Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 69.8% → 99.5%

Time: 17.5s

Alternatives: 20

Speedup: 6.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.8% 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: 99.5% accurate, 0.5× speedup

Math

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

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))
        (t_3
         (+
          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))))))
   (if (<= t_3 INFINITY)
     t_3
     (*
      (* (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 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = 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)));
	double tmp;
	if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (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(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(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))))
	tmp = 0.0
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(Float64(Float64(2.0 * x2) - 3.0), 4.0, 9.0)) * 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 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(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]}, If[LessEqual[t$95$3, Infinity], t$95$3, N[(N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $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 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 40.8%

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

   10. Applied rewrites 31.4%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 100.0%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

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

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))
        (t_3 (fma (* x1 x1) 3.0 (- (* 2.0 x2) x1)))
        (t_4 (/ t_3 (fma x1 x1 1.0))))
   (if (<=
        (+
         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))))
        INFINITY)
     (+
      (fma
       (fma x1 x1 1.0)
       (+
        x1
        (fma
         (* (* 2.0 x1) t_4)
         (- t_4 3.0)
         (* (* x1 x1) (- (* t_3 (/ 4.0 (fma x1 x1 1.0))) 6.0))))
       (* (* 3.0 x1) (* 3.0 x1)))
      (fma (fma -2.0 x2 (- x1)) 3.0 x1))
     (*
      (* (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 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = fma((x1 * x1), 3.0, ((2.0 * x2) - x1));
	double t_4 = t_3 / fma(x1, x1, 1.0);
	double tmp;
	if ((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)))) <= ((double) INFINITY)) {
		tmp = fma(fma(x1, x1, 1.0), (x1 + fma(((2.0 * x1) * t_4), (t_4 - 3.0), ((x1 * x1) * ((t_3 * (4.0 / fma(x1, x1, 1.0))) - 6.0)))), ((3.0 * x1) * (3.0 * x1))) + fma(fma(-2.0, x2, -x1), 3.0, x1);
	} else {
		tmp = (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(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = fma(Float64(x1 * x1), 3.0, Float64(Float64(2.0 * x2) - x1))
	t_4 = Float64(t_3 / fma(x1, x1, 1.0))
	tmp = 0.0
	if (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)))) <= Inf)
		tmp = Float64(fma(fma(x1, x1, 1.0), Float64(x1 + fma(Float64(Float64(2.0 * x1) * t_4), Float64(t_4 - 3.0), Float64(Float64(x1 * x1) * Float64(Float64(t_3 * Float64(4.0 / fma(x1, x1, 1.0))) - 6.0)))), Float64(Float64(3.0 * x1) * Float64(3.0 * x1))) + fma(fma(-2.0, x2, Float64(-x1)), 3.0, x1));
	else
		tmp = Float64(Float64(fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(Float64(Float64(2.0 * x2) - 3.0), 4.0, 9.0)) * 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 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] * 3.0 + N[(N[(2.0 * x2), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * N[(x1 + N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(t$95$4 - 3.0), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * N[(4.0 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 * x1), $MachinePrecision] * N[(3.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * x2 + (-x1)), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $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 99.6%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.0

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

   8. Applied rewrites 99.0%

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

   10. Applied rewrites 90.0%

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

   12. Applied rewrites 99.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 40.8%

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

   10. Applied rewrites 31.4%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 100.0%

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

Alternative 3: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq 10^{+305}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot x2\right) \cdot 8\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         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))))
        1e+305)
     (* -6.0 x2)
     (* (* (* x1 x1) x2) 8.0))))

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;
	double tmp;
	if ((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)))) <= 1e+305) {
		tmp = -6.0 * x2;
	} else {
		tmp = ((x1 * x1) * x2) * 8.0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    if ((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)))) <= 1d+305) then
        tmp = (-6.0d0) * x2
    else
        tmp = ((x1 * x1) * x2) * 8.0d0
    end if
    code = tmp
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;
	double tmp;
	if ((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)))) <= 1e+305) {
		tmp = -6.0 * x2;
	} else {
		tmp = ((x1 * x1) * x2) * 8.0;
	}
	return tmp;
}

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
	tmp = 0
	if (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)))) <= 1e+305:
		tmp = -6.0 * x2
	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(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(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)))) <= 1e+305)
		tmp = Float64(-6.0 * x2);
	else
		tmp = Float64(Float64(Float64(x1 * x1) * x2) * 8.0);
	end
	return tmp
end

MATLAB

function tmp_2 = 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 = 0.0;
	if ((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)))) <= 1e+305)
		tmp = -6.0 * x2;
	else
		tmp = ((x1 * x1) * x2) * 8.0;
	end
	tmp_2 = 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 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(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], 1e+305], N[(-6.0 * x2), $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)))))) < 9.9999999999999994e304

   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

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

      1. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

   if 9.9999999999999994e304 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 30.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 35.1%

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

Alternative 4: 95.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := 2 \cdot x2 - 3\\ t_4 := \mathsf{fma}\left(t\_3, 4, 9\right)\\ \mathbf{if}\;x1 \leq -9.5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot x1 - 3, x1, t\_4\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 8200:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot t\_3 + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot 3\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \mathsf{fma}\left(-2, x2, -x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\frac{-1 + t\_3 \cdot -6}{x1}, -1, t\_4\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \end{array} \end{array} \]

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))
        (t_3 (- (* 2.0 x2) 3.0))
        (t_4 (fma t_3 4.0 9.0)))
   (if (<= x1 -9.5e+66)
     (* (fma (- (* 6.0 x1) 3.0) x1 t_4) (* x1 x1))
     (if (<= x1 8200.0)
       (+
        x1
        (+
         (+
          (+
           (+
            (*
             (+ (* (* (* 2.0 x1) t_2) t_3) (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
             t_1)
            (* t_0 3.0))
           (* (* x1 x1) x1))
          x1)
         (* 3.0 (fma -2.0 x2 (- x1)))))
       (*
        (-
         6.0
         (/ (- 3.0 (/ (fma (/ (+ -1.0 (* t_3 -6.0)) x1) -1.0 t_4) x1)) x1))
        (pow x1 4.0))))))

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;
	double t_3 = (2.0 * x2) - 3.0;
	double t_4 = fma(t_3, 4.0, 9.0);
	double tmp;
	if (x1 <= -9.5e+66) {
		tmp = fma(((6.0 * x1) - 3.0), x1, t_4) * (x1 * x1);
	} else if (x1 <= 8200.0) {
		tmp = x1 + (((((((((2.0 * x1) * t_2) * t_3) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * 3.0)) + ((x1 * x1) * x1)) + x1) + (3.0 * fma(-2.0, x2, -x1)));
	} else {
		tmp = (6.0 - ((3.0 - (fma(((-1.0 + (t_3 * -6.0)) / x1), -1.0, t_4) / x1)) / x1)) * pow(x1, 4.0);
	}
	return tmp;
}

