Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.7% → 99.6%

Time: 23.0s

Alternatives: 23

Speedup: 9.3×

• 46.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(3, x1, -1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \frac{\mathsf{fma}\left(2, x2, t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_4 := \left(x2 \cdot 2 + t\_1\right) - x1\\ t_5 := x1 \cdot x1 - -1\\ t_6 := \frac{t\_4}{t\_5}\\ \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{t\_4}{t\_2} \cdot t\_1 - t\_2 \cdot \left(\left(3 - t\_6\right) \cdot \left(t\_6 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_6 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_1 - x2 \cdot 2\right) - x1}{t\_5} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right) + \left(t\_3 \cdot t\_1 + \left(\mathsf{fma}\left(\left(t\_3 \cdot x1\right) \cdot \left(t\_3 - 3\right), 2, \mathsf{fma}\left(t\_3, 4, -6\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \mathsf{fma}\left(x1, x1, 1\right) + x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (fma 3.0 x1 -1.0) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (/ (fma 2.0 x2 t_0) (fma x1 x1 1.0)))
        (t_4 (- (+ (* x2 2.0) t_1) x1))
        (t_5 (- (* x1 x1) -1.0))
        (t_6 (/ t_4 t_5)))
   (if (<=
        (-
         x1
         (-
          (-
           (-
            (-
             (* (/ t_4 t_2) t_1)
             (*
              t_2
              (-
               (* (- 3.0 t_6) (* t_6 (* 2.0 x1)))
               (* (- (* 4.0 t_6) 6.0) (* x1 x1)))))
            (* (* x1 x1) x1))
           x1)
          (* (/ (- (- t_1 (* x2 2.0)) x1) t_5) 3.0)))
        INFINITY)
     (+
      (fma (* x1 x1) x1 (fma (/ (fma -2.0 x2 t_0) (fma x1 x1 1.0)) 3.0 x1))
      (+
       (* t_3 t_1)
       (+
        (*
         (fma (* (* t_3 x1) (- t_3 3.0)) 2.0 (* (fma t_3 4.0 -6.0) (* x1 x1)))
         (fma x1 x1 1.0))
        x1)))
     (*
      (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0)
      (* x1 x1)))))

C

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

Julia

function code(x1, x2)
	t_0 = Float64(fma(3.0, x1, -1.0) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(fma(2.0, x2, t_0) / fma(x1, x1, 1.0))
	t_4 = Float64(Float64(Float64(x2 * 2.0) + t_1) - x1)
	t_5 = Float64(Float64(x1 * x1) - -1.0)
	t_6 = Float64(t_4 / t_5)
	tmp = 0.0
	if (Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 / t_2) * t_1) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_6) * Float64(t_6 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_6) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_1 - Float64(x2 * 2.0)) - x1) / t_5) * 3.0))) <= Inf)
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(Float64(fma(-2.0, x2, t_0) / fma(x1, x1, 1.0)), 3.0, x1)) + Float64(Float64(t_3 * t_1) + Float64(Float64(fma(Float64(Float64(t_3 * x1) * Float64(t_3 - 3.0)), 2.0, Float64(fma(t_3, 4.0, -6.0) * Float64(x1 * x1))) * fma(x1, x1, 1.0)) + x1)));
	else
		tmp = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * x2 + t$95$0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$1), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$5), $MachinePrecision]}, If[LessEqual[N[(x1 - N[(N[(N[(N[(N[(N[(t$95$4 / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$6), $MachinePrecision] * N[(t$95$6 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$6), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$1 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * t$95$1), $MachinePrecision] + N[(N[(N[(N[(N[(t$95$3 * x1), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(t$95$3 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 62.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{t\_3}{t\_4}\\ t_6 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_5\right) \cdot \left(t\_5 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_5 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\right)\\ \mathbf{if}\;t\_6 \leq -1 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_6 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (fma (* (* x2 x2) 8.0) x1 (* -6.0 x2)))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (+ (* x2 2.0) t_0) x1))
        (t_4 (- (* x1 x1) -1.0))
        (t_5 (/ t_3 t_4))
        (t_6
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_3 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_5) (* t_5 (* 2.0 x1)))
                (* (- (* 4.0 t_5) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_4) 3.0)))))
   (if (<= t_6 -1e+95)
     t_1
     (if (<= t_6 2e-5)
       (fma (fma -12.0 x1 -6.0) x2 (- x1))
       (if (<= t_6 INFINITY) t_1 (* (* (* x1 x1) x2) 8.0))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = fma(((x2 * x2) * 8.0), x1, (-6.0 * x2));
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = t_3 / t_4;
	double t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	double tmp;
	if (t_6 <= -1e+95) {
		tmp = t_1;
	} else if (t_6 <= 2e-5) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((x1 * x1) * x2) * 8.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = fma(Float64(Float64(x2 * x2) * 8.0), x1, Float64(-6.0 * x2))
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(t_3 / t_4)
	t_6 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_5) * Float64(t_5 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_5) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)))
	tmp = 0.0
	if (t_6 <= -1e+95)
		tmp = t_1;
	elseif (t_6 <= 2e-5)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	elseif (t_6 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(x1 * x1) * x2) * 8.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x2 * x2), $MachinePrecision] * 8.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$5), $MachinePrecision] * N[(t$95$5 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -1e+95], t$95$1, If[LessEqual[t$95$6, 2e-5], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], If[LessEqual[t$95$6, Infinity], t$95$1, N[(N[(N[(x1 * x1), $MachinePrecision] * x2), $MachinePrecision] * 8.0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.00000000000000002e95 or 2.00000000000000016e-5 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 19.6

