Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 70.1% → 99.3%

Time: 19.8s

Alternatives: 17

Speedup: 5.8×

• 46.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.1% 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: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(-1.0 - Float64(x1 * x1))
	t_2 = Float64(Float64(fma(Float64(x1 * x1), 3.0, Float64(x2 * 2.0)) - 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(-2.0, x1, Float64(-6.0 * x2)) + fma(fma(fma(t_2, 4.0, -6.0), Float64(x1 * x1), Float64(Float64(t_2 * Float64(2.0 * x1)) * Float64(t_2 - 3.0))), fma(x1, x1, 1.0), Float64(Float64(3.0 * 3.0) * Float64(x1 * x1))))) + x1);
	else
		tmp = fma(1.0, x1, Float64((x1 ^ 4.0) * 6.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 99.8

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

   10. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 6.2%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.3

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

   7. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 52.3%

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

   10. Taylor expanded in x1 around 0

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

   12. Applied rewrites 98.5%

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

4. Final simplification 99.5%

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

Alternative 2: 94.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

   3. Taylor expanded in x1 around inf

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x1 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 95.7

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

   10. Applied rewrites 95.7%

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

   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

   4. Applied rewrites 6.2%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 52.3

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

   7. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 52.3%

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

   10. Taylor expanded in x1 around 0

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

   12. Applied rewrites 98.5%

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

4. Final simplification 96.4%

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

Alternative 3: 95.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9 - \frac{\mathsf{fma}\left(-12, x2, 18\right)}{x1}\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 10^{+22}:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (*
            (-
             6.0
             (/
              (-
               3.0
               (/
                (fma (fma 2.0 x2 -3.0) 4.0 (- 9.0 (/ (fma -12.0 x2 18.0) x1)))
                x1))
              x1))
            (* x1 x1))
           (* x1 x1))
          x1)))
   (if (<= x1 -2.55e+30)
     t_0
     (if (<= x1 1e+22)
       (+
        (+
         (+ (* (* (* 8.0 (/ x1 (fma x1 x1 1.0))) x2) x2) x1)
         (*
          (/ (- (- (* (* 3.0 x1) x1) (* x2 2.0)) x1) (- (* x1 x1) -1.0))
          3.0))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (((6.0 - ((3.0 - (fma(fma(2.0, x2, -3.0), 4.0, (9.0 - (fma(-12.0, x2, 18.0) / x1))) / x1)) / x1)) * (x1 * x1)) * (x1 * x1)) + x1;
	double tmp;
	if (x1 <= -2.55e+30) {
		tmp = t_0;
	} else if (x1 <= 1e+22) {
		tmp = (((((8.0 * (x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + ((((((3.0 * x1) * x1) - (x2 * 2.0)) - x1) / ((x1 * x1) - -1.0)) * 3.0)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, Float64(9.0 - Float64(fma(-12.0, x2, 18.0) / x1))) / x1)) / x1)) * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -2.55e+30)
		tmp = t_0;
	elseif (x1 <= 1e+22)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(x1 / fma(x1, x1, 1.0))) * x2) * x2) + x1) + Float64(Float64(Float64(Float64(Float64(Float64(3.0 * x1) * x1) - Float64(x2 * 2.0)) - x1) / Float64(Float64(x1 * x1) - -1.0)) * 3.0)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + N[(9.0 - N[(N[(-12.0 * x2 + 18.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.55e+30], t$95$0, If[LessEqual[x1, 1e+22], N[(N[(N[(N[(N[(N[(8.0 * N[(x1 / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision] + N[(N[(N[(N[(N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision] - N[(x2 * 2.0), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(N[(x1 * x1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.55000000000000018e30 or 1e22 < x1