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)
	t_3 = Float64(Float64(2.0 * x2) - 3.0)
	t_4 = fma(t_3, 4.0, 9.0)
	tmp = 0.0
	if (x1 <= -9.5e+66)
		tmp = Float64(fma(Float64(Float64(6.0 * x1) - 3.0), x1, t_4) * Float64(x1 * x1));
	elseif (x1 <= 8200.0)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * t_3) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * 3.0)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * fma(-2.0, x2, Float64(-x1)))));
	else
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(Float64(Float64(-1.0 + Float64(t_3 * -6.0)) / x1), -1.0, t_4) / x1)) / x1)) * (x1 ^ 4.0));
	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[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * 4.0 + 9.0), $MachinePrecision]}, If[LessEqual[x1, -9.5e+66], N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + t$95$4), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8200.0], N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $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 * 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(-2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(N[(-1.0 + N[(t$95$3 * -6.0), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] * -1.0 + t$95$4), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -9.50000000000000051e66

   1. Initial program 16.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 100.0%

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

   if -9.50000000000000051e66 < x1 < 8200

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x1 around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 98.3

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

   11. Applied rewrites 98.3%

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

   if 8200 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\frac{-1 + \left(2 \cdot x2 - 3\right) \cdot -6}{x1}, -1, \mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)\right)}{x1}}{x1}\right) \cdot {x1}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot x2 - 3\\ t_1 := \mathsf{fma}\left(t\_0, 4, 9\right)\\ \mathbf{if}\;x1 \leq -42000000:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot x1 - 3, x1, t\_1\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(\left(x2 \cdot x2\right) \cdot 8\right) \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\frac{-1 + t\_0 \cdot -6}{x1}, -1, t\_1\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* 2.0 x2) 3.0)) (t_1 (fma t_0 4.0 9.0)))
   (if (<= x1 -42000000.0)
     (* (fma (- (* 6.0 x1) 3.0) x1 t_1) (* x1 x1))
     (if (<= x1 8500.0)
       (+
        (fma (fma x1 x1 1.0) x1 (* (* (* x2 x2) 8.0) (/ x1 (fma x1 x1 1.0))))
        (fma
         (/ (- (fma -2.0 x2 (* (* 3.0 x1) x1)) x1) (fma x1 x1 1.0))
         3.0
         x1))
       (*
        (-
         6.0
         (/ (- 3.0 (/ (fma (/ (+ -1.0 (* t_0 -6.0)) x1) -1.0 t_1) x1)) x1))
        (pow x1 4.0))))))

C

double code(double x1, double x2) {
	double t_0 = (2.0 * x2) - 3.0;
	double t_1 = fma(t_0, 4.0, 9.0);
	double tmp;
	if (x1 <= -42000000.0) {
		tmp = fma(((6.0 * x1) - 3.0), x1, t_1) * (x1 * x1);
	} else if (x1 <= 8500.0) {
		tmp = fma(fma(x1, x1, 1.0), x1, (((x2 * x2) * 8.0) * (x1 / fma(x1, x1, 1.0)))) + fma(((fma(-2.0, x2, ((3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, x1);
	} else {
		tmp = (6.0 - ((3.0 - (fma(((-1.0 + (t_0 * -6.0)) / x1), -1.0, t_1) / x1)) / x1)) * pow(x1, 4.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(2.0 * x2) - 3.0)
	t_1 = fma(t_0, 4.0, 9.0)
	tmp = 0.0
	if (x1 <= -42000000.0)
		tmp = Float64(fma(Float64(Float64(6.0 * x1) - 3.0), x1, t_1) * Float64(x1 * x1));
	elseif (x1 <= 8500.0)
		tmp = Float64(fma(fma(x1, x1, 1.0), x1, Float64(Float64(Float64(x2 * x2) * 8.0) * Float64(x1 / fma(x1, x1, 1.0)))) + fma(Float64(Float64(fma(-2.0, x2, Float64(Float64(3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, x1));
	else
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(Float64(Float64(-1.0 + Float64(t_0 * -6.0)) / x1), -1.0, t_1) / x1)) / x1)) * (x1 ^ 4.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 4.0 + 9.0), $MachinePrecision]}, If[LessEqual[x1, -42000000.0], N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + t$95$1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8500.0], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(x2 * x2), $MachinePrecision] * 8.0), $MachinePrecision] * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-2.0 * x2 + N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(N[(-1.0 + N[(t$95$0 * -6.0), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] * -1.0 + t$95$1), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.2e7

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -4.2e7 < x1 < 8500

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 86.7%

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

   if 8500 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\frac{-1 + \left(2 \cdot x2 - 3\right) \cdot -6}{x1}, -1, \mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)\right)}{x1}}{x1}\right) \cdot {x1}^{4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)\right)\\ \mathbf{if}\;x1 \leq -42000000:\\ \;\;\;\;t\_0 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(\left(x2 \cdot x2\right) \cdot 8\right) \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (- (* 6.0 x1) 3.0) x1 (fma (- (* 2.0 x2) 3.0) 4.0 9.0))))
   (if (<= x1 -42000000.0)
     (* t_0 (* x1 x1))
     (if (<= x1 8500.0)
       (+
        (fma (fma x1 x1 1.0) x1 (* (* (* x2 x2) 8.0) (/ x1 (fma x1 x1 1.0))))
        (fma
         (/ (- (fma -2.0 x2 (* (* 3.0 x1) x1)) x1) (fma x1 x1 1.0))
         3.0
         x1))
       (* (* t_0 x1) x1)))))