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

   5. Applied rewrites 19.6%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 47.9

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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 47.8%

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

   if -1.00000000000000002e95 < (+.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)))))) < 2.00000000000000016e-5

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 97.1

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

   8. Applied rewrites 97.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 43.0%

       \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot \color{blue}{8} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{t\_3}{t\_4}\\ t_6 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_5\right) \cdot \left(t\_5 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_5 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\right)\\ \mathbf{if}\;t\_6 \leq -1 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (* (* (* x2 x2) x1) 8.0))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (+ (* x2 2.0) t_0) x1))
        (t_4 (- (* x1 x1) -1.0))
        (t_5 (/ t_3 t_4))
        (t_6
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_3 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_5) (* t_5 (* 2.0 x1)))
                (* (- (* 4.0 t_5) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_4) 3.0)))))
   (if (<= t_6 -1e+198)
     t_1
     (if (<= t_6 5e+257)
       (fma (fma -12.0 x1 -6.0) x2 (- x1))
       (if (<= t_6 INFINITY) t_1 (* (* (* x1 x1) x2) 8.0))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = ((x2 * x2) * x1) * 8.0;
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = t_3 / t_4;
	double t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	double tmp;
	if (t_6 <= -1e+198) {
		tmp = t_1;
	} else if (t_6 <= 5e+257) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((x1 * x1) * x2) * 8.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(Float64(x2 * x2) * x1) * 8.0)
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(t_3 / t_4)
	t_6 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_5) * Float64(t_5 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_5) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)))
	tmp = 0.0
	if (t_6 <= -1e+198)
		tmp = t_1;
	elseif (t_6 <= 5e+257)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	elseif (t_6 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(x1 * x1) * x2) * 8.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$5), $MachinePrecision] * N[(t$95$5 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -1e+198], t$95$1, If[LessEqual[t$95$6, 5e+257], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], If[LessEqual[t$95$6, Infinity], t$95$1, N[(N[(N[(x1 * x1), $MachinePrecision] * x2), $MachinePrecision] * 8.0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.00000000000000002e198 or 5.00000000000000028e257 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.9%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 49.9

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

   8. Applied rewrites 49.9%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 50.0%

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

   if -1.00000000000000002e198 < (+.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)))))) < 5.00000000000000028e257

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 74.1

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 72.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 43.0%

       \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot \color{blue}{8} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq -1 \cdot 10^{+198}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(3, x1, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\\ t_6 := \frac{t\_3}{t\_4}\\ t_7 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_6\right) \cdot \left(t\_6 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_6 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - t\_5\right)\\ \mathbf{if}\;t\_7 \leq 2:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + t\_5\right) + x1\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4, t\_1, -6\right), x1 \cdot x1, \left(t\_1 \cdot \left(2 \cdot x1\right)\right) \cdot \left(t\_1 - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), -6 \cdot x2\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(fma(x1, fma(3.0, x1, -1.0), Float64(x2 * 2.0)) / fma(x1, x1, 1.0))
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)
	t_6 = Float64(t_3 / t_4)
	t_7 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_6) * Float64(t_6 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_6) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - t_5))
	tmp = 0.0
	if (t_7 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + t_5) + x1);
	elseif (t_7 <= Inf)
		tmp = Float64(fma(fma(fma(4.0, t_1, -6.0), Float64(x1 * x1), Float64(Float64(t_1 * Float64(2.0 * x1)) * Float64(t_1 - 3.0))), fma(x1, x1, 1.0), Float64(-6.0 * x2)) + x1);
	else
		tmp = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * N[(3.0 * x1 + -1.0), $MachinePrecision] + N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$6), $MachinePrecision] * N[(t$95$6 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$6), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 2.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + t$95$5), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[t$95$7, Infinity], N[(N[(N[(N[(4.0 * t$95$1 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$1 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $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

   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 x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 96.7