   1. Initial program 43.2%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 96.5%

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

   7. Applied rewrites 96.5%

      \[\leadsto x1 + \left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9 - \frac{\mathsf{fma}\left(-12, x2, 18\right)}{x1}\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \color{blue}{\left(x1 \cdot x1\right)} \]

   if -2.55000000000000018e30 < x1 < 1e22

   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 x2 around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. lower-fma.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;\left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9 - \frac{\mathsf{fma}\left(-12, x2, 18\right)}{x1}\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 10^{+22}:\\ \;\;\;\;\left(\left(\left(\left(8 \cdot \frac{x1}{\mathsf{fma}\left(x1, x1, 1\right)}\right) \cdot x2\right) \cdot x2 + x1\right) + \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x2 \cdot 2\right) - x1}{x1 \cdot x1 - -1} \cdot 3\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9 - \frac{\mathsf{fma}\left(-12, x2, 18\right)}{x1}\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9 - \frac{\mathsf{fma}\left(-12, x2, 18\right)}{x1}\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \left(\mathsf{fma}\left(x2, 8, \mathsf{fma}\left(6, x1, -12\right)\right) \cdot x1\right) \cdot x2\right) + x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (*
            (-
             6.0
             (/
              (-
               3.0
               (/
                (fma (fma 2.0 x2 -3.0) 4.0 (- 9.0 (/ (fma -12.0 x2 18.0) x1)))
                x1))
              x1))
            (* x1 x1))
           (* x1 x1))
          x1)))
   (if (<= x1 -2.55e+30)
     t_0
     (if (<= x1 3.8e+21)
       (+
        (+
         (fma
          (/ (- (fma -2.0 x2 (* (* 3.0 x1) x1)) x1) (fma x1 x1 1.0))
          3.0
          (* (* (fma x2 8.0 (fma 6.0 x1 -12.0)) x1) x2))
         x1)
        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 - (fma(-12.0, x2, 18.0) / x1))) / x1)) / x1)) * (x1 * x1)) * (x1 * x1)) + x1;
	double tmp;
	if (x1 <= -2.55e+30) {
		tmp = t_0;
	} else if (x1 <= 3.8e+21) {
		tmp = (fma(((fma(-2.0, x2, ((3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, ((fma(x2, 8.0, fma(6.0, x1, -12.0)) * x1) * x2)) + x1) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(Float64(6.0 - Float64(Float64(3.0 - Float64(fma(fma(2.0, x2, -3.0), 4.0, Float64(9.0 - Float64(fma(-12.0, x2, 18.0) / x1))) / x1)) / x1)) * Float64(x1 * x1)) * Float64(x1 * x1)) + x1)
	tmp = 0.0
	if (x1 <= -2.55e+30)
		tmp = t_0;
	elseif (x1 <= 3.8e+21)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-2.0, x2, Float64(Float64(3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, Float64(Float64(fma(x2, 8.0, fma(6.0, x1, -12.0)) * x1) * x2)) + x1) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(6.0 - N[(N[(3.0 - N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + N[(9.0 - N[(N[(-12.0 * x2 + 18.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.55e+30], t$95$0, If[LessEqual[x1, 3.8e+21], N[(N[(N[(N[(N[(N[(-2.0 * x2 + N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(x2 * 8.0 + N[(6.0 * x1 + -12.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.55000000000000018e30 or 3.8e21 < x1

   1. Initial program 43.2%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 96.5%

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

   7. Applied rewrites 96.5%

      \[\leadsto x1 + \left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9 - \frac{\mathsf{fma}\left(-12, x2, 18\right)}{x1}\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \color{blue}{\left(x1 \cdot x1\right)} \]

