Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.0% → 99.1%

Time: 19.5s

Alternatives: 26

Speedup: 7.3×

• 47.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.0% 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: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(-1.0 - Float64(x1 * x1))
	t_2 = Float64(fma(x2, 2.0, t_0) - x1)
	t_3 = Float64(t_2 / fma(x1, x1, 1.0))
	t_4 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_5 = Float64(Float64(x1 * x1) - -1.0)
	t_6 = Float64(t_4 / t_5)
	tmp = 0.0
	if (Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 / t_1) * t_0) - Float64(t_1 * Float64(Float64(Float64(3.0 - t_6) * Float64(t_6 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_6) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_5) * 3.0))) <= Inf)
		tmp = Float64(fma(Float64(Float64(fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, fma(fma(fma(4.0, t_3, -6.0), Float64(x1 * x1), Float64(Float64(t_3 * Float64(2.0 * x1)) * Float64(t_3 - 3.0))), fma(x1, x1, 1.0), fma(Float64(Float64(t_2 * x1) / fma(x1, x1, 1.0)), Float64(3.0 * x1), Float64((x1 ^ 3.0) + x1)))) + x1);
	else
		tmp = Float64(Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / 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[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x2 * 2.0 + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$5), $MachinePrecision]}, If[LessEqual[N[(x1 - N[(N[(N[(N[(N[(N[(t$95$4 / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$1 * N[(N[(N[(3.0 - t$95$6), $MachinePrecision] * N[(t$95$6 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$6), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(4.0 * t$95$3 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$3 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(t$95$2 * x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(3.0 * x1), $MachinePrecision] + N[(N[Power[x1, 3.0], $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $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 99.4%

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 99.3%

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

Alternative 2: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(9, x1, -1\right), x1, -6 \cdot x2\right)\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{t\_3}{t\_4}\\ t_6 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_5\right) \cdot \left(t\_5 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_5 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\right)\\ \mathbf{if}\;t\_6 \leq -5 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (fma (fma 9.0 x1 -1.0) x1 (* -6.0 x2)))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (+ (* x2 2.0) t_0) x1))
        (t_4 (- (* x1 x1) -1.0))
        (t_5 (/ t_3 t_4))
        (t_6
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_3 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_5) (* t_5 (* 2.0 x1)))
                (* (- (* 4.0 t_5) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_4) 3.0)))))
   (if (<= t_6 -5e+246)
     (* (* (* x2 x2) x1) 8.0)
     (if (<= t_6 5e+30)
       t_1
       (if (<= t_6 INFINITY) (fma (* (* x2 x2) 8.0) x1 (* -6.0 x2)) t_1)))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = fma(fma(9.0, x1, -1.0), x1, (-6.0 * x2));
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = t_3 / t_4;
	double t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	double tmp;
	if (t_6 <= -5e+246) {
		tmp = ((x2 * x2) * x1) * 8.0;
	} else if (t_6 <= 5e+30) {
		tmp = t_1;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = fma(((x2 * x2) * 8.0), x1, (-6.0 * x2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = fma(fma(9.0, x1, -1.0), x1, Float64(-6.0 * x2))
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(t_3 / t_4)
	t_6 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_5) * Float64(t_5 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_5) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)))
	tmp = 0.0
	if (t_6 <= -5e+246)
		tmp = Float64(Float64(Float64(x2 * x2) * x1) * 8.0);
	elseif (t_6 <= 5e+30)
		tmp = t_1;
	elseif (t_6 <= Inf)
		tmp = fma(Float64(Float64(x2 * x2) * 8.0), x1, Float64(-6.0 * x2));
	else
		tmp = t_1;
	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[(9.0 * x1 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$5), $MachinePrecision] * N[(t$95$5 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -5e+246], N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision], If[LessEqual[t$95$6, 5e+30], t$95$1, If[LessEqual[t$95$6, Infinity], N[(N[(N[(x2 * x2), $MachinePrecision] * 8.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.99999999999999976e246

   1. Initial program 99.9%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 7.2

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

   5. Applied rewrites 7.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 85.9%

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

   if -4.99999999999999976e246 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 4.9999999999999998e30 or +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 56.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 88.2%

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

   if 4.9999999999999998e30 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 18.3

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

   5. Applied rewrites 18.3%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 47.0%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 45.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq -5 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(x2 \cdot x2\right) \cdot 8, x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -1\right), x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(9, x1, -1\right), x1, -6 \cdot x2\right)\\ t_3 := -1 - x1 \cdot x1\\ t_4 := \left(x2 \cdot 2 + t\_1\right) - x1\\ t_5 := x1 \cdot x1 - -1\\ t_6 := \frac{t\_4}{t\_5}\\ t_7 := x1 - \left(\left(\left(\left(\frac{t\_4}{t\_3} \cdot t\_1 - t\_3 \cdot \left(\left(3 - t\_6\right) \cdot \left(t\_6 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_6 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_1 - x2 \cdot 2\right) - x1}{t\_5} \cdot 3\right)\\ \mathbf{if}\;t\_7 \leq -5 \cdot 10^{+246}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_7 \leq 2 \cdot 10^{+200}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* (* x2 x2) x1) 8.0))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (fma (fma 9.0 x1 -1.0) x1 (* -6.0 x2)))
        (t_3 (- -1.0 (* x1 x1)))
        (t_4 (- (+ (* x2 2.0) t_1) x1))
        (t_5 (- (* x1 x1) -1.0))
        (t_6 (/ t_4 t_5))
        (t_7
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_4 t_3) t_1)
              (*
               t_3
               (-
                (* (- 3.0 t_6) (* t_6 (* 2.0 x1)))
                (* (- (* 4.0 t_6) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_1 (* x2 2.0)) x1) t_5) 3.0)))))
   (if (<= t_7 -5e+246)
     t_0
     (if (<= t_7 2e+200) t_2 (if (<= t_7 INFINITY) t_0 t_2)))))

C

double code(double x1, double x2) {
	double t_0 = ((x2 * x2) * x1) * 8.0;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = fma(fma(9.0, x1, -1.0), x1, (-6.0 * x2));
	double t_3 = -1.0 - (x1 * x1);
	double t_4 = ((x2 * 2.0) + t_1) - x1;
	double t_5 = (x1 * x1) - -1.0;
	double t_6 = t_4 / t_5;
	double t_7 = x1 - ((((((t_4 / t_3) * t_1) - (t_3 * (((3.0 - t_6) * (t_6 * (2.0 * x1))) - (((4.0 * t_6) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_1 - (x2 * 2.0)) - x1) / t_5) * 3.0));
	double tmp;
	if (t_7 <= -5e+246) {
		tmp = t_0;
	} else if (t_7 <= 2e+200) {
		tmp = t_2;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(x2 * x2) * x1) * 8.0)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = fma(fma(9.0, x1, -1.0), x1, Float64(-6.0 * x2))
	t_3 = Float64(-1.0 - Float64(x1 * x1))
	t_4 = Float64(Float64(Float64(x2 * 2.0) + t_1) - x1)
	t_5 = Float64(Float64(x1 * x1) - -1.0)
	t_6 = Float64(t_4 / t_5)
	t_7 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_4 / t_3) * t_1) - Float64(t_3 * Float64(Float64(Float64(3.0 - t_6) * Float64(t_6 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_6) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_1 - Float64(x2 * 2.0)) - x1) / t_5) * 3.0)))
	tmp = 0.0
	if (t_7 <= -5e+246)
		tmp = t_0;
	elseif (t_7 <= 2e+200)
		tmp = t_2;
	elseif (t_7 <= Inf)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$1), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$4 / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$3 * N[(N[(N[(3.0 - t$95$6), $MachinePrecision] * N[(t$95$6 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$6), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$1 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, -5e+246], t$95$0, If[LessEqual[t$95$7, 2e+200], t$95$2, If[LessEqual[t$95$7, Infinity], t$95$0, t$95$2]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.99999999999999976e246 or 1.9999999999999999e200 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 53.0%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 52.1%