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

   5. Applied rewrites 96.7%

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

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x1 around 0

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

      1. lower-*.f64 98.8

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

   9. Applied rewrites 98.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 98.3%

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

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \frac{\mathsf{fma}\left(x1, \mathsf{fma}\left(3, x1, -1\right), x2 \cdot 2\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x1 \cdot x1\right) \cdot x1\\ t_4 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_5 := x1 \cdot x1 - -1\\ t_6 := \frac{t\_4}{t\_5}\\ \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{t\_4}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_6\right) \cdot \left(t\_6 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_6 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - t\_3\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_5} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4, t\_1, -6\right), x1 \cdot x1, \left(t\_1 \cdot \left(2 \cdot x1\right)\right) \cdot \left(t\_1 - 3\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_0\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right) + \mathsf{fma}\left(t\_1, t\_0, t\_3\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (/ (fma x1 (fma 3.0 x1 -1.0) (* x2 2.0)) (fma x1 x1 1.0)))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (* (* x1 x1) x1))
        (t_4 (- (+ (* x2 2.0) t_0) x1))
        (t_5 (- (* x1 x1) -1.0))
        (t_6 (/ t_4 t_5)))
   (if (<=
        (-
         x1
         (-
          (-
           (-
            (-
             (* (/ t_4 t_2) t_0)
             (*
              t_2
              (-
               (* (- 3.0 t_6) (* t_6 (* 2.0 x1)))
               (* (- (* 4.0 t_6) 6.0) (* x1 x1)))))
            t_3)
           x1)
          (* (/ (- (- t_0 (* x2 2.0)) x1) t_5) 3.0)))
        INFINITY)
     (+
      (fma
       (fma (fma 4.0 t_1 -6.0) (* x1 x1) (* (* t_1 (* 2.0 x1)) (- t_1 3.0)))
       (fma x1 x1 1.0)
       (+
        (fma (/ (- (fma -2.0 x2 t_0) x1) (fma x1 x1 1.0)) 3.0 x1)
        (fma t_1 t_0 t_3)))
      x1)
     (*
      (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0)
      (* x1 x1)))))

C

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

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(fma(x1, fma(3.0, x1, -1.0), Float64(x2 * 2.0)) / fma(x1, x1, 1.0))
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(x1 * x1) * x1)
	t_4 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_5 = Float64(Float64(x1 * x1) - -1.0)
	t_6 = Float64(t_4 / t_5)
	tmp = 0.0
	if (Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_6) * Float64(t_6 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_6) - 6.0) * Float64(x1 * x1))))) - t_3) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_5) * 3.0))) <= Inf)
		tmp = Float64(fma(fma(fma(4.0, t_1, -6.0), Float64(x1 * x1), Float64(Float64(t_1 * Float64(2.0 * x1)) * Float64(t_1 - 3.0))), fma(x1, x1, 1.0), Float64(fma(Float64(Float64(fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, x1) + fma(t_1, t_0, t_3))) + x1);
	else
		tmp = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * N[(3.0 * x1 + -1.0), $MachinePrecision] + N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$5), $MachinePrecision]}, If[LessEqual[N[(x1 - N[(N[(N[(N[(N[(N[(t$95$4 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$6), $MachinePrecision] * N[(t$95$6 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$6), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(4.0 * t$95$1 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$1 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision] + N[(t$95$1 * t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 6: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(3, x1, -1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \frac{\mathsf{fma}\left(2, x2, t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}\\ t_4 := \left(x2 \cdot 2 + t\_1\right) - x1\\ t_5 := x1 \cdot x1 - -1\\ t_6 := \frac{t\_4}{t\_5}\\ \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{t\_4}{t\_2} \cdot t\_1 - t\_2 \cdot \left(\left(3 - t\_6\right) \cdot \left(t\_6 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_6 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_1 - x2 \cdot 2\right) - x1}{t\_5} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_3 \cdot x1\right) \cdot \left(t\_3 - 3\right), 2, \mathsf{fma}\left(t\_3, 4, -6\right) \cdot \left(x1 \cdot x1\right)\right), \mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(x1, \mathsf{fma}\left(t\_3, 3 \cdot x1, x1 \cdot x1\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, t\_0\right)}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\right) + x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (fma 3.0 x1 -1.0) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (/ (fma 2.0 x2 t_0) (fma x1 x1 1.0)))
        (t_4 (- (+ (* x2 2.0) t_1) x1))
        (t_5 (- (* x1 x1) -1.0))
        (t_6 (/ t_4 t_5)))
   (if (<=
        (-
         x1
         (-
          (-
           (-
            (-
             (* (/ t_4 t_2) t_1)
             (*
              t_2
              (-
               (* (- 3.0 t_6) (* t_6 (* 2.0 x1)))
               (* (- (* 4.0 t_6) 6.0) (* x1 x1)))))
            (* (* x1 x1) x1))
           x1)
          (* (/ (- (- t_1 (* x2 2.0)) x1) t_5) 3.0)))
        INFINITY)
     (fma
      (fma (* (* t_3 x1) (- t_3 3.0)) 2.0 (* (fma t_3 4.0 -6.0) (* x1 x1)))
      (fma x1 x1 1.0)
      (+
       (fma
        x1
        (fma t_3 (* 3.0 x1) (* x1 x1))
        (fma (/ (fma -2.0 x2 t_0) (fma x1 x1 1.0)) 3.0 x1))
       x1))
     (*
      (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0)
      (* x1 x1)))))

