Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 69.8% → 99.5%

Time: 20.8s

Alternatives: 21

Speedup: 8.3×

• 46.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 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 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied egg-rr 99.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Simplified 100.0%

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

4. Final simplification 99.5%

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

Alternative 2: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+100}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x2 (* 8.0 (* x1 x2))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -5e+180)
     t_1
     (if (<= t_4 1e+100)
       (+ x1 (fma x1 (fma 9.0 x1 -2.0) (* x2 -6.0)))
       (if (<= t_4 INFINITY) t_1 (* x1 (fma 9.0 x1 -1.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x2 * (8.0 * (x1 * x2));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -5e+180) {
		tmp = t_1;
	} else if (t_4 <= 1e+100) {
		tmp = x1 + fma(x1, fma(9.0, x1, -2.0), (x2 * -6.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 * fma(9.0, x1, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x2 * Float64(8.0 * Float64(x1 * x2)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -5e+180)
		tmp = t_1;
	elseif (t_4 <= 1e+100)
		tmp = Float64(x1 + fma(x1, fma(9.0, x1, -2.0), Float64(x2 * -6.0)));
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 * fma(9.0, x1, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x2 * N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+180], t$95$1, If[LessEqual[t$95$4, 1e+100], N[(x1 + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.9999999999999996e180 or 1.00000000000000002e100 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 48.1%

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

   7. Applied egg-rr 48.1%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(8 \cdot x1\right) \cdot {x2}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1\right)} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 52.4

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

   10. Simplified 52.4%

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

   if -4.9999999999999996e180 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.00000000000000002e100

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 91.2

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

   8. Simplified 91.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 74.4%

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

   7. Applied egg-rr 63.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \color{blue}{-1} \cdot x1 + 9 \cdot {x1}^{2} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{9 \cdot {x1}^{2} + -1 \cdot x1} \]

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 90.8

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

   10. Simplified 90.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+180}:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+100}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x2 \cdot -6\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ t_2 := x1 \cdot x1 + 1\\ t_3 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_2}\\ t_4 := x1 + \left(\left(x1 + \left(\left(t\_2 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_3 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_3\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_2}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{+100}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (* x2 (* 8.0 (* x1 x2))))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (/ (- (+ t_0 (* 2.0 x2)) x1) t_2))
        (t_4
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_2
               (+
                (* (* (* x1 2.0) t_3) (- t_3 3.0))
                (* (* x1 x1) (- (* t_3 4.0) 6.0))))
              (* t_0 t_3))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_2))))))
   (if (<= t_4 -5e+180)
     t_1
     (if (<= t_4 1e+100)
       (- (* x2 -6.0) x1)
       (if (<= t_4 INFINITY) t_1 (* x1 (fma 9.0 x1 -1.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = x2 * (8.0 * (x1 * x2));
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = ((t_0 + (2.0 * x2)) - x1) / t_2;
	double t_4 = x1 + ((x1 + (((t_2 * ((((x1 * 2.0) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((t_3 * 4.0) - 6.0)))) + (t_0 * t_3)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_2)));
	double tmp;
	if (t_4 <= -5e+180) {
		tmp = t_1;
	} else if (t_4 <= 1e+100) {
		tmp = (x2 * -6.0) - x1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = x1 * fma(9.0, x1, -1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(x2 * Float64(8.0 * Float64(x1 * x2)))
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_2)
	t_4 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_2 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_3 * 4.0) - 6.0)))) + Float64(t_0 * t_3)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_2))))
	tmp = 0.0
	if (t_4 <= -5e+180)
		tmp = t_1;
	elseif (t_4 <= 1e+100)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	elseif (t_4 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(x1 * fma(9.0, x1, -1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x2 * N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$2 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$3 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+180], t$95$1, If[LessEqual[t$95$4, 1e+100], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$1, N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.9999999999999996e180 or 1.00000000000000002e100 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 48.1%

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

   7. Applied egg-rr 48.1%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(8 \cdot x1\right) \cdot {x2}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1\right)} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 52.4

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

   10. Simplified 52.4%

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

   if -4.9999999999999996e180 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.00000000000000002e100

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 91.2

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

   8. Simplified 91.2%

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

   9. Taylor expanded in x1 around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 90.3

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

   11. Simplified 90.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 74.4%

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

   7. Applied egg-rr 63.3%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \color{blue}{-1} \cdot x1 + 9 \cdot {x1}^{2} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{9 \cdot {x1}^{2} + -1 \cdot x1} \]