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

   if -4.99999999999999976e246 < (+.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)))))) < 1.9999999999999999e200 or +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 63.4%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 39.2

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

   5. Applied rewrites 39.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 77.9%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 82.3%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq -5 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(9, x1, -1\right), x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := \left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ t_2 := -1 - x1 \cdot x1\\ t_3 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_4 := x1 \cdot x1 - -1\\ t_5 := \frac{t\_3}{t\_4}\\ t_6 := x1 - \left(\left(\left(\left(\frac{t\_3}{t\_2} \cdot t\_0 - t\_2 \cdot \left(\left(3 - t\_5\right) \cdot \left(t\_5 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_5 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_4} \cdot 3\right)\\ \mathbf{if}\;t\_6 \leq -5 \cdot 10^{+246}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_6 \leq 2 \cdot 10^{+197}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (* (* (* x2 x2) x1) 8.0))
        (t_2 (- -1.0 (* x1 x1)))
        (t_3 (- (+ (* x2 2.0) t_0) x1))
        (t_4 (- (* x1 x1) -1.0))
        (t_5 (/ t_3 t_4))
        (t_6
         (-
          x1
          (-
           (-
            (-
             (-
              (* (/ t_3 t_2) t_0)
              (*
               t_2
               (-
                (* (- 3.0 t_5) (* t_5 (* 2.0 x1)))
                (* (- (* 4.0 t_5) 6.0) (* x1 x1)))))
             (* (* x1 x1) x1))
            x1)
           (* (/ (- (- t_0 (* x2 2.0)) x1) t_4) 3.0)))))
   (if (<= t_6 -5e+246)
     t_1
     (if (<= t_6 2e+197)
       (* -6.0 x2)
       (if (<= t_6 INFINITY) t_1 (* 9.0 (* x1 x1)))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = ((x2 * x2) * x1) * 8.0;
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = t_3 / t_4;
	double t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	double tmp;
	if (t_6 <= -5e+246) {
		tmp = t_1;
	} else if (t_6 <= 2e+197) {
		tmp = -6.0 * x2;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 9.0 * (x1 * x1);
	}
	return tmp;
}

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = ((x2 * x2) * x1) * 8.0;
	double t_2 = -1.0 - (x1 * x1);
	double t_3 = ((x2 * 2.0) + t_0) - x1;
	double t_4 = (x1 * x1) - -1.0;
	double t_5 = t_3 / t_4;
	double t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	double tmp;
	if (t_6 <= -5e+246) {
		tmp = t_1;
	} else if (t_6 <= 2e+197) {
		tmp = -6.0 * x2;
	} else if (t_6 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 9.0 * (x1 * x1);
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (3.0 * x1) * x1
	t_1 = ((x2 * x2) * x1) * 8.0
	t_2 = -1.0 - (x1 * x1)
	t_3 = ((x2 * 2.0) + t_0) - x1
	t_4 = (x1 * x1) - -1.0
	t_5 = t_3 / t_4
	t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0))
	tmp = 0
	if t_6 <= -5e+246:
		tmp = t_1
	elif t_6 <= 2e+197:
		tmp = -6.0 * x2
	elif t_6 <= math.inf:
		tmp = t_1
	else:
		tmp = 9.0 * (x1 * x1)
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(Float64(x2 * x2) * x1) * 8.0)
	t_2 = Float64(-1.0 - Float64(x1 * x1))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(t_3 / t_4)
	t_6 = Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_2) * t_0) - Float64(t_2 * Float64(Float64(Float64(3.0 - t_5) * Float64(t_5 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_5) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0)))
	tmp = 0.0
	if (t_6 <= -5e+246)
		tmp = t_1;
	elseif (t_6 <= 2e+197)
		tmp = Float64(-6.0 * x2);
	elseif (t_6 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(9.0 * Float64(x1 * x1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (3.0 * x1) * x1;
	t_1 = ((x2 * x2) * x1) * 8.0;
	t_2 = -1.0 - (x1 * x1);
	t_3 = ((x2 * 2.0) + t_0) - x1;
	t_4 = (x1 * x1) - -1.0;
	t_5 = t_3 / t_4;
	t_6 = x1 - ((((((t_3 / t_2) * t_0) - (t_2 * (((3.0 - t_5) * (t_5 * (2.0 * x1))) - (((4.0 * t_5) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_4) * 3.0));
	tmp = 0.0;
	if (t_6 <= -5e+246)
		tmp = t_1;
	elseif (t_6 <= 2e+197)
		tmp = -6.0 * x2;
	elseif (t_6 <= Inf)
		tmp = t_1;
	else
		tmp = 9.0 * (x1 * x1);
	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[(N[(x2 * x2), $MachinePrecision] * x1), $MachinePrecision] * 8.0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$2 * N[(N[(N[(3.0 - t$95$5), $MachinePrecision] * N[(t$95$5 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -5e+246], t$95$1, If[LessEqual[t$95$6, 2e+197], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[t$95$6, Infinity], t$95$1, N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.99999999999999976e246 or 1.9999999999999999e197 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.5

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

   5. Applied rewrites 3.5%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 52.4%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 51.5%

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

   if -4.99999999999999976e246 < (+.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)))))) < 1.9999999999999999e197

   1. Initial program 99.2%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 71.0%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 84.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 84.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq -5 \cdot 10^{+246}:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 2 \cdot 10^{+197}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq \infty:\\ \;\;\;\;\left(\left(x2 \cdot x2\right) \cdot x1\right) \cdot 8\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(-1.0 - Float64(x1 * x1))
	t_2 = Float64(Float64(fma(x2, 2.0, t_0) - x1) / fma(x1, x1, 1.0))
	t_3 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_4 = Float64(Float64(x1 * x1) - -1.0)
	t_5 = Float64(t_3 / t_4)
	tmp = 0.0
	if (Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_3 / t_1) * t_0) - Float64(t_1 * Float64(Float64(Float64(3.0 - t_5) * Float64(t_5 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_5) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_4) * 3.0))) <= Inf)
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(fma(Float64(Float64(fma(-2.0, x2, t_0) - x1) / fma(x1, x1, 1.0)), 3.0, x1) + fma(fma(fma(4.0, t_2, -6.0), Float64(x1 * x1), Float64(Float64(t_2 * Float64(2.0 * x1)) * Float64(t_2 - 3.0))), fma(x1, x1, 1.0), Float64(t_2 * t_0)))) + x1);
	else
		tmp = Float64(Float64((x1 ^ 4.0) * Float64(6.0 - Float64(3.0 / 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[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x2 * 2.0 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, If[LessEqual[N[(x1 - N[(N[(N[(N[(N[(N[(t$95$3 / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$1 * N[(N[(N[(3.0 - t$95$5), $MachinePrecision] * N[(t$95$5 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(N[(-2.0 * x2 + t$95$0), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + x1), $MachinePrecision] + N[(N[(N[(4.0 * t$95$2 + -6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision] + N[(N[(t$95$2 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $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 99.4%