C

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

Julia

function code(x1, x2)
	t_0 = Float64(fma(3.0, x1, -1.0) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(fma(2.0, x2, t_0) / fma(x1, x1, 1.0))
	t_4 = Float64(Float64(Float64(x2 * 2.0) + t_1) - x1)
	t_5 = Float64(Float64(x1 * x1) - -1.0)
	t_6 = Float64(t_4 / t_5)
	tmp = 0.0
	if (Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 / t_2) * t_1) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_6) * Float64(t_6 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_6) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_1 - Float64(x2 * 2.0)) - x1) / t_5) * 3.0))) <= Inf)
		tmp = fma(fma(Float64(Float64(t_3 * x1) * Float64(t_3 - 3.0)), 2.0, Float64(fma(t_3, 4.0, -6.0) * Float64(x1 * x1))), fma(x1, x1, 1.0), Float64(fma(x1, fma(t_3, Float64(3.0 * x1), Float64(x1 * x1)), fma(Float64(fma(-2.0, x2, t_0) / fma(x1, x1, 1.0)), 3.0, x1)) + x1));
	else
		tmp = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * x2 + t$95$0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$1), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$5), $MachinePrecision]}, If[LessEqual[N[(x1 - N[(N[(N[(N[(N[(N[(t$95$4 / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$6), $MachinePrecision] * N[(t$95$6 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$6), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$1 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(t$95$3 * x1), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(t$95$3 * 4.0 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(x1 * N[(t$95$3 * N[(3.0 * x1), $MachinePrecision] + N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 7: 97.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 99.5

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 98.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 98.6%

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

Alternative 8: 69.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. lower-*.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 64.4

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

   8. Applied rewrites 64.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 72.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 43.0%

       \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot \color{blue}{8} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

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

Alternative 9: 53.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := -1 - x1 \cdot x1\\ t_2 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_3 := x1 \cdot x1 - -1\\ t_4 := \frac{t\_2}{t\_3}\\ \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{t\_2}{t\_1} \cdot t\_0 - t\_1 \cdot \left(\left(3 - t\_4\right) \cdot \left(t\_4 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_4 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_3} \cdot 3\right) \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (- -1.0 (* x1 x1)))
        (t_2 (- (+ (* x2 2.0) t_0) x1))
        (t_3 (- (* x1 x1) -1.0))
        (t_4 (/ t_2 t_3)))
   (if (<=
        (-
         x1
         (-
          (-
           (-
            (-
             (* (/ t_2 t_1) t_0)
             (*
              t_1
              (-
               (* (- 3.0 t_4) (* t_4 (* 2.0 x1)))
               (* (- (* 4.0 t_4) 6.0) (* x1 x1)))))
            (* (* x1 x1) x1))
           x1)
          (* (/ (- (- t_0 (* x2 2.0)) x1) t_3) 3.0)))
        5e+257)
     (fma (fma -12.0 x1 -6.0) x2 (- x1))
     (* (* (* x1 x1) x2) 8.0))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = -1.0 - (x1 * x1);
	double t_2 = ((x2 * 2.0) + t_0) - x1;
	double t_3 = (x1 * x1) - -1.0;
	double t_4 = t_2 / t_3;
	double tmp;
	if ((x1 - ((((((t_2 / t_1) * t_0) - (t_1 * (((3.0 - t_4) * (t_4 * (2.0 * x1))) - (((4.0 * t_4) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_3) * 3.0))) <= 5e+257) {
		tmp = fma(fma(-12.0, x1, -6.0), x2, -x1);
	} else {
		tmp = ((x1 * x1) * x2) * 8.0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(-1.0 - Float64(x1 * x1))
	t_2 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_3 = Float64(Float64(x1 * x1) - -1.0)
	t_4 = Float64(t_2 / t_3)
	tmp = 0.0
	if (Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_2 / t_1) * t_0) - Float64(t_1 * Float64(Float64(Float64(3.0 - t_4) * Float64(t_4 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_4) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_3) * 3.0))) <= 5e+257)
		tmp = fma(fma(-12.0, x1, -6.0), x2, Float64(-x1));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * x2) * 8.0);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / t$95$3), $MachinePrecision]}, If[LessEqual[N[(x1 - N[(N[(N[(N[(N[(N[(t$95$2 / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$1 * N[(N[(N[(3.0 - t$95$4), $MachinePrecision] * N[(t$95$4 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$4), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+257], N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x2), $MachinePrecision] * 8.0), $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)))))) < 5.00000000000000028e257