C

double code(double x1, double x2) {
	double t_0 = fma(((6.0 * x1) - 3.0), x1, fma(((2.0 * x2) - 3.0), 4.0, 9.0));
	double tmp;
	if (x1 <= -42000000.0) {
		tmp = t_0 * (x1 * x1);
	} else if (x1 <= 8500.0) {
		tmp = fma(fma(x1, x1, 1.0), x1, (((x2 * x2) * 8.0) * (x1 / fma(x1, x1, 1.0)))) + fma(((fma(-2.0, x2, ((3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, x1);
	} else {
		tmp = (t_0 * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(Float64(Float64(2.0 * x2) - 3.0), 4.0, 9.0))
	tmp = 0.0
	if (x1 <= -42000000.0)
		tmp = Float64(t_0 * Float64(x1 * x1));
	elseif (x1 <= 8500.0)
		tmp = Float64(fma(fma(x1, x1, 1.0), x1, Float64(Float64(Float64(x2 * x2) * 8.0) * Float64(x1 / fma(x1, x1, 1.0)))) + fma(Float64(Float64(fma(-2.0, x2, Float64(Float64(3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, x1));
	else
		tmp = Float64(Float64(t_0 * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -42000000.0], N[(t$95$0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8500.0], N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(x2 * x2), $MachinePrecision] * 8.0), $MachinePrecision] * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-2.0 * x2 + N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x1), $MachinePrecision] * x1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.2e7

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -4.2e7 < x1 < 8500

   1. Initial program 99.6%

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

   3. Taylor expanded in x2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 86.7%

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

   if 8500 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 30.2%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 95.9%

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

Alternative 7: 91.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \mathsf{fma}\left(-2, x2, -x1\right)\\ t_1 := 2 \cdot x2 - 3\\ t_2 := \mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(t\_1, 4, 9\right)\right)\\ \mathbf{if}\;x1 \leq -27:\\ \;\;\;\;t\_2 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{-229}:\\ \;\;\;\;x1 + \left(\left(\mathsf{fma}\left(-22 \cdot x1, x1, \left(t\_1 \cdot x2\right) \cdot 4\right) \cdot x1 + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{-180}:\\ \;\;\;\;x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \mathsf{fma}\left(\left(6 \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \left(3 \cdot 3\right) \cdot \left(x1 \cdot x1\right)\right)\right) + t\_0\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;x1 + \left(\left(\frac{8 \cdot \left(\left(x2 \cdot x2\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} + x1\right) + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 3.0 (fma -2.0 x2 (- x1))))
        (t_1 (- (* 2.0 x2) 3.0))
        (t_2 (fma (- (* 6.0 x1) 3.0) x1 (fma t_1 4.0 9.0))))
   (if (<= x1 -27.0)
     (* t_2 (* x1 x1))
     (if (<= x1 -1.5e-229)
       (+
        x1
        (+
         (+ (* (fma (* -22.0 x1) x1 (* (* t_1 x2) 4.0)) x1) x1)
         (*
          3.0
          (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
       (if (<= x1 4.8e-180)
         (+
          x1
          (+
           (fma
            (fma x1 x1 1.0)
            x1
            (fma (* (* 6.0 x1) x1) (fma x1 x1 1.0) (* (* 3.0 3.0) (* x1 x1))))
           t_0))
         (if (<= x1 8500.0)
           (+ x1 (+ (+ (/ (* 8.0 (* (* x2 x2) x1)) (fma x1 x1 1.0)) x1) t_0))
           (* (* t_2 x1) x1)))))))

C

double code(double x1, double x2) {
	double t_0 = 3.0 * fma(-2.0, x2, -x1);
	double t_1 = (2.0 * x2) - 3.0;
	double t_2 = fma(((6.0 * x1) - 3.0), x1, fma(t_1, 4.0, 9.0));
	double tmp;
	if (x1 <= -27.0) {
		tmp = t_2 * (x1 * x1);
	} else if (x1 <= -1.5e-229) {
		tmp = x1 + (((fma((-22.0 * x1), x1, ((t_1 * x2) * 4.0)) * x1) + x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
	} else if (x1 <= 4.8e-180) {
		tmp = x1 + (fma(fma(x1, x1, 1.0), x1, fma(((6.0 * x1) * x1), fma(x1, x1, 1.0), ((3.0 * 3.0) * (x1 * x1)))) + t_0);
	} else if (x1 <= 8500.0) {
		tmp = x1 + ((((8.0 * ((x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + t_0);
	} else {
		tmp = (t_2 * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(3.0 * fma(-2.0, x2, Float64(-x1)))
	t_1 = Float64(Float64(2.0 * x2) - 3.0)
	t_2 = fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(t_1, 4.0, 9.0))
	tmp = 0.0
	if (x1 <= -27.0)
		tmp = Float64(t_2 * Float64(x1 * x1));
	elseif (x1 <= -1.5e-229)
		tmp = Float64(x1 + Float64(Float64(Float64(fma(Float64(-22.0 * x1), x1, Float64(Float64(t_1 * x2) * 4.0)) * x1) + x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))));
	elseif (x1 <= 4.8e-180)
		tmp = Float64(x1 + Float64(fma(fma(x1, x1, 1.0), x1, fma(Float64(Float64(6.0 * x1) * x1), fma(x1, x1, 1.0), Float64(Float64(3.0 * 3.0) * Float64(x1 * x1)))) + t_0));
	elseif (x1 <= 8500.0)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(8.0 * Float64(Float64(x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + t_0));
	else
		tmp = Float64(Float64(t_2 * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(3.0 * N[(-2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(t$95$1 * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -27.0], N[(t$95$2 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.5e-229], N[(x1 + N[(N[(N[(N[(N[(-22.0 * x1), $MachinePrecision] * x1 + N[(N[(t$95$1 * x2), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e-180], N[(x1 + N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(6.0 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(3.0 * 3.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8500.0], N[(x1 + N[(N[(N[(N[(8.0 * N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * x1), $MachinePrecision] * x1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -27

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -27 < x1 < -1.50000000000000001e-229

   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

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(\mathsf{fma}\left(-22 \cdot x1, x1, \left(\left(2 \cdot x2 - 3\right) \cdot x2\right) \cdot 4\right) \cdot x1 + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 92.6%

      \[\leadsto x1 + \left(\left(\mathsf{fma}\left(-22 \cdot x1, x1, \left(\left(2 \cdot x2 - 3\right) \cdot x2\right) \cdot 4\right) \cdot x1 + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   if -1.50000000000000001e-229 < x1 < 4.79999999999999959e-180

   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 inf

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.7

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 73.1%

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

   11. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 92.9

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

   13. Applied rewrites 92.9%

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

   if 4.79999999999999959e-180 < x1 < 8500

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x2 around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 94.0