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

   4. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 99.3%

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

Alternative 6: 51.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := -1 - x1 \cdot x1\\ t_2 := \left(x2 \cdot 2 + t\_0\right) - x1\\ t_3 := x1 \cdot x1 - -1\\ t_4 := \frac{t\_2}{t\_3}\\ \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{t\_2}{t\_1} \cdot t\_0 - t\_1 \cdot \left(\left(3 - t\_4\right) \cdot \left(t\_4 \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot t\_4 - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(t\_0 - x2 \cdot 2\right) - x1}{t\_3} \cdot 3\right) \leq 2 \cdot 10^{+197}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (- -1.0 (* x1 x1)))
        (t_2 (- (+ (* x2 2.0) t_0) x1))
        (t_3 (- (* x1 x1) -1.0))
        (t_4 (/ t_2 t_3)))
   (if (<=
        (-
         x1
         (-
          (-
           (-
            (-
             (* (/ t_2 t_1) t_0)
             (*
              t_1
              (-
               (* (- 3.0 t_4) (* t_4 (* 2.0 x1)))
               (* (- (* 4.0 t_4) 6.0) (* x1 x1)))))
            (* (* x1 x1) x1))
           x1)
          (* (/ (- (- t_0 (* x2 2.0)) x1) t_3) 3.0)))
        2e+197)
     (* -6.0 x2)
     (* 9.0 (* x1 x1)))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = -1.0 - (x1 * x1);
	double t_2 = ((x2 * 2.0) + t_0) - x1;
	double t_3 = (x1 * x1) - -1.0;
	double t_4 = t_2 / t_3;
	double tmp;
	if ((x1 - ((((((t_2 / t_1) * t_0) - (t_1 * (((3.0 - t_4) * (t_4 * (2.0 * x1))) - (((4.0 * t_4) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_3) * 3.0))) <= 2e+197) {
		tmp = -6.0 * x2;
	} else {
		tmp = 9.0 * (x1 * x1);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (3.0d0 * x1) * x1
    t_1 = (-1.0d0) - (x1 * x1)
    t_2 = ((x2 * 2.0d0) + t_0) - x1
    t_3 = (x1 * x1) - (-1.0d0)
    t_4 = t_2 / t_3
    if ((x1 - ((((((t_2 / t_1) * t_0) - (t_1 * (((3.0d0 - t_4) * (t_4 * (2.0d0 * x1))) - (((4.0d0 * t_4) - 6.0d0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0d0)) - x1) / t_3) * 3.0d0))) <= 2d+197) then
        tmp = (-6.0d0) * x2
    else
        tmp = 9.0d0 * (x1 * x1)
    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 = -1.0 - (x1 * x1);
	double t_2 = ((x2 * 2.0) + t_0) - x1;
	double t_3 = (x1 * x1) - -1.0;
	double t_4 = t_2 / t_3;
	double tmp;
	if ((x1 - ((((((t_2 / t_1) * t_0) - (t_1 * (((3.0 - t_4) * (t_4 * (2.0 * x1))) - (((4.0 * t_4) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_3) * 3.0))) <= 2e+197) {
		tmp = -6.0 * x2;
	} else {
		tmp = 9.0 * (x1 * x1);
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (3.0 * x1) * x1
	t_1 = -1.0 - (x1 * x1)
	t_2 = ((x2 * 2.0) + t_0) - x1
	t_3 = (x1 * x1) - -1.0
	t_4 = t_2 / t_3
	tmp = 0
	if (x1 - ((((((t_2 / t_1) * t_0) - (t_1 * (((3.0 - t_4) * (t_4 * (2.0 * x1))) - (((4.0 * t_4) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_3) * 3.0))) <= 2e+197:
		tmp = -6.0 * x2
	else:
		tmp = 9.0 * (x1 * x1)
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(-1.0 - Float64(x1 * x1))
	t_2 = Float64(Float64(Float64(x2 * 2.0) + t_0) - x1)
	t_3 = Float64(Float64(x1 * x1) - -1.0)
	t_4 = Float64(t_2 / t_3)
	tmp = 0.0
	if (Float64(x1 - Float64(Float64(Float64(Float64(Float64(Float64(t_2 / t_1) * t_0) - Float64(t_1 * Float64(Float64(Float64(3.0 - t_4) * Float64(t_4 * Float64(2.0 * x1))) - Float64(Float64(Float64(4.0 * t_4) - 6.0) * Float64(x1 * x1))))) - Float64(Float64(x1 * x1) * x1)) - x1) - Float64(Float64(Float64(Float64(t_0 - Float64(x2 * 2.0)) - x1) / t_3) * 3.0))) <= 2e+197)
		tmp = Float64(-6.0 * x2);
	else
		tmp = Float64(9.0 * Float64(x1 * x1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (3.0 * x1) * x1;
	t_1 = -1.0 - (x1 * x1);
	t_2 = ((x2 * 2.0) + t_0) - x1;
	t_3 = (x1 * x1) - -1.0;
	t_4 = t_2 / t_3;
	tmp = 0.0;
	if ((x1 - ((((((t_2 / t_1) * t_0) - (t_1 * (((3.0 - t_4) * (t_4 * (2.0 * x1))) - (((4.0 * t_4) - 6.0) * (x1 * x1))))) - ((x1 * x1) * x1)) - x1) - ((((t_0 - (x2 * 2.0)) - x1) / t_3) * 3.0))) <= 2e+197)
		tmp = -6.0 * x2;
	else
		tmp = 9.0 * (x1 * x1);
	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[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x2 * 2.0), $MachinePrecision] + t$95$0), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / t$95$3), $MachinePrecision]}, If[LessEqual[N[(x1 - N[(N[(N[(N[(N[(N[(t$95$2 / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$1 * N[(N[(N[(3.0 - t$95$4), $MachinePrecision] * N[(t$95$4 * N[(2.0 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * t$95$4), $MachinePrecision] - 6.0), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] - N[(N[(N[(N[(t$95$0 - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$3), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+197], N[(-6.0 * x2), $MachinePrecision], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.9999999999999999e197

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if 1.9999999999999999e197 < (+.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 48.2%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 56.6%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 45.6%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 45.7%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 - \left(\left(\left(\left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{-1 - x1 \cdot x1} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right) - \left(-1 - x1 \cdot x1\right) \cdot \left(\left(3 - \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1}\right) \cdot \left(\frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} \cdot \left(2 \cdot x1\right)\right) - \left(4 \cdot \frac{\left(x2 \cdot 2 + \left(3 \cdot x1\right) \cdot x1\right) - x1}{x1 \cdot x1 - -1} - 6\right) \cdot \left(x1 \cdot x1\right)\right)\right) - \left(x1 \cdot x1\right) \cdot x1\right) - x1\right) - \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) \leq 2 \cdot 10^{+197}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\\ \mathbf{if}\;x1 \leq -360000000000:\\ \;\;\;\;\left(6 - \frac{3 - \frac{t\_0}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, -3\right) \cdot -6 + -1}{x1}, -1, t\_0\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (fma 2.0 x2 -3.0) 4.0 9.0)))
   (if (<= x1 -360000000000.0)
     (* (- 6.0 (/ (- 3.0 (/ t_0 x1)) x1)) (pow x1 4.0))
     (if (<= x1 11.0)
       (fma
        (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
        x2
        (* (fma 9.0 x1 -1.0) x1))
       (*
        (-
         6.0
         (/
          (-
           3.0
           (/ (fma (/ (+ (* (fma 2.0 x2 -3.0) -6.0) -1.0) x1) -1.0 t_0) x1))
          x1))
        (pow x1 4.0))))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(2.0, x2, -3.0), 4.0, 9.0);
	double tmp;
	if (x1 <= -360000000000.0) {
		tmp = (6.0 - ((3.0 - (t_0 / x1)) / x1)) * pow(x1, 4.0);
	} else if (x1 <= 11.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = (6.0 - ((3.0 - (fma((((fma(2.0, x2, -3.0) * -6.0) + -1.0) / x1), -1.0, t_0) / x1)) / x1)) * pow(x1, 4.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(fma(2.0, x2, -3.0), 4.0, 9.0)
	tmp = 0.0
	if (x1 <= -360000000000.0)
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(t_0 / x1)) / x1)) * (x1 ^ 4.0));
	elseif (x1 <= 11.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(Float64(Float64(Float64(fma(2.0, x2, -3.0) * -6.0) + -1.0) / x1), -1.0, t_0) / x1)) / x1)) * (x1 ^ 4.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]}, If[LessEqual[x1, -360000000000.0], N[(N[(6.0 - N[(N[(3.0 - N[(t$95$0 / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 11.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * -6.0), $MachinePrecision] + -1.0), $MachinePrecision] / x1), $MachinePrecision] * -1.0 + t$95$0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.6e11