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 40.9

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

   5. Applied rewrites 40.9%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 64.4%

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

   if 5.00000000000000028e257 < (+.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 49.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 27.9%

       \[\leadsto \left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot \color{blue}{8} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -750000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 29000000000000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -750000000.0)
   (* (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0) (* x1 x1))
   (if (<= x1 29000000000000.0)
     (+
      (+
       (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
       (* (/ (- (- (* (* 3.0 x1) x1) (* x2 2.0)) x1) (- (* x1 x1) -1.0)) 3.0))
      x1)
     (+
      (*
       (pow x1 4.0)
       (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1)))
      x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -750000000.0) {
		tmp = (fma(fma(6.0, x1, -3.0), x1, (fma(2.0, x2, -3.0) * 4.0)) + 9.0) * (x1 * x1);
	} else if (x1 <= 29000000000000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((((3.0 * x1) * x1) - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = (pow(x1, 4.0) * (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1))) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -750000000.0)
		tmp = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1));
	elseif (x1 <= 29000000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = Float64(Float64((x1 ^ 4.0) * Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1))) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -750000000.0], N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 29000000000000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.5e8

   1. Initial program 38.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 95.6%

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

   if -7.5e8 < x1 < 2.9e13

   1. Initial program 99.4%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 2.9e13 < x1

   1. Initial program 65.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -750000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 29000000000000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -750000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 29000000000000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -750000000.0)
   (* (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0) (* x1 x1))
   (if (<= x1 29000000000000.0)
     (+
      (+
       (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
       (* (/ (- (- (* (* 3.0 x1) x1) (* x2 2.0)) x1) (- (* x1 x1) -1.0)) 3.0))
      x1)
     (*
      (pow x1 4.0)
      (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -750000000.0) {
		tmp = (fma(fma(6.0, x1, -3.0), x1, (fma(2.0, x2, -3.0) * 4.0)) + 9.0) * (x1 * x1);
	} else if (x1 <= 29000000000000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((((3.0 * x1) * x1) - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = pow(x1, 4.0) * (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -750000000.0)
		tmp = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1));
	elseif (x1 <= 29000000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -750000000.0], N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 29000000000000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -7.5e8

   1. Initial program 38.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.6%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 95.6%

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

   if -7.5e8 < x1 < 2.9e13

   1. Initial program 99.4%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 2.9e13 < x1

   1. Initial program 65.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -750000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 29000000000000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -750000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 29000000000000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0)
          (* x1 x1))))
   (if (<= x1 -750000000.0)
     t_0
     (if (<= x1 29000000000000.0)
       (+
        (+
         (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
         (*
          (/ (- (- (* (* 3.0 x1) x1) (* x2 2.0)) x1) (- (* x1 x1) -1.0))
          3.0))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(6.0, x1, -3.0), x1, (fma(2.0, x2, -3.0) * 4.0)) + 9.0) * (x1 * x1);
	double tmp;
	if (x1 <= -750000000.0) {
		tmp = t_0;
	} else if (x1 <= 29000000000000.0) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((((3.0 * x1) * x1) - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -750000000.0)
		tmp = t_0;
	elseif (x1 <= 29000000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -750000000.0], t$95$0, If[LessEqual[x1, 29000000000000.0], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -7.5e8 or 2.9e13 < x1

   1. Initial program 50.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 96.7%

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

   if -7.5e8 < x1 < 2.9e13

   1. Initial program 99.4%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -750000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 29000000000000:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 6400000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(3 - -2 \cdot x2\right) + x1, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -12\right), x1, \left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\right) + x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0)
          (* x1 x1))))
   (if (<= x1 -600000000.0)
     t_0
     (if (<= x1 6400000000000.0)
       (+
        (* (fma (fma (+ (- 3.0 (* -2.0 x2)) x1) x1 -1.0) x1 (* -2.0 x2)) 3.0)
        (+ (+ (* (fma (fma 6.0 x1 -12.0) x1 (* (* x2 x1) 8.0)) x2) x1) x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(6.0, x1, -3.0), x1, (fma(2.0, x2, -3.0) * 4.0)) + 9.0) * (x1 * x1);
	double tmp;
	if (x1 <= -600000000.0) {
		tmp = t_0;
	} else if (x1 <= 6400000000000.0) {
		tmp = (fma(fma(((3.0 - (-2.0 * x2)) + x1), x1, -1.0), x1, (-2.0 * x2)) * 3.0) + (((fma(fma(6.0, x1, -12.0), x1, ((x2 * x1) * 8.0)) * x2) + x1) + x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -600000000.0)
		tmp = t_0;
	elseif (x1 <= 6400000000000.0)
		tmp = Float64(Float64(fma(fma(Float64(Float64(3.0 - Float64(-2.0 * x2)) + x1), x1, -1.0), x1, Float64(-2.0 * x2)) * 3.0) + Float64(Float64(Float64(fma(fma(6.0, x1, -12.0), x1, Float64(Float64(x2 * x1) * 8.0)) * x2) + x1) + x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -600000000.0], t$95$0, If[LessEqual[x1, 6400000000000.0], N[(N[(N[(N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(N[(N[(6.0 * x1 + -12.0), $MachinePrecision] * x1 + N[(N[(x2 * x1), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -6e8 or 6.4e12 < x1

