\right) \]

      19. lower-fma.f64 97.5

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}\right) \]

   8. Applied rewrites 97.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 95.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), x1\right)\\ \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (fma
          (* x1 x1)
          (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
          x1)))
   (if (<= x1 -31000.0)
     t_0
     (if (<= x1 4.8e+21)
       (+
        x1
        (fma
         x2
         (fma x2 (* x1 8.0) (fma x1 (fma 12.0 x1 -12.0) -6.0))
         (* x1 (fma 9.0 x1 -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), x1);
	double tmp;
	if (x1 <= -31000.0) {
		tmp = t_0;
	} else if (x1 <= 4.8e+21) {
		tmp = x1 + fma(x2, fma(x2, (x1 * 8.0), fma(x1, fma(12.0, x1, -12.0), -6.0)), (x1 * fma(9.0, x1, -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * x1), fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), x1)
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = t_0;
	elseif (x1 <= 4.8e+21)
		tmp = Float64(x1 + fma(x2, fma(x2, Float64(x1 * 8.0), fma(x1, fma(12.0, x1, -12.0), -6.0)), Float64(x1 * fma(9.0, x1, -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -31000.0], t$95$0, If[LessEqual[x1, 4.8e+21], N[(x1 + N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + N[(x1 * N[(12.0 * x1 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -31000 or 4.8e21 < x1

   1. Initial program 39.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto x1 + \color{blue}{{x1}^{4}} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right) \]

      3. +-commutative N/A

         \[\leadsto x1 + {x1}^{4} \cdot \left(\color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + 6\right)} - 3 \cdot \frac{1}{x1}\right) \]

      4. associate--l+ N/A

         \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + \left(6 - 3 \cdot \frac{1}{x1}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + \left(6 - 3 \cdot \frac{1}{x1}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), \frac{4}{x1 \cdot x1}, \frac{9}{x1 \cdot x1}\right) + \left(6 + \frac{-3}{x1}\right)\right)} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{x1 \cdot \left(1 + x1 \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right)\right) + x1 \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x1 \cdot x1\right) \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right)} + x1 \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \color{blue}{{x1}^{2}} \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right) + x1 \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto {x1}^{2} \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right) + \color{blue}{x1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x1}^{2}, 9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right), x1\right)} \]

   8. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), x1\right)} \]

   if -31000 < x1 < 4.8e21

   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

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(9 \cdot x1 - 2\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right) + x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6, x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      3. associate--l+ N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{8 \cdot \left(x1 \cdot x2\right) + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x1 \cdot x2\right) \cdot 8} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x2 \cdot x1\right)} \cdot 8 + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{x2 \cdot \left(x1 \cdot 8\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, x2 \cdot \color{blue}{\left(8 \cdot x1\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, \color{blue}{8 \cdot x1}, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      10. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{x1 \cdot \left(12 \cdot x1 - 12\right) + \left(\mathsf{neg}\left(6\right)\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) + \color{blue}{-6}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{\mathsf{fma}\left(x1, 12 \cdot x1 - 12, -6\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      13. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{12 \cdot x1 + \left(\mathsf{neg}\left(12\right)\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, 12 \cdot x1 + \color{blue}{-12}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(12, x1, -12\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \left(9 \cdot x1 + \color{blue}{-2}\right)\right) \]

      19. lower-fma.f64 97.5

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}\right) \]

   8. Applied rewrites 97.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), x1\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 95.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), x1\right)\\ \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (fma
          (* x1 x1)
          (fma x1 (fma x1 6.0 -3.0) (fma 4.0 (fma x2 2.0 -3.0) 9.0))
          x1)))
   (if (<= x1 -31000.0)
     t_0
     (if (<= x1 4.8e+21)
       (+ x1 (fma x2 (fma x2 (* x1 8.0) -6.0) (* x1 (fma 9.0 x1 -2.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), x1);
	double tmp;
	if (x1 <= -31000.0) {
		tmp = t_0;
	} else if (x1 <= 4.8e+21) {
		tmp = x1 + fma(x2, fma(x2, (x1 * 8.0), -6.0), (x1 * fma(9.0, x1, -2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(x1 * x1), fma(x1, fma(x1, 6.0, -3.0), fma(4.0, fma(x2, 2.0, -3.0), 9.0)), x1)
	tmp = 0.0
	if (x1 <= -31000.0)
		tmp = t_0;
	elseif (x1 <= 4.8e+21)
		tmp = Float64(x1 + fma(x2, fma(x2, Float64(x1 * 8.0), -6.0), Float64(x1 * fma(9.0, x1, -2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(x1 * 6.0 + -3.0), $MachinePrecision] + N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -31000.0], t$95$0, If[LessEqual[x1, 4.8e+21], N[(x1 + N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -31000 or 4.8e21 < x1