   1. Initial program 37.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.2%

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

   if -3.6e11 < x1 < 11

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.9%

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

   if 11 < x1

   1. Initial program 59.9%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in x1 around -inf

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

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

   8. Applied rewrites 96.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -360000000000:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x2, -3\right) \cdot -6 + -1}{x1}, -1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\\ \mathbf{if}\;x1 \leq -360000000000:\\ \;\;\;\;\left(6 - \frac{3 - \frac{t\_0}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{t\_0 - \frac{\mathsf{fma}\left(2, x2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right) \cdot {x1}^{4} + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (fma 2.0 x2 -3.0) 4.0 9.0)))
   (if (<= x1 -360000000000.0)
     (* (- 6.0 (/ (- 3.0 (/ t_0 x1)) x1)) (pow x1 4.0))
     (if (<= x1 11.0)
       (fma
        (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
        x2
        (* (fma 9.0 x1 -1.0) x1))
       (+
        (*
         (-
          6.0
          (/ (- 3.0 (/ (- t_0 (/ (* (fma 2.0 x2 -3.0) -6.0) x1)) x1)) x1))
         (pow x1 4.0))
        x1)))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(2.0, x2, -3.0), 4.0, 9.0);
	double tmp;
	if (x1 <= -360000000000.0) {
		tmp = (6.0 - ((3.0 - (t_0 / x1)) / x1)) * pow(x1, 4.0);
	} else if (x1 <= 11.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = ((6.0 - ((3.0 - ((t_0 - ((fma(2.0, x2, -3.0) * -6.0) / x1)) / x1)) / x1)) * pow(x1, 4.0)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(fma(2.0, x2, -3.0), 4.0, 9.0)
	tmp = 0.0
	if (x1 <= -360000000000.0)
		tmp = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(t_0 / x1)) / x1)) * (x1 ^ 4.0));
	elseif (x1 <= 11.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = Float64(Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(Float64(t_0 - Float64(Float64(fma(2.0, x2, -3.0) * -6.0) / x1)) / x1)) / x1)) * (x1 ^ 4.0)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]}, If[LessEqual[x1, -360000000000.0], N[(N[(6.0 - N[(N[(3.0 - N[(t$95$0 / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 11.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(6.0 - N[(N[(3.0 - N[(N[(t$95$0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -3.6e11

   1. Initial program 37.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.2%

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

   if -3.6e11 < x1 < 11

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.9%

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

   if 11 < x1

   1. Initial program 59.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -360000000000:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right) - \frac{\mathsf{fma}\left(2, x2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right) \cdot {x1}^{4} + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right)\\ t_1 := \left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ \mathbf{if}\;x1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -360000000000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, t\_0 + \mathsf{fma}\left(t\_1, \mathsf{fma}\left(x1, x1, 1\right), 3 \cdot t\_2\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(t\_1, \mathsf{fma}\left(x1, x1, 1\right), \left(3 - \frac{1 - \frac{\mathsf{fma}\left(2, x2, -3\right)}{x1}}{x1}\right) \cdot t\_2\right) + t\_0\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (fma -2.0 x2 (- x1)) 3.0 x1))
        (t_1
         (*
          (* (- 6.0 (/ (- 4.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 -6.0) x1)) x1)) x1)
          x1))
        (t_2 (* (* 3.0 x1) x1)))
   (if (<= x1 -5e+154)
     (* 9.0 (* x1 x1))
     (if (<= x1 -360000000000.0)
       (+ (fma (* x1 x1) x1 (+ t_0 (fma t_1 (fma x1 x1 1.0) (* 3.0 t_2)))) x1)
       (if (<= x1 11.0)
         (fma
          (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
          x2
          (* (fma 9.0 x1 -1.0) x1))
         (+
          (fma
           (* x1 x1)
           x1
           (+
            (fma
             t_1
             (fma x1 x1 1.0)
             (* (- 3.0 (/ (- 1.0 (/ (fma 2.0 x2 -3.0) x1)) x1)) t_2))
            t_0))
          x1))))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(-2.0, x2, -x1), 3.0, x1);
	double t_1 = ((6.0 - ((4.0 - (fma(fma(2.0, x2, -3.0), 4.0, -6.0) / x1)) / x1)) * x1) * x1;
	double t_2 = (3.0 * x1) * x1;
	double tmp;
	if (x1 <= -5e+154) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= -360000000000.0) {
		tmp = fma((x1 * x1), x1, (t_0 + fma(t_1, fma(x1, x1, 1.0), (3.0 * t_2)))) + x1;
	} else if (x1 <= 11.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = fma((x1 * x1), x1, (fma(t_1, fma(x1, x1, 1.0), ((3.0 - ((1.0 - (fma(2.0, x2, -3.0) / x1)) / x1)) * t_2)) + t_0)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(fma(-2.0, x2, Float64(-x1)), 3.0, x1)
	t_1 = Float64(Float64(Float64(6.0 - Float64(Float64(4.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, -6.0) / x1)) / x1)) * x1) * x1)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	tmp = 0.0
	if (x1 <= -5e+154)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= -360000000000.0)
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(t_0 + fma(t_1, fma(x1, x1, 1.0), Float64(3.0 * t_2)))) + x1);
	elseif (x1 <= 11.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(fma(t_1, fma(x1, x1, 1.0), Float64(Float64(3.0 - Float64(Float64(1.0 - Float64(fma(2.0, x2, -3.0) / x1)) / x1)) * t_2)) + t_0)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(-2.0 * x2 + (-x1)), $MachinePrecision] * 3.0 + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(6.0 - N[(N[(4.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, If[LessEqual[x1, -5e+154], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -360000000000.0], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(t$95$0 + N[(t$95$1 * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(3.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 11.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(t$95$1 * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(3.0 - N[(N[(1.0 - N[(N[(2.0 * x2 + -3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -5.00000000000000004e154

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -5.00000000000000004e154 < x1 < -3.6e11

   1. Initial program 68.3%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 96.7

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

   10. Applied rewrites 96.7%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 96.7%

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

   if -3.6e11 < x1 < 11

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.9%

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

   if 11 < x1

   1. Initial program 59.9%

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 56.6%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 56.6

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

   10. Applied rewrites 56.6%

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

   11. Taylor expanded in x1 around -inf

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. mul-1-neg N/A

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

       6. unsub-neg N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. sub-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. metadata-eval 96.3

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

   13. Applied rewrites 96.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -360000000000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right) + \mathsf{fma}\left(\left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), 3 \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \left(3 - \frac{1 - \frac{\mathsf{fma}\left(2, x2, -3\right)}{x1}}{x1}\right) \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right)\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{if}\;x1 \leq -360000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (- 6.0 (/ (- 3.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 9.0) x1)) x1))
          (pow x1 4.0))))
   (if (<= x1 -360000000000.0)
     t_0
     (if (<= x1 11.0)
       (fma
        (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
        x2
        (* (fma 9.0 x1 -1.0) x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * pow(x1, 4.0);
	double tmp;
	if (x1 <= -360000000000.0) {
		tmp = t_0;
	} else if (x1 <= 11.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, 9.0) / x1)) / x1)) * (x1 ^ 4.0))
	tmp = 0.0
	if (x1 <= -360000000000.0)
		tmp = t_0;
	elseif (x1 <= 11.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[Power[x1, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -360000000000.0], t$95$0, If[LessEqual[x1, 11.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -3.6e11 or 11 < x1