   if -2.55000000000000018e30 < x1 < 3.8e21

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 96.3%

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

   10. Applied rewrites 96.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;\left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9 - \frac{\mathsf{fma}\left(-12, x2, 18\right)}{x1}\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \mathbf{elif}\;x1 \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \left(\mathsf{fma}\left(x2, 8, \mathsf{fma}\left(6, x1, -12\right)\right) \cdot x1\right) \cdot x2\right) + x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(6 - \frac{3 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9 - \frac{\mathsf{fma}\left(-12, x2, 18\right)}{x1}\right)}{x1}}{x1}\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right) + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \left(\mathsf{fma}\left(x2, 8, \mathsf{fma}\left(6, x1, -12\right)\right) \cdot x1\right) \cdot x2\right) + x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (fma
            (fma (fma 6.0 x1 -3.0) x1 (fma (fma 2.0 x2 -3.0) 4.0 9.0))
            x1
            (* (fma -2.0 x2 3.0) -6.0))
           x1)
          x1)))
   (if (<= x1 -2.55e+30)
     t_0
     (if (<= x1 3.8e+21)
       (+
        (+
         (fma
          (/ (- (fma -2.0 x2 (* (* 3.0 x1) x1)) x1) (fma x1 x1 1.0))
          3.0
          (* (* (fma x2 8.0 (fma 6.0 x1 -12.0)) x1) x2))
         x1)
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)), x1, (fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1;
	double tmp;
	if (x1 <= -2.55e+30) {
		tmp = t_0;
	} else if (x1 <= 3.8e+21) {
		tmp = (fma(((fma(-2.0, x2, ((3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, ((fma(x2, 8.0, fma(6.0, x1, -12.0)) * x1) * x2)) + x1) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)), x1, Float64(fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1)
	tmp = 0.0
	if (x1 <= -2.55e+30)
		tmp = t_0;
	elseif (x1 <= 3.8e+21)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-2.0, x2, Float64(Float64(3.0 * x1) * x1)) - x1) / fma(x1, x1, 1.0)), 3.0, Float64(Float64(fma(x2, 8.0, fma(6.0, x1, -12.0)) * x1) * x2)) + x1) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.55e+30], t$95$0, If[LessEqual[x1, 3.8e+21], N[(N[(N[(N[(N[(N[(-2.0 * x2 + N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / N[(x1 * x1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0 + N[(N[(N[(x2 * 8.0 + N[(6.0 * x1 + -12.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.55000000000000018e30 or 3.8e21 < x1

   1. Initial program 43.2%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 96.5%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.5%

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

   if -2.55000000000000018e30 < x1 < 3.8e21

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 96.3%

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

   10. Applied rewrites 96.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2, x2, \left(3 \cdot x1\right) \cdot x1\right) - x1}{\mathsf{fma}\left(x1, x1, 1\right)}, 3, \left(\mathsf{fma}\left(x2, 8, \mathsf{fma}\left(6, x1, -12\right)\right) \cdot x1\right) \cdot x2\right) + x1\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.1 \cdot 10^{+21}:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, x2, -x1\right) \cdot 3 + \left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -12\right), x1, \left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (fma
            (fma (fma 6.0 x1 -3.0) x1 (fma (fma 2.0 x2 -3.0) 4.0 9.0))
            x1
            (* (fma -2.0 x2 3.0) -6.0))
           x1)
          x1)))
   (if (<= x1 -2.55e+30)
     t_0
     (if (<= x1 3.1e+21)
       (+
        (+
         (* (fma -2.0 x2 (- x1)) 3.0)
         (+ (* (fma (fma 6.0 x1 -12.0) x1 (* (* x2 x1) 8.0)) x2) x1))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)), x1, (fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1;
	double tmp;
	if (x1 <= -2.55e+30) {
		tmp = t_0;
	} else if (x1 <= 3.1e+21) {
		tmp = ((fma(-2.0, x2, -x1) * 3.0) + ((fma(fma(6.0, x1, -12.0), x1, ((x2 * x1) * 8.0)) * x2) + x1)) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)), x1, Float64(fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1)
	tmp = 0.0
	if (x1 <= -2.55e+30)
		tmp = t_0;
	elseif (x1 <= 3.1e+21)
		tmp = Float64(Float64(Float64(fma(-2.0, x2, Float64(-x1)) * 3.0) + Float64(Float64(fma(fma(6.0, x1, -12.0), x1, Float64(Float64(x2 * x1) * 8.0)) * x2) + x1)) + x1);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.55e+30], t$95$0, If[LessEqual[x1, 3.1e+21], N[(N[(N[(N[(-2.0 * x2 + (-x1)), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(N[(6.0 * x1 + -12.0), $MachinePrecision] * x1 + N[(N[(x2 * x1), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision] + x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.55000000000000018e30 or 3.1e21 < x1