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

   11. Applied rewrites 94.0%

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

   if 8500 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 30.2%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 95.9%

       \[\leadsto \left(\mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)\right) \cdot x1\right) \cdot \color{blue}{x1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \mathsf{fma}\left(-2, x2, -x1\right)\\ t_1 := 2 \cdot x2 - 3\\ t_2 := \mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(t\_1, 4, 9\right)\right)\\ \mathbf{if}\;x1 \leq -27:\\ \;\;\;\;t\_2 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-4, x2, \mathsf{fma}\left(-2, t\_1, \mathsf{fma}\left(3 - -2 \cdot x2, 3, x2 \cdot 14\right) - 6\right)\right), x1, \left(t\_1 \cdot x2\right) \cdot 4 - 1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{-180}:\\ \;\;\;\;x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \mathsf{fma}\left(\left(6 \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \left(3 \cdot 3\right) \cdot \left(x1 \cdot x1\right)\right)\right) + t\_0\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;x1 + \left(\left(\frac{8 \cdot \left(\left(x2 \cdot x2\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} + x1\right) + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 3.0 (fma -2.0 x2 (- x1))))
        (t_1 (- (* 2.0 x2) 3.0))
        (t_2 (fma (- (* 6.0 x1) 3.0) x1 (fma t_1 4.0 9.0))))
   (if (<= x1 -27.0)
     (* t_2 (* x1 x1))
     (if (<= x1 -1.5e-229)
       (fma
        (fma
         (fma
          -4.0
          x2
          (fma -2.0 t_1 (- (fma (- 3.0 (* -2.0 x2)) 3.0 (* x2 14.0)) 6.0)))
         x1
         (- (* (* t_1 x2) 4.0) 1.0))
        x1
        (* -6.0 x2))
       (if (<= x1 4.8e-180)
         (+
          x1
          (+
           (fma
            (fma x1 x1 1.0)
            x1
            (fma (* (* 6.0 x1) x1) (fma x1 x1 1.0) (* (* 3.0 3.0) (* x1 x1))))
           t_0))
         (if (<= x1 8500.0)
           (+ x1 (+ (+ (/ (* 8.0 (* (* x2 x2) x1)) (fma x1 x1 1.0)) x1) t_0))
           (* (* t_2 x1) x1)))))))

C

double code(double x1, double x2) {
	double t_0 = 3.0 * fma(-2.0, x2, -x1);
	double t_1 = (2.0 * x2) - 3.0;
	double t_2 = fma(((6.0 * x1) - 3.0), x1, fma(t_1, 4.0, 9.0));
	double tmp;
	if (x1 <= -27.0) {
		tmp = t_2 * (x1 * x1);
	} else if (x1 <= -1.5e-229) {
		tmp = fma(fma(fma(-4.0, x2, fma(-2.0, t_1, (fma((3.0 - (-2.0 * x2)), 3.0, (x2 * 14.0)) - 6.0))), x1, (((t_1 * x2) * 4.0) - 1.0)), x1, (-6.0 * x2));
	} else if (x1 <= 4.8e-180) {
		tmp = x1 + (fma(fma(x1, x1, 1.0), x1, fma(((6.0 * x1) * x1), fma(x1, x1, 1.0), ((3.0 * 3.0) * (x1 * x1)))) + t_0);
	} else if (x1 <= 8500.0) {
		tmp = x1 + ((((8.0 * ((x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + t_0);
	} else {
		tmp = (t_2 * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(3.0 * fma(-2.0, x2, Float64(-x1)))
	t_1 = Float64(Float64(2.0 * x2) - 3.0)
	t_2 = fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(t_1, 4.0, 9.0))
	tmp = 0.0
	if (x1 <= -27.0)
		tmp = Float64(t_2 * Float64(x1 * x1));
	elseif (x1 <= -1.5e-229)
		tmp = fma(fma(fma(-4.0, x2, fma(-2.0, t_1, Float64(fma(Float64(3.0 - Float64(-2.0 * x2)), 3.0, Float64(x2 * 14.0)) - 6.0))), x1, Float64(Float64(Float64(t_1 * x2) * 4.0) - 1.0)), x1, Float64(-6.0 * x2));
	elseif (x1 <= 4.8e-180)
		tmp = Float64(x1 + Float64(fma(fma(x1, x1, 1.0), x1, fma(Float64(Float64(6.0 * x1) * x1), fma(x1, x1, 1.0), Float64(Float64(3.0 * 3.0) * Float64(x1 * x1)))) + t_0));
	elseif (x1 <= 8500.0)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(8.0 * Float64(Float64(x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + t_0));
	else
		tmp = Float64(Float64(t_2 * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(3.0 * N[(-2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(t$95$1 * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -27.0], N[(t$95$2 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.5e-229], N[(N[(N[(-4.0 * x2 + N[(-2.0 * t$95$1 + N[(N[(N[(3.0 - N[(-2.0 * x2), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(x2 * 14.0), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(N[(t$95$1 * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e-180], N[(x1 + N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(6.0 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(3.0 * 3.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8500.0], N[(x1 + N[(N[(N[(N[(8.0 * N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * x1), $MachinePrecision] * x1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -27

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -27 < x1 < -1.50000000000000001e-229

   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

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

   5. Applied rewrites 92.5%

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

   if -1.50000000000000001e-229 < x1 < 4.79999999999999959e-180

   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 inf

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.7

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 73.1%

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

   11. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 92.9

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

   13. Applied rewrites 92.9%

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

   if 4.79999999999999959e-180 < x1 < 8500

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x2 around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 94.0

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

   11. Applied rewrites 94.0%

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

   if 8500 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 30.2%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 95.9%

       \[\leadsto \left(\mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)\right) \cdot x1\right) \cdot \color{blue}{x1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot x2 - 3\\ t_1 := \mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(t\_0, 4, 9\right)\right)\\ t_2 := 3 \cdot \mathsf{fma}\left(-2, x2, -x1\right)\\ \mathbf{if}\;x1 \leq -27:\\ \;\;\;\;t\_1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{-229}:\\ \;\;\;\;x1 + \left(\mathsf{fma}\left(t\_0 \cdot x2, 4, 1\right) \cdot x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{-180}:\\ \;\;\;\;x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \mathsf{fma}\left(\left(6 \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \left(3 \cdot 3\right) \cdot \left(x1 \cdot x1\right)\right)\right) + t\_2\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;x1 + \left(\left(\frac{8 \cdot \left(\left(x2 \cdot x2\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} + x1\right) + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* 2.0 x2) 3.0))
        (t_1 (fma (- (* 6.0 x1) 3.0) x1 (fma t_0 4.0 9.0)))
        (t_2 (* 3.0 (fma -2.0 x2 (- x1)))))
   (if (<= x1 -27.0)
     (* t_1 (* x1 x1))
     (if (<= x1 -1.5e-229)
       (+
        x1
        (+
         (* (fma (* t_0 x2) 4.0 1.0) x1)
         (*
          3.0
          (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
       (if (<= x1 4.8e-180)
         (+
          x1
          (+
           (fma
            (fma x1 x1 1.0)
            x1
            (fma (* (* 6.0 x1) x1) (fma x1 x1 1.0) (* (* 3.0 3.0) (* x1 x1))))
           t_2))
         (if (<= x1 8500.0)
           (+ x1 (+ (+ (/ (* 8.0 (* (* x2 x2) x1)) (fma x1 x1 1.0)) x1) t_2))
           (* (* t_1 x1) x1)))))))