   1. Initial program 39.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto x1 + \color{blue}{{x1}^{4}} \cdot \left(\left(6 + \left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right)\right) - 3 \cdot \frac{1}{x1}\right) \]

      3. +-commutative N/A

         \[\leadsto x1 + {x1}^{4} \cdot \left(\color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + 6\right)} - 3 \cdot \frac{1}{x1}\right) \]

      4. associate--l+ N/A

         \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + \left(6 - 3 \cdot \frac{1}{x1}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto x1 + {x1}^{4} \cdot \color{blue}{\left(\left(4 \cdot \frac{2 \cdot x2 - 3}{{x1}^{2}} + \frac{9}{{x1}^{2}}\right) + \left(6 - 3 \cdot \frac{1}{x1}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x2, 2, -3\right), \frac{4}{x1 \cdot x1}, \frac{9}{x1 \cdot x1}\right) + \left(6 + \frac{-3}{x1}\right)\right)} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{x1 \cdot \left(1 + x1 \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right)\right) + x1 \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x1 \cdot x1\right) \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right)} + x1 \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \color{blue}{{x1}^{2}} \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right) + x1 \cdot 1 \]

      5. *-rgt-identity N/A

         \[\leadsto {x1}^{2} \cdot \left(9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right)\right) + \color{blue}{x1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x1}^{2}, 9 + \left(4 \cdot \left(2 \cdot x2 - 3\right) + x1 \cdot \left(6 \cdot x1 - 3\right)\right), x1\right)} \]

   8. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), x1\right)} \]

   if -31000 < x1 < 4.8e21

   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

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(9 \cdot x1 - 2\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right) + x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6, x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      3. associate--l+ N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{8 \cdot \left(x1 \cdot x2\right) + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x1 \cdot x2\right) \cdot 8} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x2 \cdot x1\right)} \cdot 8 + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{x2 \cdot \left(x1 \cdot 8\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, x2 \cdot \color{blue}{\left(8 \cdot x1\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, \color{blue}{8 \cdot x1}, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      10. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{x1 \cdot \left(12 \cdot x1 - 12\right) + \left(\mathsf{neg}\left(6\right)\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) + \color{blue}{-6}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{\mathsf{fma}\left(x1, 12 \cdot x1 - 12, -6\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      13. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{12 \cdot x1 + \left(\mathsf{neg}\left(12\right)\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, 12 \cdot x1 + \color{blue}{-12}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(12, x1, -12\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \left(9 \cdot x1 + \color{blue}{-2}\right)\right) \]

      19. lower-fma.f64 97.5

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}\right) \]

   8. Applied rewrites 97.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)} \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{-6}\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 97.5%

       \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{-6}\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -31000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), x1\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x1, 6, -3\right), \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 93.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -108000:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right), 6, x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -108000.0)
   (* x1 (* x1 (* x1 (* x1 6.0))))
   (if (<= x1 4.8e+21)
     (+ x1 (fma x2 (fma x2 (* x1 8.0) -6.0) (* x1 (fma 9.0 x1 -2.0))))
     (fma (* x1 (* x1 (* x1 x1))) 6.0 x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -108000.0) {
		tmp = x1 * (x1 * (x1 * (x1 * 6.0)));
	} else if (x1 <= 4.8e+21) {
		tmp = x1 + fma(x2, fma(x2, (x1 * 8.0), -6.0), (x1 * fma(9.0, x1, -2.0)));
	} else {
		tmp = fma((x1 * (x1 * (x1 * x1))), 6.0, x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -108000.0)
		tmp = Float64(x1 * Float64(x1 * Float64(x1 * Float64(x1 * 6.0))));
	elseif (x1 <= 4.8e+21)
		tmp = Float64(x1 + fma(x2, fma(x2, Float64(x1 * 8.0), -6.0), Float64(x1 * fma(9.0, x1, -2.0))));
	else
		tmp = fma(Float64(x1 * Float64(x1 * Float64(x1 * x1))), 6.0, x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -108000.0], N[(x1 * N[(x1 * N[(x1 * N[(x1 * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e+21], N[(x1 + N[(x2 * N[(x2 * N[(x1 * 8.0), $MachinePrecision] + -6.0), $MachinePrecision] + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x1 * N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -108000

   1. Initial program 37.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]
   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. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{x1 \cdot \left(1 + 6 \cdot {x1}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \color{blue}{\left(6 \cdot {x1}^{3} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(6 \cdot {x1}^{3}\right) \cdot x1 + 1 \cdot x1} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({x1}^{3} \cdot 6\right)} \cdot x1 + 1 \cdot x1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x1}^{3} \cdot \left(6 \cdot x1\right)} + 1 \cdot x1 \]