   1. Initial program 43.2%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 96.5%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.5%

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

   if -2.55000000000000018e30 < x1 < 3.1e21

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in x2 around 0

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

   8. Applied rewrites 96.3%

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

   9. Taylor expanded in x1 around 0

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

       1. lower-*.f64 72.0

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

   11. Applied rewrites 72.0%

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

   12. Taylor expanded in x1 around 0

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

       1. lower-fma.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 95.8

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

   14. Applied rewrites 95.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 3.1 \cdot 10^{+21}:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, x2, -x1\right) \cdot 3 + \left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -12\right), x1, \left(x2 \cdot x1\right) \cdot 8\right) \cdot x2 + x1\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (fma
            (fma (fma 6.0 x1 -3.0) x1 (fma (fma 2.0 x2 -3.0) 4.0 9.0))
            x1
            (* (fma -2.0 x2 3.0) -6.0))
           x1)
          x1)))
   (if (<= x1 -2.55e+30)
     t_0
     (if (<= x1 3.8e+21)
       (+
        (fma
         (* x1 x1)
         x1
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0) x1 (* -6.0 x2)))
        x1)
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (fma(fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)), x1, (fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1;
	double tmp;
	if (x1 <= -2.55e+30) {
		tmp = t_0;
	} else if (x1 <= 3.8e+21) {
		tmp = fma((x1 * x1), x1, fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(fma(fma(fma(6.0, x1, -3.0), x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)), x1, Float64(fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1)
	tmp = 0.0
	if (x1 <= -2.55e+30)
		tmp = t_0;
	elseif (x1 <= 3.8e+21)
		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);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(N[(N[(6.0 * x1 + -3.0), $MachinePrecision] * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision]}, If[LessEqual[x1, -2.55e+30], t$95$0, If[LessEqual[x1, 3.8e+21], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.55000000000000018e30 or 3.8e21 < x1

   1. Initial program 43.2%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 96.5%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 96.5%

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

   if -2.55000000000000018e30 < x1 < 3.8e21

   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

   4. Applied rewrites 99.0%

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

   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. lower-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.55 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(6, x1, -3\right), x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-3, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6\right)\\ \mathbf{elif}\;x1 \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.35e+154)
   (+
    (*
     (fma
      (fma -3.0 x1 (fma (fma 2.0 x2 -3.0) 4.0 9.0))
      x1
      (* (fma -2.0 x2 3.0) -6.0))
     x1)
    x1)
   (if (<= x1 -7.8e+31)
     (fma (fma x1 x1 1.0) x1 (* (* (* x1 x1) (* x1 x1)) 6.0))
     (if (<= x1 4.1e+21)
       (+
        (fma
         (* x1 x1)
         x1
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0) x1 (* -6.0 x2)))
        x1)
       (fma (fma x1 x1 1.0) x1 (* (* 6.0 (* x1 x1)) (* x1 x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.35e+154) {
		tmp = (fma(fma(-3.0, x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)), x1, (fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1;
	} else if (x1 <= -7.8e+31) {
		tmp = fma(fma(x1, x1, 1.0), x1, (((x1 * x1) * (x1 * x1)) * 6.0));
	} else if (x1 <= 4.1e+21) {
		tmp = fma((x1 * x1), x1, fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = fma(fma(x1, x1, 1.0), x1, ((6.0 * (x1 * x1)) * (x1 * x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.35e+154)
		tmp = Float64(Float64(fma(fma(-3.0, x1, fma(fma(2.0, x2, -3.0), 4.0, 9.0)), x1, Float64(fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1);
	elseif (x1 <= -7.8e+31)
		tmp = fma(fma(x1, x1, 1.0), x1, Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * 6.0));
	elseif (x1 <= 4.1e+21)
		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);
	else
		tmp = fma(fma(x1, x1, 1.0), x1, Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.35e+154], N[(N[(N[(N[(-3.0 * x1 + N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * 4.0 + 9.0), $MachinePrecision]), $MachinePrecision] * x1 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -7.8e+31], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.1e+21], 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], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.35000000000000003e154