C

double code(double x1, double x2) {
	double t_0 = (2.0 * x2) - 3.0;
	double t_1 = fma(((6.0 * x1) - 3.0), x1, fma(t_0, 4.0, 9.0));
	double t_2 = 3.0 * fma(-2.0, x2, -x1);
	double tmp;
	if (x1 <= -27.0) {
		tmp = t_1 * (x1 * x1);
	} else if (x1 <= -1.5e-229) {
		tmp = x1 + ((fma((t_0 * x2), 4.0, 1.0) * x1) + (3.0 * (((((3.0 * x1) * x1) - (2.0 * x2)) - x1) / ((x1 * x1) + 1.0))));
	} else if (x1 <= 4.8e-180) {
		tmp = x1 + (fma(fma(x1, x1, 1.0), x1, fma(((6.0 * x1) * x1), fma(x1, x1, 1.0), ((3.0 * 3.0) * (x1 * x1)))) + t_2);
	} else if (x1 <= 8500.0) {
		tmp = x1 + ((((8.0 * ((x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + t_2);
	} else {
		tmp = (t_1 * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(2.0 * x2) - 3.0)
	t_1 = fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(t_0, 4.0, 9.0))
	t_2 = Float64(3.0 * fma(-2.0, x2, Float64(-x1)))
	tmp = 0.0
	if (x1 <= -27.0)
		tmp = Float64(t_1 * Float64(x1 * x1));
	elseif (x1 <= -1.5e-229)
		tmp = Float64(x1 + Float64(Float64(fma(Float64(t_0 * x2), 4.0, 1.0) * x1) + Float64(3.0 * Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(2.0 * x2)) - x1) / Float64(Float64(x1 * x1) + 1.0)))));
	elseif (x1 <= 4.8e-180)
		tmp = Float64(x1 + Float64(fma(fma(x1, x1, 1.0), x1, fma(Float64(Float64(6.0 * x1) * x1), fma(x1, x1, 1.0), Float64(Float64(3.0 * 3.0) * Float64(x1 * x1)))) + t_2));
	elseif (x1 <= 8500.0)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(8.0 * Float64(Float64(x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + t_2));
	else
		tmp = Float64(Float64(t_1 * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(t$95$0 * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 * N[(-2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -27.0], N[(t$95$1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.5e-229], N[(x1 + N[(N[(N[(N[(t$95$0 * x2), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision] * x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e-180], N[(x1 + N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(6.0 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(3.0 * 3.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8500.0], N[(x1 + N[(N[(N[(N[(8.0 * N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * x1), $MachinePrecision] * x1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -27

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -27 < x1 < -1.50000000000000001e-229

   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 inf

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 92.0

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

   8. Applied rewrites 92.0%

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

   if -1.50000000000000001e-229 < x1 < 4.79999999999999959e-180

   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 inf

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.7

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 73.1%

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

   11. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 92.9

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

   13. Applied rewrites 92.9%

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

   if 4.79999999999999959e-180 < x1 < 8500

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x2 around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 94.0

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

   11. Applied rewrites 94.0%

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

   if 8500 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 30.2%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 95.9%

       \[\leadsto \left(\mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)\right) \cdot x1\right) \cdot \color{blue}{x1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \mathsf{fma}\left(-2, x2, -x1\right)\\ t_1 := 2 \cdot x2 - 3\\ t_2 := \mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(t\_1, 4, 9\right)\right)\\ \mathbf{if}\;x1 \leq -27:\\ \;\;\;\;t\_2 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -1.5 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_1 \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 4.8 \cdot 10^{-180}:\\ \;\;\;\;x1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \mathsf{fma}\left(\left(6 \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \left(3 \cdot 3\right) \cdot \left(x1 \cdot x1\right)\right)\right) + t\_0\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;x1 + \left(\left(\frac{8 \cdot \left(\left(x2 \cdot x2\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} + x1\right) + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 3.0 (fma -2.0 x2 (- x1))))
        (t_1 (- (* 2.0 x2) 3.0))
        (t_2 (fma (- (* 6.0 x1) 3.0) x1 (fma t_1 4.0 9.0))))
   (if (<= x1 -27.0)
     (* t_2 (* x1 x1))
     (if (<= x1 -1.5e-229)
       (fma (- (* (* t_1 x2) 4.0) 1.0) x1 (* -6.0 x2))
       (if (<= x1 4.8e-180)
         (+
          x1
          (+
           (fma
            (fma x1 x1 1.0)
            x1
            (fma (* (* 6.0 x1) x1) (fma x1 x1 1.0) (* (* 3.0 3.0) (* x1 x1))))
           t_0))
         (if (<= x1 8500.0)
           (+ x1 (+ (+ (/ (* 8.0 (* (* x2 x2) x1)) (fma x1 x1 1.0)) x1) t_0))
           (* (* t_2 x1) x1)))))))