      5. *-lft-identity N/A

         \[\leadsto {x1}^{3} \cdot \left(6 \cdot x1\right) + \color{blue}{x1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x1}^{3}, 6 \cdot x1, x1\right)} \]

      7. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{{x1}^{2}}, 6 \cdot x1, x1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot {x1}^{2}}, 6 \cdot x1, x1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), \color{blue}{x1 \cdot 6}, x1\right) \]

      13. lower-*.f64 89.0

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), \color{blue}{x1 \cdot 6}, x1\right) \]

   8. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)} \]

   9. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto 6 \cdot {x1}^{\color{blue}{\left(3 + 1\right)}} \]

       2. pow-plus N/A

          \[\leadsto 6 \cdot \color{blue}{\left({x1}^{3} \cdot x1\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(6 \cdot {x1}^{3}\right) \cdot x1} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{x1 \cdot \left(6 \cdot {x1}^{3}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{x1 \cdot \left(6 \cdot {x1}^{3}\right)} \]

       6. *-commutative N/A

          \[\leadsto x1 \cdot \color{blue}{\left({x1}^{3} \cdot 6\right)} \]

       7. cube-mult N/A

          \[\leadsto x1 \cdot \left(\color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)} \cdot 6\right) \]

       8. unpow2 N/A

          \[\leadsto x1 \cdot \left(\left(x1 \cdot \color{blue}{{x1}^{2}}\right) \cdot 6\right) \]

       9. associate-*l* N/A

          \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left({x1}^{2} \cdot 6\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto x1 \cdot \color{blue}{\left(x1 \cdot \left({x1}^{2} \cdot 6\right)\right)} \]

       11. unpow2 N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot 6\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot 6\right)\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(6 \cdot x1\right)}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot \left(6 \cdot x1\right)\right)}\right) \]

       15. *-commutative N/A

           \[\leadsto x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot 6\right)}\right)\right) \]

       16. lower-*.f64 89.1

           \[\leadsto x1 \cdot \left(x1 \cdot \left(x1 \cdot \color{blue}{\left(x1 \cdot 6\right)}\right)\right) \]

   11. Applied rewrites 89.1%

       \[\leadsto \color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)} \]

   if -108000 < x1 < 4.8e21

   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

      \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

   5. Applied rewrites 82.7%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \color{blue}{\left(x1 \cdot \left(9 \cdot x1 - 2\right) + x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 + \color{blue}{\left(x2 \cdot \left(\left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6\right) + x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \left(8 \cdot \left(x1 \cdot x2\right) + x1 \cdot \left(12 \cdot x1 - 12\right)\right) - 6, x1 \cdot \left(9 \cdot x1 - 2\right)\right)} \]

      3. associate--l+ N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{8 \cdot \left(x1 \cdot x2\right) + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x1 \cdot x2\right) \cdot 8} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\left(x2 \cdot x1\right)} \cdot 8 + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{x2 \cdot \left(x1 \cdot 8\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, x2 \cdot \color{blue}{\left(8 \cdot x1\right)} + \left(x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \color{blue}{\mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right)}, x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, \color{blue}{8 \cdot x1}, x1 \cdot \left(12 \cdot x1 - 12\right) - 6\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      10. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{x1 \cdot \left(12 \cdot x1 - 12\right) + \left(\mathsf{neg}\left(6\right)\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, x1 \cdot \left(12 \cdot x1 - 12\right) + \color{blue}{-6}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{\mathsf{fma}\left(x1, 12 \cdot x1 - 12, -6\right)}\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      13. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{12 \cdot x1 + \left(\mathsf{neg}\left(12\right)\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, 12 \cdot x1 + \color{blue}{-12}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(12, x1, -12\right)}, -6\right)\right), x1 \cdot \left(9 \cdot x1 - 2\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), \color{blue}{x1 \cdot \left(9 \cdot x1 - 2\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\left(9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      18. metadata-eval N/A

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \left(9 \cdot x1 + \color{blue}{-2}\right)\right) \]

      19. lower-fma.f64 97.5

          \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}\right) \]

   8. Applied rewrites 97.5%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(12, x1, -12\right), -6\right)\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)} \]

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{-6}\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 97.5%

       \[\leadsto x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8 \cdot x1, \color{blue}{-6}\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right) \]

   if 4.8e21 < x1

   1. Initial program 41.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   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 94.8

         \[\leadsto x1 + 6 \cdot \color{blue}{{x1}^{4}} \]

   5. Applied rewrites 94.8%

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]

   6. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{x1 \cdot \left(1 + 6 \cdot {x1}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x1 \cdot \color{blue}{\left(6 \cdot {x1}^{3} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(6 \cdot {x1}^{3}\right) \cdot x1 + 1 \cdot x1} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({x1}^{3} \cdot 6\right)} \cdot x1 + 1 \cdot x1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{{x1}^{3} \cdot \left(6 \cdot x1\right)} + 1 \cdot x1 \]