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 93.3%

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

   if -1.35000000000000003e154 < x1 < -7.79999999999999999e31

   1. Initial program 84.3%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 80.6

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

   7. Applied rewrites 80.6%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 80.6%

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

   11. Applied rewrites 80.7%

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

   if -7.79999999999999999e31 < x1 < 4.1e21

   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

   4. Applied rewrites 99.0%

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

   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. lower-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if 4.1e21 < x1

   1. Initial program 47.3%

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 94.9

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

   7. Applied rewrites 94.9%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 94.9%

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

   11. Applied rewrites 94.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-3, x1, \mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right), 4, 9\right)\right), x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6\right)\\ \mathbf{elif}\;x1 \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(8 \cdot x2, x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6\right)\\ \mathbf{elif}\;x1 \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.35e+154)
   (+ (* (fma (* 8.0 x2) x1 (* (fma -2.0 x2 3.0) -6.0)) x1) x1)
   (if (<= x1 -7.8e+31)
     (fma (fma x1 x1 1.0) x1 (* (* (* x1 x1) (* x1 x1)) 6.0))
     (if (<= x1 4.1e+21)
       (+
        (fma
         (* x1 x1)
         x1
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -2.0) x1 (* -6.0 x2)))
        x1)
       (fma (fma x1 x1 1.0) x1 (* (* 6.0 (* x1 x1)) (* x1 x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.35e+154) {
		tmp = (fma((8.0 * x2), x1, (fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1;
	} else if (x1 <= -7.8e+31) {
		tmp = fma(fma(x1, x1, 1.0), x1, (((x1 * x1) * (x1 * x1)) * 6.0));
	} else if (x1 <= 4.1e+21) {
		tmp = fma((x1 * x1), x1, fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -2.0), x1, (-6.0 * x2))) + x1;
	} else {
		tmp = fma(fma(x1, x1, 1.0), x1, ((6.0 * (x1 * x1)) * (x1 * x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.35e+154)
		tmp = Float64(Float64(fma(Float64(8.0 * x2), x1, Float64(fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1);
	elseif (x1 <= -7.8e+31)
		tmp = fma(fma(x1, x1, 1.0), x1, Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * 6.0));
	elseif (x1 <= 4.1e+21)
		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);
	else
		tmp = fma(fma(x1, x1, 1.0), x1, Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.35e+154], N[(N[(N[(N[(8.0 * x2), $MachinePrecision] * x1 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -7.8e+31], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.1e+21], 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], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.35000000000000003e154

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 40.0%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 54.4%

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

   if -1.35000000000000003e154 < x1 < -7.79999999999999999e31

   1. Initial program 84.3%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 80.6

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

   7. Applied rewrites 80.6%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 80.6%

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

   11. Applied rewrites 80.7%

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

   if -7.79999999999999999e31 < x1 < 4.1e21

   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

   4. Applied rewrites 99.0%

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

   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. lower-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if 4.1e21 < x1