   1. Initial program 49.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.2%

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

   if -3.6e11 < x1 < 11

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -360000000000:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)}{x1}}{x1}\right) \cdot {x1}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right)\\ t_1 := \left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ \mathbf{if}\;x1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -360000000000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, t\_0 + \mathsf{fma}\left(t\_1, \mathsf{fma}\left(x1, x1, 1\right), 3 \cdot t\_2\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(t\_1, \mathsf{fma}\left(x1, x1, 1\right), \left(3 - \frac{1}{x1}\right) \cdot t\_2\right) + t\_0\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (fma -2.0 x2 (- x1)) 3.0 x1))
        (t_1
         (*
          (* (- 6.0 (/ (- 4.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 -6.0) x1)) x1)) x1)
          x1))
        (t_2 (* (* 3.0 x1) x1)))
   (if (<= x1 -5e+154)
     (* 9.0 (* x1 x1))
     (if (<= x1 -360000000000.0)
       (+ (fma (* x1 x1) x1 (+ t_0 (fma t_1 (fma x1 x1 1.0) (* 3.0 t_2)))) x1)
       (if (<= x1 11.0)
         (fma
          (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
          x2
          (* (fma 9.0 x1 -1.0) x1))
         (+
          (fma
           (* x1 x1)
           x1
           (+ (fma t_1 (fma x1 x1 1.0) (* (- 3.0 (/ 1.0 x1)) t_2)) t_0))
          x1))))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(-2.0, x2, -x1), 3.0, x1);
	double t_1 = ((6.0 - ((4.0 - (fma(fma(2.0, x2, -3.0), 4.0, -6.0) / x1)) / x1)) * x1) * x1;
	double t_2 = (3.0 * x1) * x1;
	double tmp;
	if (x1 <= -5e+154) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= -360000000000.0) {
		tmp = fma((x1 * x1), x1, (t_0 + fma(t_1, fma(x1, x1, 1.0), (3.0 * t_2)))) + x1;
	} else if (x1 <= 11.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = fma((x1 * x1), x1, (fma(t_1, fma(x1, x1, 1.0), ((3.0 - (1.0 / x1)) * t_2)) + t_0)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(fma(-2.0, x2, Float64(-x1)), 3.0, x1)
	t_1 = Float64(Float64(Float64(6.0 - Float64(Float64(4.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, -6.0) / x1)) / x1)) * x1) * x1)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	tmp = 0.0
	if (x1 <= -5e+154)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= -360000000000.0)
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(t_0 + fma(t_1, fma(x1, x1, 1.0), Float64(3.0 * t_2)))) + x1);
	elseif (x1 <= 11.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(fma(t_1, fma(x1, x1, 1.0), Float64(Float64(3.0 - Float64(1.0 / x1)) * t_2)) + t_0)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(-2.0 * x2 + (-x1)), $MachinePrecision] * 3.0 + x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(6.0 - N[(N[(4.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, If[LessEqual[x1, -5e+154], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -360000000000.0], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(t$95$0 + N[(t$95$1 * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(3.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 11.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(t$95$1 * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(3.0 - N[(1.0 / x1), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -5.00000000000000004e154

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -5.00000000000000004e154 < x1 < -3.6e11

   1. Initial program 68.3%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 96.7

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

   10. Applied rewrites 96.7%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 96.7%

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

   if -3.6e11 < x1 < 11

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.9%

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

   if 11 < x1

   1. Initial program 59.9%

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 56.6%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 56.6

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

   10. Applied rewrites 56.6%

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

   11. Taylor expanded in x1 around inf

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

       1. lower--.f64 N/A

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

       2. lower-/.f64 96.3

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

   13. Applied rewrites 96.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -360000000000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right) + \mathsf{fma}\left(\left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), 3 \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \left(3 - \frac{1}{x1}\right) \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right)\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 9 \cdot \left(x1 \cdot x1\right)\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\left(6 \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \frac{\mathsf{fma}\left(x2, 2, t\_1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot t\_1\right) + \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right)\right) + x1\\ \mathbf{if}\;x1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -58000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x1 \leq 12:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 9.0 (* x1 x1)))
        (t_1 (* (* 3.0 x1) x1))
        (t_2
         (+
          (fma
           (* x1 x1)
           x1
           (+
            (fma
             (* (* 6.0 x1) x1)
             (fma x1 x1 1.0)
             (* (/ (- (fma x2 2.0 t_1) x1) (fma x1 x1 1.0)) t_1))
            (fma (fma -2.0 x2 (- x1)) 3.0 x1)))
          x1)))
   (if (<= x1 -5e+154)
     t_0
     (if (<= x1 -58000000000000.0)
       t_2
       (if (<= x1 12.0)
         (fma
          (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
          x2
          (* (fma 9.0 x1 -1.0) x1))
         (if (<= x1 1e+153) t_2 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = 9.0 * (x1 * x1);
	double t_1 = (3.0 * x1) * x1;
	double t_2 = fma((x1 * x1), x1, (fma(((6.0 * x1) * x1), fma(x1, x1, 1.0), (((fma(x2, 2.0, t_1) - x1) / fma(x1, x1, 1.0)) * t_1)) + fma(fma(-2.0, x2, -x1), 3.0, x1))) + x1;
	double tmp;
	if (x1 <= -5e+154) {
		tmp = t_0;
	} else if (x1 <= -58000000000000.0) {
		tmp = t_2;
	} else if (x1 <= 12.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else if (x1 <= 1e+153) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(9.0 * Float64(x1 * x1))
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(fma(Float64(x1 * x1), x1, Float64(fma(Float64(Float64(6.0 * x1) * x1), fma(x1, x1, 1.0), Float64(Float64(Float64(fma(x2, 2.0, t_1) - x1) / fma(x1, x1, 1.0)) * t_1)) + fma(fma(-2.0, x2, Float64(-x1)), 3.0, x1))) + x1)
	tmp = 0.0
	if (x1 <= -5e+154)
		tmp = t_0;
	elseif (x1 <= -58000000000000.0)
		tmp = t_2;
	elseif (x1 <= 12.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	elseif (x1 <= 1e+153)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(6.0 * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(N[(N[(N[(x2 * 2.0 + t$95$1), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * x2 + (-x1)), $MachinePrecision] * 3.0 + x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -5e+154], t$95$0, If[LessEqual[x1, -58000000000000.0], t$95$2, If[LessEqual[x1, 12.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1e+153], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5.00000000000000004e154 or 1e153 < x1

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -5.00000000000000004e154 < x1 < -5.8e13 or 12 < x1 < 1e153

   1. Initial program 85.7%

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 95.1

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

   10. Applied rewrites 95.1%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 82.7%

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

   if -5.8e13 < x1 < 12

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -58000000000000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\left(6 \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 12:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\left(6 \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), \frac{\mathsf{fma}\left(x2, 2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)} \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right) + \mathsf{fma}\left(\left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), 3 \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)\right) + x1\\ \mathbf{if}\;x1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -360000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (fma
           (* x1 x1)
           x1
           (+
            (fma (fma -2.0 x2 (- x1)) 3.0 x1)
            (fma
             (*
              (*
               (- 6.0 (/ (- 4.0 (/ (fma (fma 2.0 x2 -3.0) 4.0 -6.0) x1)) x1))
               x1)
              x1)
             (fma x1 x1 1.0)
             (* 3.0 (* (* 3.0 x1) x1)))))
          x1)))
   (if (<= x1 -5e+154)
     (* 9.0 (* x1 x1))
     (if (<= x1 -360000000000.0)
       t_0
       (if (<= x1 11.0)
         (fma
          (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
          x2
          (* (fma 9.0 x1 -1.0) x1))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = fma((x1 * x1), x1, (fma(fma(-2.0, x2, -x1), 3.0, x1) + fma((((6.0 - ((4.0 - (fma(fma(2.0, x2, -3.0), 4.0, -6.0) / x1)) / x1)) * x1) * x1), fma(x1, x1, 1.0), (3.0 * ((3.0 * x1) * x1))))) + x1;
	double tmp;
	if (x1 <= -5e+154) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= -360000000000.0) {
		tmp = t_0;
	} else if (x1 <= 11.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(Float64(x1 * x1), x1, Float64(fma(fma(-2.0, x2, Float64(-x1)), 3.0, x1) + fma(Float64(Float64(Float64(6.0 - Float64(Float64(4.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, -6.0) / x1)) / x1)) * x1) * x1), fma(x1, x1, 1.0), Float64(3.0 * Float64(Float64(3.0 * x1) * x1))))) + x1)
	tmp = 0.0
	if (x1 <= -5e+154)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= -360000000000.0)
		tmp = t_0;
	elseif (x1 <= 11.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(-2.0 * x2 + (-x1)), $MachinePrecision] * 3.0 + x1), $MachinePrecision] + N[(N[(N[(N[(6.0 - N[(N[(4.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x1), $MachinePrecision] * N[(x1 * x1 + 1.0), $MachinePrecision] + N[(3.0 * N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -5e+154], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -360000000000.0], t$95$0, If[LessEqual[x1, 11.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -5.00000000000000004e154