      5. *-lft-identity N/A

         \[\leadsto {x1}^{3} \cdot \left(6 \cdot x1\right) + \color{blue}{x1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x1}^{3}, 6 \cdot x1, x1\right)} \]

      7. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{{x1}^{2}}, 6 \cdot x1, x1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot {x1}^{2}}, 6 \cdot x1, x1\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}, 6 \cdot x1, x1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), \color{blue}{x1 \cdot 6}, x1\right) \]

      13. lower-*.f64 94.7

          \[\leadsto \mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), \color{blue}{x1 \cdot 6}, x1\right) \]

   8. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot x1\right), x1 \cdot 6, x1\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot \color{blue}{\left(x1 \cdot x1\right)}\right) \cdot \left(x1 \cdot 6\right) + x1 \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)} \cdot \left(x1 \cdot 6\right) + x1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \color{blue}{\left(x1 \cdot 6\right)} + x1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \color{blue}{\left(x1 \cdot 6\right)} + x1 \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot x1\right) \cdot 6} + x1 \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)\right)} \cdot 6 + x1 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)}\right) \cdot 6 + x1 \]

      8. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \cdot 6 + x1 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x1 \cdot x1\right)} \cdot \left(x1 \cdot x1\right)\right) \cdot 6 + x1 \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right), 6, x1\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x1 \cdot x1\right)} \cdot \left(x1 \cdot x1\right), 6, x1\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1 \cdot \color{blue}{\left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

      14. lower-*.f64 94.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right)}, 6, x1\right) \]

   10. Applied rewrites 94.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right), 6, x1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -108000:\\ \;\;\;\;x1 \cdot \left(x1 \cdot \left(x1 \cdot \left(x1 \cdot 6\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{+21}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, x1 \cdot 8, -6\right), x1 \cdot \mathsf{fma}\left(9, x1, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right), 6, x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.7% accurate, 33.1× speedup

Math

\[\begin{array}{l} \\ x2 \cdot -6 - x1 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (- (* x2 -6.0) x1))

C

double code(double x1, double x2) {
	return (x2 * -6.0) - x1;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = (x2 * (-6.0d0)) - x1
end function

Java

public static double code(double x1, double x2) {
	return (x2 * -6.0) - x1;
}

Python

def code(x1, x2):
	return (x2 * -6.0) - x1

Julia

function code(x1, x2)
	return Float64(Float64(x2 * -6.0) - x1)
end

MATLAB

function tmp = code(x1, x2)
	tmp = (x2 * -6.0) - x1;
end

Wolfram

code[x1_, x2_] := N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

5. Applied rewrites 70.1%

   \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

6. Taylor expanded in x2 around 0

   \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 - 2}, x2 \cdot -6\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)}, x2 \cdot -6\right) \]

   2. metadata-eval N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, 9 \cdot x1 + \color{blue}{-2}, x2 \cdot -6\right) \]

   3. lower-fma.f64 65.9

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

8. Applied rewrites 65.9%

   \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

9. Taylor expanded in x1 around 0

   \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto -6 \cdot x2 + \color{blue}{\left(\mathsf{neg}\left(x1\right)\right)} \]

    2. unsub-neg N/A

       \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]

    3. lower--.f64 N/A

       \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]

    4. *-commutative N/A

       \[\leadsto \color{blue}{x2 \cdot -6} - x1 \]

    5. lower-*.f64 40.4

       \[\leadsto \color{blue}{x2 \cdot -6} - x1 \]

11. Applied rewrites 40.4%

    \[\leadsto \color{blue}{x2 \cdot -6 - x1} \]
12. Add Preprocessing

Alternative 19: 25.9% 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 69.5%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   2. lower-*.f64 26.7

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Applied rewrites 26.7%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
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 26.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1\right)} \]

7. Applied rewrites 26.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1\right)} \]
8. Add Preprocessing

Alternative 20: 25.7% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ x2 \cdot -6 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (* x2 -6.0))

C

double code(double x1, double x2) {
	return x2 * -6.0;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x2 * (-6.0d0)
end function

Java

public static double code(double x1, double x2) {
	return x2 * -6.0;
}

Python

def code(x1, x2):
	return x2 * -6.0

Julia

function code(x1, x2)
	return Float64(x2 * -6.0)
end

MATLAB

function tmp = code(x1, x2)
	tmp = x2 * -6.0;
end

Wolfram

code[x1_, x2_] := N[(x2 * -6.0), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   2. lower-*.f64 26.7

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Applied rewrites 26.7%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

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 26.5

      \[\leadsto \color{blue}{x2 \cdot -6} \]

8. Applied rewrites 26.5%

   \[\leadsto \color{blue}{x2 \cdot -6} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))))