C

double code(double x1, double x2) {
	double t_0 = 3.0 * fma(-2.0, x2, -x1);
	double t_1 = (2.0 * x2) - 3.0;
	double t_2 = fma(((6.0 * x1) - 3.0), x1, fma(t_1, 4.0, 9.0));
	double tmp;
	if (x1 <= -27.0) {
		tmp = t_2 * (x1 * x1);
	} else if (x1 <= -1.5e-229) {
		tmp = fma((((t_1 * x2) * 4.0) - 1.0), x1, (-6.0 * x2));
	} else if (x1 <= 4.8e-180) {
		tmp = x1 + (fma(fma(x1, x1, 1.0), x1, fma(((6.0 * x1) * x1), fma(x1, x1, 1.0), ((3.0 * 3.0) * (x1 * x1)))) + t_0);
	} else if (x1 <= 8500.0) {
		tmp = x1 + ((((8.0 * ((x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + t_0);
	} else {
		tmp = (t_2 * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(3.0 * fma(-2.0, x2, Float64(-x1)))
	t_1 = Float64(Float64(2.0 * x2) - 3.0)
	t_2 = fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(t_1, 4.0, 9.0))
	tmp = 0.0
	if (x1 <= -27.0)
		tmp = Float64(t_2 * Float64(x1 * x1));
	elseif (x1 <= -1.5e-229)
		tmp = fma(Float64(Float64(Float64(t_1 * x2) * 4.0) - 1.0), x1, Float64(-6.0 * x2));
	elseif (x1 <= 4.8e-180)
		tmp = Float64(x1 + Float64(fma(fma(x1, x1, 1.0), x1, fma(Float64(Float64(6.0 * x1) * x1), fma(x1, x1, 1.0), Float64(Float64(3.0 * 3.0) * Float64(x1 * x1)))) + t_0));
	elseif (x1 <= 8500.0)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(8.0 * Float64(Float64(x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + t_0));
	else
		tmp = Float64(Float64(t_2 * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(3.0 * N[(-2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(t$95$1 * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -27.0], N[(t$95$2 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -1.5e-229], N[(N[(N[(N[(t$95$1 * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.8e-180], N[(x1 + N[(N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(6.0 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(3.0 * 3.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8500.0], N[(x1 + N[(N[(N[(N[(8.0 * N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * x1), $MachinePrecision] * x1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x1 < -27

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -27 < x1 < -1.50000000000000001e-229

   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

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   4. 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. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if -1.50000000000000001e-229 < x1 < 4.79999999999999959e-180

   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 inf

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.7

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 73.1%

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

   11. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 92.9

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

   13. Applied rewrites 92.9%

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

   if 4.79999999999999959e-180 < x1 < 8500

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x2 around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 94.0

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

   11. Applied rewrites 94.0%

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

   if 8500 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 30.2%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 95.9%

       \[\leadsto \left(\mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)\right) \cdot x1\right) \cdot \color{blue}{x1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 90.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(2 \cdot x2 - 3, 4, 9\right)\right)\\ \mathbf{if}\;x1 \leq -3700000:\\ \;\;\;\;t\_0 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 8500:\\ \;\;\;\;x1 + \left(\left(\frac{8 \cdot \left(\left(x2 \cdot x2\right) \cdot x1\right)}{\mathsf{fma}\left(x1, x1, 1\right)} + x1\right) + 3 \cdot \mathsf{fma}\left(-2, x2, -x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (- (* 6.0 x1) 3.0) x1 (fma (- (* 2.0 x2) 3.0) 4.0 9.0))))
   (if (<= x1 -3700000.0)
     (* t_0 (* x1 x1))
     (if (<= x1 8500.0)
       (+
        x1
        (+
         (+ (/ (* 8.0 (* (* x2 x2) x1)) (fma x1 x1 1.0)) x1)
         (* 3.0 (fma -2.0 x2 (- x1)))))
       (* (* t_0 x1) x1)))))

C

double code(double x1, double x2) {
	double t_0 = fma(((6.0 * x1) - 3.0), x1, fma(((2.0 * x2) - 3.0), 4.0, 9.0));
	double tmp;
	if (x1 <= -3700000.0) {
		tmp = t_0 * (x1 * x1);
	} else if (x1 <= 8500.0) {
		tmp = x1 + ((((8.0 * ((x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + (3.0 * fma(-2.0, x2, -x1)));
	} else {
		tmp = (t_0 * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(Float64(Float64(2.0 * x2) - 3.0), 4.0, 9.0))
	tmp = 0.0
	if (x1 <= -3700000.0)
		tmp = Float64(t_0 * Float64(x1 * x1));
	elseif (x1 <= 8500.0)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(8.0 * Float64(Float64(x2 * x2) * x1)) / fma(x1, x1, 1.0)) + x1) + Float64(3.0 * fma(-2.0, x2, Float64(-x1)))));
	else
		tmp = Float64(Float64(t_0 * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -3700000.0], N[(t$95$0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8500.0], N[(x1 + N[(N[(N[(N[(8.0 * N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(-2.0 * x2 + (-x1)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * x1), $MachinePrecision] * x1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.7e6

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -3.7e6 < x1 < 8500

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-neg.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x2 around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 85.8

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

   11. Applied rewrites 85.8%

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

   if 8500 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 30.2%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 95.9%

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

Alternative 12: 90.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot x2 - 3\\ t_1 := \mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(t\_0, 4, 9\right)\right)\\ \mathbf{if}\;x1 \leq -27:\\ \;\;\;\;t\_1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 8200:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* 2.0 x2) 3.0))
        (t_1 (fma (- (* 6.0 x1) 3.0) x1 (fma t_0 4.0 9.0))))
   (if (<= x1 -27.0)
     (* t_1 (* x1 x1))
     (if (<= x1 8200.0)
       (fma (- (* (* t_0 x2) 4.0) 1.0) x1 (* -6.0 x2))
       (* (* t_1 x1) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (2.0 * x2) - 3.0;
	double t_1 = fma(((6.0 * x1) - 3.0), x1, fma(t_0, 4.0, 9.0));
	double tmp;
	if (x1 <= -27.0) {
		tmp = t_1 * (x1 * x1);
	} else if (x1 <= 8200.0) {
		tmp = fma((((t_0 * x2) * 4.0) - 1.0), x1, (-6.0 * x2));
	} else {
		tmp = (t_1 * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(2.0 * x2) - 3.0)
	t_1 = fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(t_0, 4.0, 9.0))
	tmp = 0.0
	if (x1 <= -27.0)
		tmp = Float64(t_1 * Float64(x1 * x1));
	elseif (x1 <= 8200.0)
		tmp = fma(Float64(Float64(Float64(t_0 * x2) * 4.0) - 1.0), x1, Float64(-6.0 * x2));
	else
		tmp = Float64(Float64(t_1 * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(t$95$0 * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -27.0], N[(t$95$1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8200.0], N[(N[(N[(N[(t$95$0 * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * x1), $MachinePrecision] * x1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -27

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.8%

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

   if -27 < x1 < 8200

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   4. 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. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

   if 8200 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 30.2%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 95.9%

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

Alternative 13: 89.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot x2 - 3\\ \mathbf{if}\;x1 \leq -27000:\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 8200:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_0 \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(6 \cdot x1 - 3, x1, \mathsf{fma}\left(t\_0, 4, 9\right)\right) \cdot x1\right) \cdot x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* 2.0 x2) 3.0)))
   (if (<= x1 -27000.0)
     (* (* (- 6.0 (/ 3.0 x1)) (* x1 x1)) (* x1 x1))
     (if (<= x1 8200.0)
       (fma (- (* (* t_0 x2) 4.0) 1.0) x1 (* -6.0 x2))
       (* (* (fma (- (* 6.0 x1) 3.0) x1 (fma t_0 4.0 9.0)) x1) x1)))))