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -5.00000000000000004e154 < x1 < -3.6e11 or 11 < x1

   1. Initial program 62.6%

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in x1 around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in x1 around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 69.4

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

   10. Applied rewrites 69.4%

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

   11. Taylor expanded in x1 around inf

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

   13. Applied rewrites 96.4%

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

   if -3.6e11 < x1 < 11

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -360000000000:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right) + \mathsf{fma}\left(\left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), 3 \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)\right) + x1\\ \mathbf{elif}\;x1 \leq 11:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(-2, x2, -x1\right), 3, x1\right) + \mathsf{fma}\left(\left(\left(6 - \frac{4 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, -6\right)}{x1}}{x1}\right) \cdot x1\right) \cdot x1, \mathsf{fma}\left(x1, x1, 1\right), 3 \cdot \left(\left(3 \cdot x1\right) \cdot x1\right)\right)\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 93.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x1}^{4} \cdot 6\\ \mathbf{if}\;x1 \leq -58000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 12:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (pow x1 4.0) 6.0)))
   (if (<= x1 -58000000000000.0)
     t_0
     (if (<= x1 12.0)
       (fma
        (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
        x2
        (* (fma 9.0 x1 -1.0) x1))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = pow(x1, 4.0) * 6.0;
	double tmp;
	if (x1 <= -58000000000000.0) {
		tmp = t_0;
	} else if (x1 <= 12.0) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64((x1 ^ 4.0) * 6.0)
	tmp = 0.0
	if (x1 <= -58000000000000.0)
		tmp = t_0;
	elseif (x1 <= 12.0)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[Power[x1, 4.0], $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[x1, -58000000000000.0], t$95$0, If[LessEqual[x1, 12.0], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -5.8e13 or 12 < x1

   1. Initial program 49.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 89.9

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

   8. Applied rewrites 89.9%

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

   if -5.8e13 < x1 < 12

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 97.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -58000000000000:\\ \;\;\;\;{x1}^{4} \cdot 6\\ \mathbf{elif}\;x1 \leq 12:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot 6\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.4 \cdot 10^{+153}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -7.5 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(6, x1, -12\right) \cdot x2\right) \cdot x1 + x1\right) - \frac{\mathsf{fma}\left(x1, \frac{\mathsf{fma}\left(3, x1, -1\right)}{x2}, -2\right) \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4.4e+153)
   (* 9.0 (* x1 x1))
   (if (<= x1 -7.5e+80)
     (+
      (-
       (+ (* (* (fma 6.0 x1 -12.0) x2) x1) x1)
       (*
        (/ (* (fma x1 (/ (fma 3.0 x1 -1.0) x2) -2.0) x2) (- -1.0 (* x1 x1)))
        3.0))
      x1)
     (if (<= x1 0.1)
       (fma
        (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
        x2
        (* (fma 9.0 x1 -1.0) x1))
       (+
        (fma
         (* x1 x1)
         x1
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0) x1 (* -6.0 x2)))
        x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4.4e+153) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= -7.5e+80) {
		tmp = ((((fma(6.0, x1, -12.0) * x2) * x1) + x1) - (((fma(x1, (fma(3.0, x1, -1.0) / x2), -2.0) * x2) / (-1.0 - (x1 * x1))) * 3.0)) + x1;
	} else if (x1 <= 0.1) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = fma((x1 * x1), x1, fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, (-6.0 * x2))) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4.4e+153)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= -7.5e+80)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(6.0, x1, -12.0) * x2) * x1) + x1) - Float64(Float64(Float64(fma(x1, Float64(fma(3.0, x1, -1.0) / x2), -2.0) * x2) / Float64(-1.0 - Float64(x1 * x1))) * 3.0)) + x1);
	elseif (x1 <= 0.1)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, Float64(-6.0 * x2))) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4.4e+153], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -7.5e+80], N[(N[(N[(N[(N[(N[(6.0 * x1 + -12.0), $MachinePrecision] * x2), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision] - N[(N[(N[(N[(x1 * N[(N[(3.0 * x1 + -1.0), $MachinePrecision] / x2), $MachinePrecision] + -2.0), $MachinePrecision] * x2), $MachinePrecision] / N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 0.1], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -4.3999999999999999e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -4.3999999999999999e153 < x1 < -7.49999999999999994e80

   1. Initial program 41.2%

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

   3. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in x2 around 0

      \[\leadsto x1 + \left(\left(\left(x2 \cdot \left(6 \cdot x1 - 12\right)\right) \cdot x1 + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 36.0%

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

   9. Taylor expanded in x2 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 70.9%

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

   if -7.49999999999999994e80 < x1 < 0.10000000000000001

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 89.6%

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

   if 0.10000000000000001 < x1

   1. Initial program 59.9%

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.4 \cdot 10^{+153}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -7.5 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(6, x1, -12\right) \cdot x2\right) \cdot x1 + x1\right) - \frac{\mathsf{fma}\left(x1, \frac{\mathsf{fma}\left(3, x1, -1\right)}{x2}, -2\right) \cdot x2}{-1 - x1 \cdot x1} \cdot 3\right) + x1\\ \mathbf{elif}\;x1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 84.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.05 \cdot 10^{+158}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -5.1 \cdot 10^{+67}:\\ \;\;\;\;\left(x2 \cdot x2\right) \cdot \mathsf{fma}\left(8, x1, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, 12\right), x1, \mathsf{fma}\left(-x1, \frac{\mathsf{fma}\left(9, x1, -1\right)}{x2}, 6\right)\right)}{-x2}\right)\\ \mathbf{elif}\;x1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.05e+158)
   (* 9.0 (* x1 x1))
   (if (<= x1 -5.1e+67)
     (*
      (* x2 x2)
      (fma
       8.0
       x1
       (/
        (fma (fma -12.0 x1 12.0) x1 (fma (- x1) (/ (fma 9.0 x1 -1.0) x2) 6.0))
        (- x2))))
     (if (<= x1 0.1)
       (fma
        (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
        x2
        (* (fma 9.0 x1 -1.0) x1))
       (+
        (fma
         (* x1 x1)
         x1
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0) x1 (* -6.0 x2)))
        x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.05e+158) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= -5.1e+67) {
		tmp = (x2 * x2) * fma(8.0, x1, (fma(fma(-12.0, x1, 12.0), x1, fma(-x1, (fma(9.0, x1, -1.0) / x2), 6.0)) / -x2));
	} else if (x1 <= 0.1) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = fma((x1 * x1), x1, fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, (-6.0 * x2))) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.05e+158)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= -5.1e+67)
		tmp = Float64(Float64(x2 * x2) * fma(8.0, x1, Float64(fma(fma(-12.0, x1, 12.0), x1, fma(Float64(-x1), Float64(fma(9.0, x1, -1.0) / x2), 6.0)) / Float64(-x2))));
	elseif (x1 <= 0.1)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, Float64(-6.0 * x2))) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.05e+158], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -5.1e+67], N[(N[(x2 * x2), $MachinePrecision] * N[(8.0 * x1 + N[(N[(N[(-12.0 * x1 + 12.0), $MachinePrecision] * x1 + N[((-x1) * N[(N[(9.0 * x1 + -1.0), $MachinePrecision] / x2), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision] / (-x2)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 0.1], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.0499999999999999e158

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.1

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -1.0499999999999999e158 < x1 < -5.1000000000000002e67