   1. Initial program 47.3%

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 94.9

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

   7. Applied rewrites 94.9%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 94.9%

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

   11. Applied rewrites 94.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(8 \cdot x2, x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6\right)\\ \mathbf{elif}\;x1 \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -2\right), x1, -6 \cdot x2\right)\right) + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(8 \cdot x2, x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6\right)\\ \mathbf{elif}\;x1 \leq 3.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -1.35e+154)
   (+ (* (fma (* 8.0 x2) x1 (* (fma -2.0 x2 3.0) -6.0)) x1) x1)
   (if (<= x1 -7.8e+31)
     (fma (fma x1 x1 1.0) x1 (* (* (* x1 x1) (* x1 x1)) 6.0))
     (if (<= x1 3.1e+21)
       (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
       (fma (fma x1 x1 1.0) x1 (* (* 6.0 (* x1 x1)) (* x1 x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -1.35e+154) {
		tmp = (fma((8.0 * x2), x1, (fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1;
	} else if (x1 <= -7.8e+31) {
		tmp = fma(fma(x1, x1, 1.0), x1, (((x1 * x1) * (x1 * x1)) * 6.0));
	} else if (x1 <= 3.1e+21) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else {
		tmp = fma(fma(x1, x1, 1.0), x1, ((6.0 * (x1 * x1)) * (x1 * x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -1.35e+154)
		tmp = Float64(Float64(fma(Float64(8.0 * x2), x1, Float64(fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1);
	elseif (x1 <= -7.8e+31)
		tmp = fma(fma(x1, x1, 1.0), x1, Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * 6.0));
	elseif (x1 <= 3.1e+21)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = fma(fma(x1, x1, 1.0), x1, Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -1.35e+154], N[(N[(N[(N[(8.0 * x2), $MachinePrecision] * x1 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -7.8e+31], N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 3.1e+21], 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[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -1.35000000000000003e154

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 40.0%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 54.4%

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

   if -1.35000000000000003e154 < x1 < -7.79999999999999999e31

   1. Initial program 84.3%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 80.6

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

   7. Applied rewrites 80.6%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 80.6%

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

   11. Applied rewrites 80.7%

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

   if -7.79999999999999999e31 < x1 < 3.1e21

   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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

   if 3.1e21 < x1

   1. Initial program 47.3%

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 94.9

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

   7. Applied rewrites 94.9%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 94.9%

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

   11. Applied rewrites 94.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(8 \cdot x2, x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot 6\right)\\ \mathbf{elif}\;x1 \leq 3.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \mathbf{if}\;x1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(8 \cdot x2, x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 3.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma (fma x1 x1 1.0) x1 (* (* 6.0 (* x1 x1)) (* x1 x1)))))
   (if (<= x1 -1.35e+154)
     (+ (* (fma (* 8.0 x2) x1 (* (fma -2.0 x2 3.0) -6.0)) x1) x1)
     (if (<= x1 -7.8e+31)
       t_0
       (if (<= x1 3.1e+21)
         (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
         t_0)))))

C

double code(double x1, double x2) {
	double t_0 = fma(fma(x1, x1, 1.0), x1, ((6.0 * (x1 * x1)) * (x1 * x1)));
	double tmp;
	if (x1 <= -1.35e+154) {
		tmp = (fma((8.0 * x2), x1, (fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1;
	} else if (x1 <= -7.8e+31) {
		tmp = t_0;
	} else if (x1 <= 3.1e+21) {
		tmp = fma(fma((fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, (-6.0 * x2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(fma(x1, x1, 1.0), x1, Float64(Float64(6.0 * Float64(x1 * x1)) * Float64(x1 * x1)))
	tmp = 0.0
	if (x1 <= -1.35e+154)
		tmp = Float64(Float64(fma(Float64(8.0 * x2), x1, Float64(fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1);
	elseif (x1 <= -7.8e+31)
		tmp = t_0;
	elseif (x1 <= 3.1e+21)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1 + 1.0), $MachinePrecision] * x1 + N[(N[(6.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.35e+154], N[(N[(N[(N[(8.0 * x2), $MachinePrecision] * x1 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, -7.8e+31], t$95$0, If[LessEqual[x1, 3.1e+21], N[(N[(N[(N[(2.0 * x2 + -3.0), $MachinePrecision] * x2), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -1.35000000000000003e154

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 40.0%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 54.4%