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 90.8

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

   10. Simplified 90.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+180}:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+100}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq \infty:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(9, x1, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+180}:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+100}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(6, x1, -3\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_1
               (+
                (* (* (* x1 2.0) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* t_2 4.0) 6.0))))
              (* t_0 t_2))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -5e+180)
     (* x2 (* 8.0 (* x1 x2)))
     (if (<= t_3 1e+100)
       (+ x1 (fma x1 (fma 9.0 x1 -2.0) (* x2 -6.0)))
       (fma (* x1 x1) (* x1 (fma 6.0 x1 -3.0)) x1)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -5e+180) {
		tmp = x2 * (8.0 * (x1 * x2));
	} else if (t_3 <= 1e+100) {
		tmp = x1 + fma(x1, fma(9.0, x1, -2.0), (x2 * -6.0));
	} else {
		tmp = fma((x1 * x1), (x1 * fma(6.0, x1, -3.0)), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -5e+180)
		tmp = Float64(x2 * Float64(8.0 * Float64(x1 * x2)));
	elseif (t_3 <= 1e+100)
		tmp = Float64(x1 + fma(x1, fma(9.0, x1, -2.0), Float64(x2 * -6.0)));
	else
		tmp = fma(Float64(x1 * x1), Float64(x1 * fma(6.0, x1, -3.0)), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+180], N[(x2 * N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+100], N[(x1 + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(6.0 * x1 + -3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.9999999999999996e180

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 73.9%

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

   7. Applied egg-rr 73.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(8 \cdot x1\right) \cdot {x2}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1\right)} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.6

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

   10. Simplified 78.6%

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

   if -4.9999999999999996e180 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.00000000000000002e100

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 91.2

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

   8. Simplified 91.2%

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

   if 1.00000000000000002e100 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 41.4%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 80.6

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

   5. Simplified 80.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-+l+ N/A

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

      10. lower-+.f64 N/A

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

   7. Applied egg-rr 46.1%

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

   8. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 80.6

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

   10. Simplified 80.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+180}:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+100}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(6, x1, -3\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+180}:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+100}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            x1
            (+
             (+
              (*
               t_1
               (+
                (* (* (* x1 2.0) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* t_2 4.0) 6.0))))
              (* t_0 t_2))
             (* x1 (* x1 x1))))
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -5e+180)
     (* x2 (* 8.0 (* x1 x2)))
     (if (<= t_3 1e+100)
       (+ x1 (fma x1 (fma 9.0 x1 -2.0) (* x2 -6.0)))
       (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -5e+180) {
		tmp = x2 * (8.0 * (x1 * x2));
	} else if (t_3 <= 1e+100) {
		tmp = x1 + fma(x1, fma(9.0, x1, -2.0), (x2 * -6.0));
	} else {
		tmp = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -5e+180)
		tmp = Float64(x2 * Float64(8.0 * Float64(x1 * x2)));
	elseif (t_3 <= 1e+100)
		tmp = Float64(x1 + fma(x1, fma(9.0, x1, -2.0), Float64(x2 * -6.0)));
	else
		tmp = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+180], N[(x2 * N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+100], N[(x1 + N[(x1 * N[(9.0 * x1 + -2.0), $MachinePrecision] + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -4.9999999999999996e180

   1. Initial program 99.7%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 73.9%

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

   7. Applied egg-rr 73.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around inf

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(8 \cdot x1\right) \cdot {x2}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x2}^{2} \cdot \left(8 \cdot x1\right)} \]

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 78.6

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

   10. Simplified 78.6%

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

   if -4.9999999999999996e180 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.00000000000000002e100

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 91.2

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

   8. Simplified 91.2%

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

   if 1.00000000000000002e100 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 41.4%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 80.6

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

   5. Simplified 80.6%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 80.6

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

   8. Simplified 80.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq -5 \cdot 10^{+180}:\\ \;\;\;\;x2 \cdot \left(8 \cdot \left(x1 \cdot x2\right)\right)\\ \mathbf{elif}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 10^{+100}:\\ \;\;\;\;x1 + \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -2\right), x2 \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 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 99.2%

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

   3. Taylor expanded in x1 around inf

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

   5. Simplified 98.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Simplified 100.0%

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

4. Final simplification 98.7%

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

Alternative 7: 94.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 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 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-neg N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied egg-rr 99.3%