   1. Initial program 47.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 20.5%

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

   9. Taylor expanded in x2 around -inf

      \[\leadsto {x2}^{2} \cdot \color{blue}{\left(-1 \cdot \frac{6 + \left(-1 \cdot \frac{x1 \cdot \left(9 \cdot x1 - 1\right)}{x2} + x1 \cdot \left(12 + -12 \cdot x1\right)\right)}{x2} + 8 \cdot x1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.9%

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

   if -5.1000000000000002e67 < x1 < 0.10000000000000001

   1. Initial program 98.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 48.4

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

   5. Applied rewrites 48.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 92.0%

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

   if 0.10000000000000001 < x1

   1. Initial program 59.9%

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.05 \cdot 10^{+158}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -5.1 \cdot 10^{+67}:\\ \;\;\;\;\left(x2 \cdot x2\right) \cdot \mathsf{fma}\left(8, x1, \frac{\mathsf{fma}\left(\mathsf{fma}\left(-12, x1, 12\right), x1, \mathsf{fma}\left(-x1, \frac{\mathsf{fma}\left(9, x1, -1\right)}{x2}, 6\right)\right)}{-x2}\right)\\ \mathbf{elif}\;x1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.3 \cdot 10^{+153}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.3e+153)
   (* 9.0 (* x1 x1))
   (if (<= x1 0.1)
     (fma
      (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
      x2
      (* (fma 9.0 x1 -1.0) x1))
     (+
      (fma
       (* x1 x1)
       x1
       (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0) x1 (* -6.0 x2)))
      x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.3e+153) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= 0.1) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = fma((x1 * x1), x1, fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, (-6.0 * x2))) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.3e+153)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= 0.1)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, Float64(-6.0 * x2))) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.3e+153], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 0.1], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.2999999999999999e153

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -1.2999999999999999e153 < x1 < 0.10000000000000001

   1. Initial program 92.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 42.2

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 83.0%

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

   if 0.10000000000000001 < x1

   1. Initial program 59.9%

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.3 \cdot 10^{+153}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 83.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.95 \cdot 10^{+144}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.95e+144)
   (* 9.0 (* x1 x1))
   (if (<= x1 5.5e+102)
     (fma
      (fma (* x2 x1) 8.0 (fma (fma 12.0 x1 -12.0) x1 -6.0))
      x2
      (* (fma 9.0 x1 -1.0) x1))
     (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.95e+144) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= 5.5e+102) {
		tmp = fma(fma((x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, (fma(9.0, x1, -1.0) * x1));
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.95e+144)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= 5.5e+102)
		tmp = fma(fma(Float64(x2 * x1), 8.0, fma(fma(12.0, x1, -12.0), x1, -6.0)), x2, Float64(fma(9.0, x1, -1.0) * x1));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.95e+144], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 5.5e+102], N[(N[(N[(x2 * x1), $MachinePrecision] * 8.0 + N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.94999999999999994e144

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 61.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 93.6%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 93.6%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -2.94999999999999994e144 < x1 < 5.49999999999999981e102

   1. Initial program 94.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 75.8%

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

   if 5.49999999999999981e102 < x1

   1. Initial program 32.5%

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.95 \cdot 10^{+144}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x2 \cdot x1, 8, \mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right)\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 77.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.05 \cdot 10^{+158}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -2.8 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.05e+158)
   (* 9.0 (* x1 x1))
   (if (<= x1 -2.8e+15)
     (fma (fma (fma 12.0 x1 -12.0) x1 -6.0) x2 (* (fma 9.0 x1 -1.0) x1))
     (if (<= x1 5.5e+102)
       (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
       (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.05e+158) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= -2.8e+15) {
		tmp = fma(fma(fma(12.0, x1, -12.0), x1, -6.0), x2, (fma(9.0, x1, -1.0) * x1));
	} else if (x1 <= 5.5e+102) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.05e+158)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= -2.8e+15)
		tmp = fma(fma(fma(12.0, x1, -12.0), x1, -6.0), x2, Float64(fma(9.0, x1, -1.0) * x1));
	elseif (x1 <= 5.5e+102)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.05e+158], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -2.8e+15], N[(N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x1 + -6.0), $MachinePrecision] * x2 + N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 5.5e+102], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.0499999999999999e158

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.1

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

   5. Applied rewrites 1.1%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 70.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 100.0%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -1.0499999999999999e158 < x1 < -2.8e15

   1. Initial program 64.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 33.0%

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

   if -2.8e15 < x1 < 5.49999999999999981e102

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 43.8

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

   5. Applied rewrites 43.8%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

   if 5.49999999999999981e102 < x1

   1. Initial program 32.5%

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.05 \cdot 10^{+158}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq -2.8 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x1, -6\right), x2, \mathsf{fma}\left(9, x1, -1\right) \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 73.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x2, \mathsf{fma}\left(9, x1, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.8e+15)
   (fma (fma (fma 12.0 x1 -12.0) x2 (fma 9.0 x1 -1.0)) x1 (* -6.0 x2))
   (if (<= x1 5.5e+102)
     (+ (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0) x1 (* -6.0 x2)) x1)
     (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.8e+15) {
		tmp = fma(fma(fma(12.0, x1, -12.0), x2, fma(9.0, x1, -1.0)), x1, (-6.0 * x2));
	} else if (x1 <= 5.5e+102) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, (-6.0 * x2)) + x1;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.8e+15)
		tmp = fma(fma(fma(12.0, x1, -12.0), x2, fma(9.0, x1, -1.0)), x1, Float64(-6.0 * x2));
	elseif (x1 <= 5.5e+102)
		tmp = Float64(fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, Float64(-6.0 * x2)) + x1);
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.8e+15], N[(N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x2 + N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 5.5e+102], N[(N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.8e15

   1. Initial program 37.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 41.7%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 53.8%

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

   if -2.8e15 < x1 < 5.49999999999999981e102

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if 5.49999999999999981e102 < x1

   1. Initial program 32.5%

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x2, \mathsf{fma}\left(9, x1, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 73.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x2, \mathsf{fma}\left(9, x1, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.8e+15)
   (fma (fma (fma 12.0 x1 -12.0) x2 (fma 9.0 x1 -1.0)) x1 (* -6.0 x2))
   (if (<= x1 5.5e+102)
     (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
     (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.8e+15) {
		tmp = fma(fma(fma(12.0, x1, -12.0), x2, fma(9.0, x1, -1.0)), x1, (-6.0 * x2));
	} else if (x1 <= 5.5e+102) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.8e+15)
		tmp = fma(fma(fma(12.0, x1, -12.0), x2, fma(9.0, x1, -1.0)), x1, Float64(-6.0 * x2));
	elseif (x1 <= 5.5e+102)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.8e+15], N[(N[(N[(12.0 * x1 + -12.0), $MachinePrecision] * x2 + N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 5.5e+102], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.8e15

   1. Initial program 37.7%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 41.7%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 53.8%

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

   if -2.8e15 < x1 < 5.49999999999999981e102

   1. Initial program 98.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 43.8

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

   5. Applied rewrites 43.8%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

   if 5.49999999999999981e102 < x1

   1. Initial program 32.5%

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.8 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(12, x1, -12\right), x2, \mathsf{fma}\left(9, x1, -1\right)\right), x1, -6 \cdot x2\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 76.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.85 \cdot 10^{+143}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.85e+143)
   (* 9.0 (* x1 x1))
   (if (<= x1 5.5e+102)
     (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
     (+ (fma (* x1 x1) x1 (* -6.0 x2)) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.85e+143) {
		tmp = 9.0 * (x1 * x1);
	} else if (x1 <= 5.5e+102) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2)) + x1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.85e+143)
		tmp = Float64(9.0 * Float64(x1 * x1));
	elseif (x1 <= 5.5e+102)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = Float64(fma(Float64(x1 * x1), x1, Float64(-6.0 * x2)) + x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.85e+143], N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 5.5e+102], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.8500000000000001e143