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

   if -1.35000000000000003e154 < x1 < -7.79999999999999999e31 or 3.1e21 < x1

   1. Initial program 58.9%

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

   4. Applied rewrites 63.7%

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

   5. Taylor expanded in x1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 90.4

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

   7. Applied rewrites 90.4%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 90.4%

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

   11. Applied rewrites 90.5%

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

   if -7.79999999999999999e31 < x1 < 3.1e21

   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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(8 \cdot x2, x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \mathbf{elif}\;x1 \leq 3.1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x1, x1, 1\right), x1, \left(6 \cdot \left(x1 \cdot x1\right)\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.25 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(8 \cdot x2, x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 9.2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.25e+76)
   (+ (* (fma (* 8.0 x2) x1 (* (fma -2.0 x2 3.0) -6.0)) x1) x1)
   (if (<= x1 9.2e+60)
     (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
     (fma (* x1 x1) x1 (* -6.0 x2)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.25e+76) {
		tmp = (fma((8.0 * x2), x1, (fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1;
	} else if (x1 <= 9.2e+60) {
		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));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.25e+76)
		tmp = Float64(Float64(fma(Float64(8.0 * x2), x1, Float64(fma(-2.0, x2, 3.0) * -6.0)) * x1) + x1);
	elseif (x1 <= 9.2e+60)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.25e+76], N[(N[(N[(N[(8.0 * x2), $MachinePrecision] * x1 + N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 9.2e+60], 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[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -2.2499999999999999e76

   1. Initial program 22.7%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 36.8%

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

   9. Taylor expanded in x2 around inf

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

   11. Applied rewrites 46.8%

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

   if -2.2499999999999999e76 < x1 < 9.20000000000000068e60

   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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if 9.20000000000000068e60 < x1

   1. Initial program 38.8%

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

   4. Applied rewrites 38.8%

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

   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. lower-*.f64 92.3

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

   7. Applied rewrites 92.3%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 92.3%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 92.3

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

   12. Applied rewrites 92.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.25 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(8 \cdot x2, x1, \mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 9.2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -6.4 \cdot 10^{+76}:\\ \;\;\;\;\left(\mathsf{fma}\left(8, x1, 12\right) \cdot x2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 9.2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -6.4e+76)
   (+ (* (* (fma 8.0 x1 12.0) x2) x1) x1)
   (if (<= x1 9.2e+60)
     (fma (fma (* (fma 2.0 x2 -3.0) x2) 4.0 -1.0) x1 (* -6.0 x2))
     (fma (* x1 x1) x1 (* -6.0 x2)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -6.4e+76) {
		tmp = ((fma(8.0, x1, 12.0) * x2) * x1) + x1;
	} else if (x1 <= 9.2e+60) {
		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));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -6.4e+76)
		tmp = Float64(Float64(Float64(fma(8.0, x1, 12.0) * x2) * x1) + x1);
	elseif (x1 <= 9.2e+60)
		tmp = fma(fma(Float64(fma(2.0, x2, -3.0) * x2), 4.0, -1.0), x1, Float64(-6.0 * x2));
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -6.4e+76], N[(N[(N[(N[(8.0 * x1 + 12.0), $MachinePrecision] * x2), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 9.2e+60], 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[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -6.39999999999999953e76

   1. Initial program 22.7%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 100.0%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 36.8%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x2 \cdot \left(12 + 8 \cdot x1\right)\right) \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.9%

       \[\leadsto x1 + \left(\mathsf{fma}\left(8, x1, 12\right) \cdot x2\right) \cdot x1 \]

   if -6.39999999999999953e76 < x1 < 9.20000000000000068e60

   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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if 9.20000000000000068e60 < x1

   1. Initial program 38.8%

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

   4. Applied rewrites 38.8%

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

   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. lower-*.f64 92.3

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

   7. Applied rewrites 92.3%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 92.3%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 92.3