   1. Initial program 50.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 96.7%

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

   if -6e8 < x1 < 6.4e12

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 83.7

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 94.6%

       \[\leadsto x1 + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -12\right), x1, \left(x2 \cdot x1\right) \cdot 8\right) \cdot \color{blue}{x2} + x1\right) + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x1 + \left(3 - -2 \cdot x2\right), x1, -1\right), x1, -2 \cdot x2\right)\right) \]

   13. Applied rewrites 94.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 6400000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(3 - -2 \cdot x2\right) + x1, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -12\right), x1, \left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\right) + x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 95.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 6400000000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(3 - -2 \cdot x2\right) + x1, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -12\right), x1, \left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0)
          (* x1 x1))))
   (if (<= x1 -600000000.0)
     t_0
     (if (<= x1 6400000000000.0)
       (+
        (+
         (* (fma (fma (+ (- 3.0 (* -2.0 x2)) x1) x1 -1.0) x1 (* -2.0 x2)) 3.0)
         (+ (* (fma (fma 6.0 x1 -12.0) x1 (* (* x2 x1) 8.0)) x2) x1))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(6.0, x1, -3.0), x1, (fma(2.0, x2, -3.0) * 4.0)) + 9.0) * (x1 * x1);
	double tmp;
	if (x1 <= -600000000.0) {
		tmp = t_0;
	} else if (x1 <= 6400000000000.0) {
		tmp = ((fma(fma(((3.0 - (-2.0 * x2)) + x1), x1, -1.0), x1, (-2.0 * x2)) * 3.0) + ((fma(fma(6.0, x1, -12.0), x1, ((x2 * x1) * 8.0)) * x2) + x1)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -600000000.0)
		tmp = t_0;
	elseif (x1 <= 6400000000000.0)
		tmp = Float64(Float64(Float64(fma(fma(Float64(Float64(3.0 - Float64(-2.0 * x2)) + x1), x1, -1.0), x1, Float64(-2.0 * x2)) * 3.0) + Float64(Float64(fma(fma(6.0, x1, -12.0), x1, Float64(Float64(x2 * x1) * 8.0)) * x2) + x1)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -600000000.0], t$95$0, If[LessEqual[x1, 6400000000000.0], N[(N[(N[(N[(N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] * x1 + -1.0), $MachinePrecision] * x1 + N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(N[(6.0 * x1 + -12.0), $MachinePrecision] * x1 + N[(N[(x2 * x1), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -6e8 or 6.4e12 < x1

   1. Initial program 50.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 96.7%

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

   if -6e8 < x1 < 6.4e12

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. cancel-sign-sub-inv N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 83.7

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 94.6%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 6400000000000:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(3 - -2 \cdot x2\right) + x1, x1, -1\right), x1, -2 \cdot x2\right) \cdot 3 + \left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -12\right), x1, \left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 95.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(2, x2, -3\right) \cdot 4\right) + 9\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, \mathsf{fma}\left(8 \cdot x1, x2, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (+ (fma (fma 6.0 x1 -3.0) x1 (* (fma 2.0 x2 -3.0) 4.0)) 9.0)
          (* x1 x1))))
   (if (<= x1 -600000000.0)
     t_0
     (if (<= x1 1.4)
       (fma (fma -12.0 x1 (fma (* 8.0 x1) x2 -6.0)) x2 (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(6.0, x1, -3.0), x1, (fma(2.0, x2, -3.0) * 4.0)) + 9.0) * (x1 * x1);
	double tmp;
	if (x1 <= -600000000.0) {
		tmp = t_0;
	} else if (x1 <= 1.4) {
		tmp = fma(fma(-12.0, x1, fma((8.0 * x1), x2, -6.0)), x2, -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(6.0, x1, -3.0), x1, Float64(fma(2.0, x2, -3.0) * 4.0)) + 9.0) * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -600000000.0)
		tmp = t_0;
	elseif (x1 <= 1.4)
		tmp = fma(fma(-12.0, x1, fma(Float64(8.0 * x1), x2, -6.0)), x2, Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + 9.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -600000000.0], t$95$0, If[LessEqual[x1, 1.4], N[(N[(-12.0 * x1 + N[(N[(8.0 * x1), $MachinePrecision] * x2 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -6e8 or 1.3999999999999999 < x1