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 1.0

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

   5. Applied rewrites 1.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 61.4%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 93.6%

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

   12. Taylor expanded in x1 around inf

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

   14. Applied rewrites 93.6%

       \[\leadsto \left(x1 \cdot x1\right) \cdot 9 \]

   if -1.8500000000000001e143 < x1 < 5.49999999999999981e102

   1. Initial program 94.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if 5.49999999999999981e102 < x1

   1. Initial program 32.5%

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.85 \cdot 10^{+143}:\\ \;\;\;\;9 \cdot \left(x1 \cdot x1\right)\\ \mathbf{elif}\;x1 \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 31.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -1 \cdot 10^{-205}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 10^{-213}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2 + x1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* x2 2.0) -1e-205)
   (* -6.0 x2)
   (if (<= (* x2 2.0) 1e-213) (- x1) (+ (* -6.0 x2) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -1e-205) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 1e-213) {
		tmp = -x1;
	} else {
		tmp = (-6.0 * x2) + x1;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x2 * 2.0d0) <= (-1d-205)) then
        tmp = (-6.0d0) * x2
    else if ((x2 * 2.0d0) <= 1d-213) then
        tmp = -x1
    else
        tmp = ((-6.0d0) * x2) + x1
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -1e-205) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 1e-213) {
		tmp = -x1;
	} else {
		tmp = (-6.0 * x2) + x1;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x2 * 2.0) <= -1e-205:
		tmp = -6.0 * x2
	elif (x2 * 2.0) <= 1e-213:
		tmp = -x1
	else:
		tmp = (-6.0 * x2) + x1
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(x2 * 2.0) <= -1e-205)
		tmp = Float64(-6.0 * x2);
	elseif (Float64(x2 * 2.0) <= 1e-213)
		tmp = Float64(-x1);
	else
		tmp = Float64(Float64(-6.0 * x2) + x1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x2 * 2.0) <= -1e-205)
		tmp = -6.0 * x2;
	elseif ((x2 * 2.0) <= 1e-213)
		tmp = -x1;
	else
		tmp = (-6.0 * x2) + x1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(x2 * 2.0), $MachinePrecision], -1e-205], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[N[(x2 * 2.0), $MachinePrecision], 1e-213], (-x1), N[(N[(-6.0 * x2), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -1e-205

   1. Initial program 74.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 35.2

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

   5. Applied rewrites 35.2%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 35.3

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

   8. Applied rewrites 35.3%

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

   if -1e-205 < (*.f64 #s(literal 2 binary64) x2) < 9.9999999999999995e-214

   1. Initial program 92.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 7.8

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

   5. Applied rewrites 7.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 65.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 60.3%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 52.8%

       \[\leadsto -x1 \]

   if 9.9999999999999995e-214 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 69.3%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 27.7

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

   5. Applied rewrites 27.7%

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -1 \cdot 10^{-205}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 10^{-213}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 31.1% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -1 \cdot 10^{-205}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 10^{-213}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= (* x2 2.0) -1e-205)
   (* -6.0 x2)
   (if (<= (* x2 2.0) 1e-213) (- x1) (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -1e-205) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 1e-213) {
		tmp = -x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: tmp
    if ((x2 * 2.0d0) <= (-1d-205)) then
        tmp = (-6.0d0) * x2
    else if ((x2 * 2.0d0) <= 1d-213) then
        tmp = -x1
    else
        tmp = (-6.0d0) * x2
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double tmp;
	if ((x2 * 2.0) <= -1e-205) {
		tmp = -6.0 * x2;
	} else if ((x2 * 2.0) <= 1e-213) {
		tmp = -x1;
	} else {
		tmp = -6.0 * x2;
	}
	return tmp;
}

Python

def code(x1, x2):
	tmp = 0
	if (x2 * 2.0) <= -1e-205:
		tmp = -6.0 * x2
	elif (x2 * 2.0) <= 1e-213:
		tmp = -x1
	else:
		tmp = -6.0 * x2
	return tmp

Julia

function code(x1, x2)
	tmp = 0.0
	if (Float64(x2 * 2.0) <= -1e-205)
		tmp = Float64(-6.0 * x2);
	elseif (Float64(x2 * 2.0) <= 1e-213)
		tmp = Float64(-x1);
	else
		tmp = Float64(-6.0 * x2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	tmp = 0.0;
	if ((x2 * 2.0) <= -1e-205)
		tmp = -6.0 * x2;
	elseif ((x2 * 2.0) <= 1e-213)
		tmp = -x1;
	else
		tmp = -6.0 * x2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := If[LessEqual[N[(x2 * 2.0), $MachinePrecision], -1e-205], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[N[(x2 * 2.0), $MachinePrecision], 1e-213], (-x1), N[(-6.0 * x2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) x2) < -1e-205 or 9.9999999999999995e-214 < (*.f64 #s(literal 2 binary64) x2)

   1. Initial program 71.9%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.9

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

   8. Applied rewrites 30.9%

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

   if -1e-205 < (*.f64 #s(literal 2 binary64) x2) < 9.9999999999999995e-214

   1. Initial program 92.8%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 7.8

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

   5. Applied rewrites 7.8%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 65.8%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 60.3%

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

   12. Taylor expanded in x1 around 0

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

   14. Applied rewrites 52.8%

       \[\leadsto -x1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x2 \cdot 2 \leq -1 \cdot 10^{-205}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \cdot 2 \leq 10^{-213}:\\ \;\;\;\;-x1\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot x2\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 54.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(9, x1, -1\right) \cdot x1\\ \mathbf{if}\;x1 \leq -1.7 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 2.8 \cdot 10^{-31}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (fma 9.0 x1 -1.0) x1)))
   (if (<= x1 -1.7e-117) t_0 (if (<= x1 2.8e-31) (* -6.0 x2) t_0))))

C

double code(double x1, double x2) {
	double t_0 = fma(9.0, x1, -1.0) * x1;
	double tmp;
	if (x1 <= -1.7e-117) {
		tmp = t_0;
	} else if (x1 <= 2.8e-31) {
		tmp = -6.0 * x2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(fma(9.0, x1, -1.0) * x1)
	tmp = 0.0
	if (x1 <= -1.7e-117)
		tmp = t_0;
	elseif (x1 <= 2.8e-31)
		tmp = Float64(-6.0 * x2);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(9.0 * x1 + -1.0), $MachinePrecision] * x1), $MachinePrecision]}, If[LessEqual[x1, -1.7e-117], t$95$0, If[LessEqual[x1, 2.8e-31], N[(-6.0 * x2), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.70000000000000017e-117 or 2.7999999999999999e-31 < x1

   1. Initial program 57.9%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 56.6%

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

   9. Taylor expanded in x2 around 0

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

   11. Applied rewrites 43.1%

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

   if -1.70000000000000017e-117 < x1 < 2.7999999999999999e-31

   1. Initial program 98.6%

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

   3. Taylor expanded in x1 around 0

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

      1. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.5

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

   8. Applied rewrites 64.5%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.7 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(9, x1, -1\right) \cdot x1\\ \mathbf{elif}\;x1 \leq 2.8 \cdot 10^{-31}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, x1, -1\right) \cdot x1\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 13.5% accurate, 99.3× speedup

Math

\[\begin{array}{l} \\ -x1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x1, x2):
	return -x1

Julia

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

MATLAB

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

Wolfram

code[x1_, x2_] := (-x1)

Derivation

1. Initial program 74.6%

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

3. Taylor expanded in x1 around 0

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

   1. lower-*.f64 28.2

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

5. Applied rewrites 28.2%

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

6. Taylor expanded in x1 around 0

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

8. Applied rewrites 70.2%

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

9. Taylor expanded in x2 around 0

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

11. Applied rewrites 32.4%

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

12. Taylor expanded in x1 around 0

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

14. Applied rewrites 11.1%

    \[\leadsto -x1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024241 
(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))))))