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

   12. Applied rewrites 92.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -6.4 \cdot 10^{+76}:\\ \;\;\;\;\left(\mathsf{fma}\left(8, x1, 12\right) \cdot x2\right) \cdot x1 + x1\\ \mathbf{elif}\;x1 \leq 9.2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, x2, -3\right) \cdot x2, 4, -1\right), x1, -6 \cdot x2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.8% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -4.9 \cdot 10^{+30}:\\ \;\;\;\;\left(\mathsf{fma}\left(8, x1, 12\right) \cdot x2\right) \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -4.9e+30)
   (+ (* (* (fma 8.0 x1 12.0) x2) x1) x1)
   (fma (* x1 x1) x1 (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -4.9e+30) {
		tmp = ((fma(8.0, x1, 12.0) * x2) * x1) + x1;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -4.9e+30)
		tmp = Float64(Float64(Float64(fma(8.0, x1, 12.0) * x2) * x1) + x1);
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -4.9e+30], N[(N[(N[(N[(8.0 * x1 + 12.0), $MachinePrecision] * x2), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -4.89999999999999984e30

   1. Initial program 39.1%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 96.6%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 31.1%

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

   9. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(x2 \cdot \left(12 + 8 \cdot x1\right)\right) \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 38.3%

       \[\leadsto x1 + \left(\mathsf{fma}\left(8, x1, 12\right) \cdot x2\right) \cdot x1 \]

   if -4.89999999999999984e30 < x1

   1. Initial program 84.1%

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

   4. Applied rewrites 84.3%

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

   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. lower-*.f64 61.1

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

   7. Applied rewrites 61.1%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 61.1%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 61.4

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

   12. Applied rewrites 61.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -4.9 \cdot 10^{+30}:\\ \;\;\;\;\left(\mathsf{fma}\left(8, x1, 12\right) \cdot x2\right) \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.4% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -2.05 \cdot 10^{-65}:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -2.05e-65)
   (+ (* (* (fma -2.0 x2 3.0) -6.0) x1) x1)
   (fma (* x1 x1) x1 (* -6.0 x2))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -2.05e-65) {
		tmp = ((fma(-2.0, x2, 3.0) * -6.0) * x1) + x1;
	} else {
		tmp = fma((x1 * x1), x1, (-6.0 * x2));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -2.05e-65)
		tmp = Float64(Float64(Float64(fma(-2.0, x2, 3.0) * -6.0) * x1) + x1);
	else
		tmp = fma(Float64(x1 * x1), x1, Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -2.05e-65], N[(N[(N[(N[(-2.0 * x2 + 3.0), $MachinePrecision] * -6.0), $MachinePrecision] * x1), $MachinePrecision] + x1), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x1 < -2.04999999999999994e-65

   1. Initial program 52.5%

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

   3. Taylor expanded in x1 around -inf

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{2 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
   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 78.4%

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

   6. Taylor expanded in x1 around 0

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

   8. Applied rewrites 26.1%

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

   9. Taylor expanded in x1 around 0

      \[\leadsto x1 + \left(-6 \cdot \left(3 + -2 \cdot x2\right)\right) \cdot x1 \]
   10. Step-by-step derivation

   11. Applied rewrites 9.1%

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

   if -2.04999999999999994e-65 < x1

   1. Initial program 82.7%

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in x1 around 0

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

      1. lower-*.f64 66.3

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

   7. Applied rewrites 66.3%

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

      1. lift-+.f64 N/A

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

   9. Applied rewrites 66.3%

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

   10. Taylor expanded in x1 around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 66.5

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

   12. Applied rewrites 66.5%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -2.05 \cdot 10^{-65}:\\ \;\;\;\;\left(\mathsf{fma}\left(-2, x2, 3\right) \cdot -6\right) \cdot x1 + x1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1, -6 \cdot x2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.1% accurate, 17.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

4. Applied rewrites 76.0%

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

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. lower-*.f64 48.2

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

7. Applied rewrites 48.2%

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

   1. lift-+.f64 N/A

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

9. Applied rewrites 48.2%

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

10. Taylor expanded in x1 around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 48.4

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

12. Applied rewrites 48.4%

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

Alternative 17: 26.3% accurate, 33.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
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 31.3

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

5. Applied rewrites 31.3%

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

6. Final simplification 31.3%

   \[\leadsto -6 \cdot x2 + x1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024327 
(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))))))