   1. Initial program 52.2%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in x1 around 0

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

   11. Applied rewrites 94.6%

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

   if -6e8 < x1 < 1.3999999999999999

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 84.6

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

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 95.9%

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

Alternative 16: 93.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(6, x1, -4\right) \cdot x1, -6 \cdot x2\right) + x1\\ \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 6400000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, \mathsf{fma}\left(8 \cdot x1, x2, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+ (fma (fma x1 x1 1.0) (* (fma 6.0 x1 -4.0) x1) (* -6.0 x2)) x1)))
   (if (<= x1 -600000000.0)
     t_0
     (if (<= x1 6400000000000.0)
       (fma (fma -12.0 x1 (fma (* 8.0 x1) x2 -6.0)) x2 (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(x1, x1, 1.0), (fma(6.0, x1, -4.0) * x1), (-6.0 * x2)) + x1;
	double tmp;
	if (x1 <= -600000000.0) {
		tmp = t_0;
	} else if (x1 <= 6400000000000.0) {
		tmp = fma(fma(-12.0, x1, fma((8.0 * x1), x2, -6.0)), x2, -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(fma(x1, x1, 1.0), Float64(fma(6.0, x1, -4.0) * x1), Float64(-6.0 * x2)) + x1)
	tmp = 0.0
	if (x1 <= -600000000.0)
		tmp = t_0;
	elseif (x1 <= 6400000000000.0)
		tmp = fma(fma(-12.0, x1, fma(Float64(8.0 * x1), x2, -6.0)), x2, Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(6.0 * x1 + -4.0), $MachinePrecision] * x1), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -600000000.0], t$95$0, If[LessEqual[x1, 6400000000000.0], N[(N[(-12.0 * x1 + N[(N[(8.0 * x1), $MachinePrecision] * x2 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -6e8 or 6.4e12 < x1

   1. Initial program 50.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 44.6%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-*.f64 92.4

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

   10. Applied rewrites 92.4%

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

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 92.4%

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

   if -6e8 < x1 < 6.4e12

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 93.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(6, x1, -4\right) \cdot x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 6400000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, \mathsf{fma}\left(8 \cdot x1, x2, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \mathsf{fma}\left(6, x1, -4\right) \cdot x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 93.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(6 \cdot x1\right) \cdot x1, -6 \cdot x2\right) + x1\\ \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 6400000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, \mathsf{fma}\left(8 \cdot x1, x2, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (+ (fma (fma x1 x1 1.0) (* (* 6.0 x1) x1) (* -6.0 x2)) x1)))
   (if (<= x1 -600000000.0)
     t_0
     (if (<= x1 6400000000000.0)
       (fma (fma -12.0 x1 (fma (* 8.0 x1) x2 -6.0)) x2 (- x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(x1, x1, 1.0), ((6.0 * x1) * x1), (-6.0 * x2)) + x1;
	double tmp;
	if (x1 <= -600000000.0) {
		tmp = t_0;
	} else if (x1 <= 6400000000000.0) {
		tmp = fma(fma(-12.0, x1, fma((8.0 * x1), x2, -6.0)), x2, -x1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(fma(x1, x1, 1.0), Float64(Float64(6.0 * x1) * x1), Float64(-6.0 * x2)) + x1)
	tmp = 0.0
	if (x1 <= -600000000.0)
		tmp = t_0;
	elseif (x1 <= 6400000000000.0)
		tmp = fma(fma(-12.0, x1, fma(Float64(8.0 * x1), x2, -6.0)), x2, Float64(-x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(N[(6.0 * x1), $MachinePrecision] * x1), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -600000000.0], t$95$0, If[LessEqual[x1, 6400000000000.0], N[(N[(-12.0 * x1 + N[(N[(8.0 * x1), $MachinePrecision] * x2 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -6e8 or 6.4e12 < x1

   1. Initial program 50.3%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 44.6%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-*.f64 92.4

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

   10. Applied rewrites 92.4%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 92.4%

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

   if -6e8 < x1 < 6.4e12

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 41.6

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 93.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -600000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(6 \cdot x1\right) \cdot x1, -6 \cdot x2\right) + x1\\ \mathbf{elif}\;x1 \leq 6400000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, \mathsf{fma}\left(8 \cdot x1, x2, -6\right)\right), x2, -x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), \left(6 \cdot x1\right) \cdot x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 77.1% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -5.2 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), -4 \cdot x1, -6 \cdot x2\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, \mathsf{fma}\left(8 \cdot x1, x2, -6\right)\right), x2, -x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -5.2e+101)
   (+ (fma (fma x1 x1 1.0) (* -4.0 x1) (* -6.0 x2)) x1)
   (fma (fma -12.0 x1 (fma (* 8.0 x1) x2 -6.0)) x2 (- x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -5.2e+101) {
		tmp = fma(fma(x1, x1, 1.0), (-4.0 * x1), (-6.0 * x2)) + x1;
	} else {
		tmp = fma(fma(-12.0, x1, fma((8.0 * x1), x2, -6.0)), x2, -x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -5.2e+101)
		tmp = Float64(fma(fma(x1, x1, 1.0), Float64(-4.0 * x1), Float64(-6.0 * x2)) + x1);
	else
		tmp = fma(fma(-12.0, x1, fma(Float64(8.0 * x1), x2, -6.0)), x2, Float64(-x1));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -5.2e+101], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(-4.0 * x1), $MachinePrecision] + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(-12.0 * x1 + N[(N[(8.0 * x1), $MachinePrecision] * x2 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + (-x1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -5.2e101