C

double code(double x1, double x2) {
	double t_0 = (2.0 * x2) - 3.0;
	double tmp;
	if (x1 <= -27000.0) {
		tmp = ((6.0 - (3.0 / x1)) * (x1 * x1)) * (x1 * x1);
	} else if (x1 <= 8200.0) {
		tmp = fma((((t_0 * x2) * 4.0) - 1.0), x1, (-6.0 * x2));
	} else {
		tmp = (fma(((6.0 * x1) - 3.0), x1, fma(t_0, 4.0, 9.0)) * x1) * x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(2.0 * x2) - 3.0)
	tmp = 0.0
	if (x1 <= -27000.0)
		tmp = Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * Float64(x1 * x1)) * Float64(x1 * x1));
	elseif (x1 <= 8200.0)
		tmp = fma(Float64(Float64(Float64(t_0 * x2) * 4.0) - 1.0), x1, Float64(-6.0 * x2));
	else
		tmp = Float64(Float64(fma(Float64(Float64(6.0 * x1) - 3.0), x1, fma(t_0, 4.0, 9.0)) * x1) * x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[x1, -27000.0], N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8200.0], N[(N[(N[(N[(t$95$0 * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision] * x1 + N[(t$95$0 * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -27000

   1. Initial program 24.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(\left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 96.7%

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

   if -27000 < x1 < 8200

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   4. 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. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

   if 8200 < x1

   1. Initial program 52.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in x2 around inf

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 30.2%

       \[\leadsto \left(\left(x2 \cdot x1\right) \cdot x1\right) \cdot 8 \]

   11. Taylor expanded in x1 around 0

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

   13. Applied rewrites 95.9%

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

Alternative 14: 88.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -27000 \lor \neg \left(x1 \leq 8200\right):\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(2 \cdot x2 - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -27000.0) (not (<= x1 8200.0)))
   (* (* (- 6.0 (/ 3.0 x1)) (* x1 x1)) (* x1 x1))
   (fma (- (* (* (- (* 2.0 x2) 3.0) x2) 4.0) 1.0) x1 (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -27000.0) || !(x1 <= 8200.0)) {
		tmp = ((6.0 - (3.0 / x1)) * (x1 * x1)) * (x1 * x1);
	} else {
		tmp = fma((((((2.0 * x2) - 3.0) * x2) * 4.0) - 1.0), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -27000.0) || !(x1 <= 8200.0))
		tmp = Float64(Float64(Float64(6.0 - Float64(3.0 / x1)) * Float64(x1 * x1)) * Float64(x1 * x1));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(2.0 * x2) - 3.0) * x2) * 4.0) - 1.0), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -27000.0], N[Not[LessEqual[x1, 8200.0]], $MachinePrecision]], N[(N[(N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -27000 or 8200 < x1

   1. Initial program 39.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in x1 around inf

      \[\leadsto \left(\left(6 - 3 \cdot \frac{1}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 93.5%

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

   if -27000 < x1 < 8200

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   4. 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. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -27000 \lor \neg \left(x1 \leq 8200\right):\\ \;\;\;\;\left(\left(6 - \frac{3}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(2 \cdot x2 - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 88.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -3700000 \lor \neg \left(x1 \leq 8200\right):\\ \;\;\;\;6 \cdot \left(\left(\left(x1 \cdot x1\right) \cdot \left|x1\right|\right) \cdot \left|x1\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(2 \cdot x2 - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -3700000.0) (not (<= x1 8200.0)))
   (* 6.0 (* (* (* x1 x1) (fabs x1)) (fabs x1)))
   (fma (- (* (* (- (* 2.0 x2) 3.0) x2) 4.0) 1.0) x1 (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -3700000.0) || !(x1 <= 8200.0)) {
		tmp = 6.0 * (((x1 * x1) * fabs(x1)) * fabs(x1));
	} else {
		tmp = fma((((((2.0 * x2) - 3.0) * x2) * 4.0) - 1.0), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -3700000.0) || !(x1 <= 8200.0))
		tmp = Float64(6.0 * Float64(Float64(Float64(x1 * x1) * abs(x1)) * abs(x1)));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(2.0 * x2) - 3.0) * x2) * 4.0) - 1.0), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -3700000.0], N[Not[LessEqual[x1, 8200.0]], $MachinePrecision]], N[(6.0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[Abs[x1], $MachinePrecision]), $MachinePrecision] * N[Abs[x1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.7e6 or 8200 < x1

   1. Initial program 39.1%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in x1 around inf

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

   8. Applied rewrites 92.5%

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

   10. Applied rewrites 92.5%

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

   if -3.7e6 < x1 < 8200

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2 + x1 \cdot \left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) - 1\right)} \]
   4. 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. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -3700000 \lor \neg \left(x1 \leq 8200\right):\\ \;\;\;\;6 \cdot \left(\left(\left(x1 \cdot x1\right) \cdot \left|x1\right|\right) \cdot \left|x1\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(2 \cdot x2 - 3\right) \cdot x2\right) \cdot 4 - 1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.6 \cdot 10^{-17} \lor \neg \left(x1 \leq 8.2 \cdot 10^{-42}\right):\\ \;\;\;\;6 \cdot \left(\left(\left(x1 \cdot x1\right) \cdot \left|x1\right|\right) \cdot \left|x1\right|\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -4.6e-17) (not (<= x1 8.2e-42)))
   (* 6.0 (* (* (* x1 x1) (fabs x1)) (fabs x1)))
   (* -6.0 x2)))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -4.6e-17) || !(x1 <= 8.2e-42)) {
		tmp = 6.0 * (((x1 * x1) * fabs(x1)) * fabs(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 <= (-4.6d-17)) .or. (.not. (x1 <= 8.2d-42))) then
        tmp = 6.0d0 * (((x1 * x1) * abs(x1)) * abs(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 <= -4.6e-17) || !(x1 <= 8.2e-42)) {
		tmp = 6.0 * (((x1 * x1) * Math.abs(x1)) * Math.abs(x1));
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x1 <= -4.6e-17) or not (x1 <= 8.2e-42):
		tmp = 6.0 * (((x1 * x1) * math.fabs(x1)) * math.fabs(x1))
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -4.6e-17) || !(x1 <= 8.2e-42))
		tmp = Float64(6.0 * Float64(Float64(Float64(x1 * x1) * abs(x1)) * abs(x1)));
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x1 <= -4.6e-17) || ~((x1 <= 8.2e-42)))
		tmp = 6.0 * (((x1 * x1) * abs(x1)) * abs(x1));
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -4.6e-17], N[Not[LessEqual[x1, 8.2e-42]], $MachinePrecision]], N[(6.0 * N[(N[(N[(x1 * x1), $MachinePrecision] * N[Abs[x1], $MachinePrecision]), $MachinePrecision] * N[Abs[x1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-6.0 * x2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -4.60000000000000018e-17 or 8.2000000000000003e-42 < x1

   1. Initial program 44.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x1 around inf

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

   8. Applied rewrites 85.0%

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

   10. Applied rewrites 85.0%

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

   if -4.60000000000000018e-17 < x1 < 8.2000000000000003e-42

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.6 \cdot 10^{-17} \lor \neg \left(x1 \leq 8.2 \cdot 10^{-42}\right):\\ \;\;\;\;6 \cdot \left(\left(\left(x1 \cdot x1\right) \cdot \left|x1\right|\right) \cdot \left|x1\right|\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.6 \cdot 10^{-17} \lor \neg \left(x1 \leq 8.2 \cdot 10^{-42}\right):\\ \;\;\;\;6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (or (<= x1 -4.6e-17) (not (<= x1 8.2e-42)))
   (* 6.0 (* (* x1 x1) (* x1 x1)))
   (* -6.0 x2)))

C

double code(double x1, double x2) {
	double tmp;
	if ((x1 <= -4.6e-17) || !(x1 <= 8.2e-42)) {
		tmp = 6.0 * ((x1 * x1) * (x1 * 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 <= (-4.6d-17)) .or. (.not. (x1 <= 8.2d-42))) then
        tmp = 6.0d0 * ((x1 * x1) * (x1 * 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 <= -4.6e-17) || !(x1 <= 8.2e-42)) {
		tmp = 6.0 * ((x1 * x1) * (x1 * x1));
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x1 <= -4.6e-17) or not (x1 <= 8.2e-42):
		tmp = 6.0 * ((x1 * x1) * (x1 * x1))
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if ((x1 <= -4.6e-17) || !(x1 <= 8.2e-42))
		tmp = Float64(6.0 * Float64(Float64(x1 * x1) * Float64(x1 * x1)));
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x1 <= -4.6e-17) || ~((x1 <= 8.2e-42)))
		tmp = 6.0 * ((x1 * x1) * (x1 * x1));
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[Or[LessEqual[x1, -4.6e-17], N[Not[LessEqual[x1, 8.2e-42]], $MachinePrecision]], N[(6.0 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-6.0 * x2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -4.60000000000000018e-17 or 8.2000000000000003e-42 < x1

   1. Initial program 44.2%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in x1 around inf

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

   8. Applied rewrites 85.0%

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

   10. Applied rewrites 85.0%

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

   if -4.60000000000000018e-17 < x1 < 8.2000000000000003e-42

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.6 \cdot 10^{-17} \lor \neg \left(x1 \leq 8.2 \cdot 10^{-42}\right):\\ \;\;\;\;6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 69.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.6 \cdot 10^{-17}:\\ \;\;\;\;6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)\\ \mathbf{elif}\;x1 \leq 8.2 \cdot 10^{-42}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4.6e-17)
   (* 6.0 (* (* x1 x1) (* x1 x1)))
   (if (<= x1 8.2e-42) (* -6.0 x2) (* (* 6.0 (* x1 x1)) (* x1 x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4.6e-17) {
		tmp = 6.0 * ((x1 * x1) * (x1 * x1));
	} else if (x1 <= 8.2e-42) {
		tmp = -6.0 * x2;
	} else {
		tmp = (6.0 * (x1 * x1)) * (x1 * 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 <= (-4.6d-17)) then
        tmp = 6.0d0 * ((x1 * x1) * (x1 * x1))
    else if (x1 <= 8.2d-42) then
        tmp = (-6.0d0) * x2
    else
        tmp = (6.0d0 * (x1 * x1)) * (x1 * x1)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4.6e-17) {
		tmp = 6.0 * ((x1 * x1) * (x1 * x1));
	} else if (x1 <= 8.2e-42) {
		tmp = -6.0 * x2;
	} else {
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if x1 <= -4.6e-17:
		tmp = 6.0 * ((x1 * x1) * (x1 * x1))
	elif x1 <= 8.2e-42:
		tmp = -6.0 * x2
	else:
		tmp = (6.0 * (x1 * x1)) * (x1 * x1)
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4.6e-17)
		tmp = Float64(6.0 * Float64(Float64(x1 * x1) * Float64(x1 * x1)));
	elseif (x1 <= 8.2e-42)
		tmp = Float64(-6.0 * x2);
	else
		tmp = Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if (x1 <= -4.6e-17)
		tmp = 6.0 * ((x1 * x1) * (x1 * x1));
	elseif (x1 <= 8.2e-42)
		tmp = -6.0 * x2;
	else
		tmp = (6.0 * (x1 * x1)) * (x1 * x1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4.6e-17], N[(6.0 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 8.2e-42], N[(-6.0 * x2), $MachinePrecision], N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -4.60000000000000018e-17

   1. Initial program 28.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in x1 around inf

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

   8. Applied rewrites 91.1%

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

   10. Applied rewrites 91.1%

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

   if -4.60000000000000018e-17 < x1 < 8.2000000000000003e-42

   1. Initial program 99.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   if 8.2000000000000003e-42 < x1

   1. Initial program 57.7%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in x1 around inf

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

   8. Applied rewrites 79.8%

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

   10. Applied rewrites 79.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.6 \cdot 10^{-17}:\\ \;\;\;\;6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)\\ \mathbf{elif}\;x1 \leq 8.2 \cdot 10^{-42}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;\left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 26.5% accurate, 33.1× speedup

Math

\[\begin{array}{l} \\ x1 + -6 \cdot x2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

3. Taylor expanded in x1 around inf

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

5. Applied rewrites 68.3%

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

6. Taylor expanded in x1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-neg.f64 68.5

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

8. Applied rewrites 68.5%

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

10. Applied rewrites 66.1%

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

11. Taylor expanded in x1 around 0

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

    1. lower-*.f64 28.3

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

13. Applied rewrites 28.3%

    \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
14. Add Preprocessing

Alternative 20: 26.3% accurate, 49.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 28.2

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

5. Applied rewrites 28.2%

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

Reproduce

herbie shell --seed 2024337 
(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))))))