   1. Initial program 6.8%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 4.5

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

   5. Applied rewrites 4.5%

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

   7. Applied rewrites 6.8%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-*.f64 97.7

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

   10. Applied rewrites 97.7%

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

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 93.9%

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

   if -5.2e101 < x1

   1. Initial program 91.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 28.2

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 62.7

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

   8. Applied rewrites 62.7%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 69.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5.2 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), -4 \cdot x1, -6 \cdot x2\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, \mathsf{fma}\left(8 \cdot x1, x2, -6\right)\right), x2, -x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.3% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{-131}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2 + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.8e-131) (- x1) (+ (* -6.0 x2) x1)))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.8e-131) {
		tmp = -x1;
	} else {
		tmp = (-6.0 * x2) + x1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if (x1 <= (-2.8d-131)) then
        tmp = -x1
    else
        tmp = ((-6.0d0) * x2) + x1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.8e-131) {
		tmp = -x1;
	} else {
		tmp = (-6.0 * x2) + x1;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x1 <= -2.8e-131:
		tmp = -x1
	else:
		tmp = (-6.0 * x2) + x1
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.8e-131)
		tmp = Float64(-x1);
	else
		tmp = Float64(Float64(-6.0 * x2) + x1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x1 <= -2.8e-131)
		tmp = -x1;
	else
		tmp = (-6.0 * x2) + x1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.8e-131], (-x1), N[(N[(-6.0 * x2), $MachinePrecision] + x1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.8e-131

   1. Initial program 58.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 35.6

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

   8. Applied rewrites 35.6%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 18.2%

       \[\leadsto -x1 \]

   if -2.8e-131 < x1

   1. Initial program 88.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 36.4

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

   5. Applied rewrites 36.4%

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{-131}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.0% accurate, 19.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-12, x1, -6\right), x2, -x1\right) \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (fma (fma -12.0 x1 -6.0) x2 (- x1)))

C

double code(double x1, double x2) {
	return fma(fma(-12.0, x1, -6.0), x2, -x1);
}

Julia

function code(x1, x2)
	return fma(fma(-12.0, x1, -6.0), x2, Float64(-x1))
end

Wolfram

code[x1_, x2_] := N[(N[(-12.0 * x1 + -6.0), $MachinePrecision] * x2 + (-x1)), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 23.5

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

5. Applied rewrites 23.5%

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

6. Taylor expanded in x1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 53.0

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

8. Applied rewrites 53.0%

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

9. Taylor expanded in x2 around 0

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

11. Applied rewrites 39.2%

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

Alternative 21: 29.7% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3.4 \cdot 10^{-131}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (if (<= x1 -3.4e-131) (- x1) (* -6.0 x2)))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.4e-131) {
		tmp = -x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if (x1 <= (-3.4d-131)) then
        tmp = -x1
    else
        tmp = (-6.0d0) * x2
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x1 <= -3.4e-131) {
		tmp = -x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x1 <= -3.4e-131:
		tmp = -x1
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -3.4e-131)
		tmp = Float64(-x1);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x1 <= -3.4e-131)
		tmp = -x1;
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -3.4e-131], (-x1), N[(-6.0 * x2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.39999999999999995e-131

   1. Initial program 58.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 35.6

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

   8. Applied rewrites 35.6%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 18.2%

       \[\leadsto -x1 \]

   if -3.39999999999999995e-131 < x1

   1. Initial program 88.1%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 36.4

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 35.4

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

   8. Applied rewrites 35.4%

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

Alternative 22: 39.2% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, x1, -6 \cdot x2\right) \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (fma -1.0 x1 (* -6.0 x2)))

C

double code(double x1, double x2) {
	return fma(-1.0, x1, (-6.0 * x2));
}

Julia

function code(x1, x2)
	return fma(-1.0, x1, Float64(-6.0 * x2))
end

Wolfram

code[x1_, x2_] := N[(-1.0 * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 23.5

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

5. Applied rewrites 23.5%

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

6. Taylor expanded in x1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 53.0

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

8. Applied rewrites 53.0%

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

9. Taylor expanded in x2 around 0

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

11. Applied rewrites 35.8%

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

Alternative 23: 13.9% accurate, 99.3× speedup

Math

\[\begin{array}{l} \\ -x1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x1, x2):
	return -x1

Julia

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

MATLAB

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

Wolfram

code[x1_, x2_] := (-x1)

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 23.5

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

5. Applied rewrites 23.5%

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

6. Taylor expanded in x1 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 53.0

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

8. Applied rewrites 53.0%

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

9. Taylor expanded in x2 around 0

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

11. Applied rewrites 13.8%

    \[\leadsto -x1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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))))))