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

   6. Applied egg-rr 94.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Simplified 100.0%

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

4. Final simplification 96.4%

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

Alternative 8: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq 2 \cdot 10^{+283}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_1
              (+
               (* (* (* x1 2.0) t_2) (- t_2 3.0))
               (* (* x1 x1) (- (* t_2 4.0) 6.0))))
             (* t_0 t_2))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))
        2e+283)
     (- (* x2 -6.0) x1)
     (+ x1 (* (* x1 x1) 9.0)))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 2e+283) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x1 * (x1 * 3.0d0)
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0d0) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((t_2 * 4.0d0) - 6.0d0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))) <= 2d+283) then
        tmp = (x2 * (-6.0d0)) - x1
    else
        tmp = x1 + ((x1 * x1) * 9.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 2e+283) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = x1 + ((x1 * x1) * 9.0);
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 3.0)
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	tmp = 0
	if (x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 2e+283:
		tmp = (x2 * -6.0) - x1
	else:
		tmp = x1 + ((x1 * x1) * 9.0)
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= 2e+283)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = Float64(x1 + Float64(Float64(x1 * x1) * 9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 3.0);
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = 0.0;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 2e+283)
		tmp = (x2 * -6.0) - x1;
	else
		tmp = x1 + ((x1 * x1) * 9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+283], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(x1 + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.99999999999999991e283

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 75.5%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 61.8

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

   8. Simplified 61.8%

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

   9. Taylor expanded in x1 around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 60.9

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

   11. Simplified 60.9%

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

   if 1.99999999999999991e283 < (+.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 22.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 70.9%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 71.4

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

   8. Simplified 71.4%

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

   9. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

          \[\leadsto x1 + \color{blue}{9 \cdot {x1}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 71.8

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

   11. Simplified 71.8%

       \[\leadsto x1 + \color{blue}{9 \cdot \left(x1 \cdot x1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{+283}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot 3\right)\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(t\_1 \cdot \left(\left(\left(x1 \cdot 2\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(t\_2 \cdot 4 - 6\right)\right) + t\_0 \cdot t\_2\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \leq 2 \cdot 10^{+283}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 3.0)))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (if (<=
        (+
         x1
         (+
          (+
           x1
           (+
            (+
             (*
              t_1
              (+
               (* (* (* x1 2.0) t_2) (- t_2 3.0))
               (* (* x1 x1) (- (* t_2 4.0) 6.0))))
             (* t_0 t_2))
            (* x1 (* x1 x1))))
          (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))
        2e+283)
     (- (* x2 -6.0) x1)
     (* (* x1 x1) 9.0))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 2e+283) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	return tmp;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x1 * (x1 * 3.0d0)
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0d0) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((t_2 * 4.0d0) - 6.0d0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))) <= 2d+283) then
        tmp = (x2 * (-6.0d0)) - x1
    else
        tmp = (x1 * x1) * 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x1, double x2) {
	double t_0 = x1 * (x1 * 3.0);
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double tmp;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 2e+283) {
		tmp = (x2 * -6.0) - x1;
	} else {
		tmp = (x1 * x1) * 9.0;
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = x1 * (x1 * 3.0)
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	tmp = 0
	if (x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 2e+283:
		tmp = (x2 * -6.0) - x1
	else:
		tmp = (x1 * x1) * 9.0
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * 3.0))
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(x1 + Float64(Float64(Float64(t_1 * Float64(Float64(Float64(Float64(x1 * 2.0) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(t_2 * 4.0) - 6.0)))) + Float64(t_0 * t_2)) + Float64(x1 * Float64(x1 * x1)))) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1)))) <= 2e+283)
		tmp = Float64(Float64(x2 * -6.0) - x1);
	else
		tmp = Float64(Float64(x1 * x1) * 9.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = x1 * (x1 * 3.0);
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = 0.0;
	if ((x1 + ((x1 + (((t_1 * ((((x1 * 2.0) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((t_2 * 4.0) - 6.0)))) + (t_0 * t_2)) + (x1 * (x1 * x1)))) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))) <= 2e+283)
		tmp = (x2 * -6.0) - x1;
	else
		tmp = (x1 * x1) * 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(x1 + N[(N[(N[(t$95$1 * N[(N[(N[(N[(x1 * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(t$95$2 * 4.0), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+283], N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.99999999999999991e283

   1. Initial program 99.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 75.5%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 61.8

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

   8. Simplified 61.8%

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

   9. Taylor expanded in x1 around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 60.9

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

   11. Simplified 60.9%

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

   if 1.99999999999999991e283 < (+.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 22.1%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 70.9%

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

   6. Taylor expanded in x2 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 71.4

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

   8. Simplified 71.4%

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

   9. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{9 \cdot {x1}^{2}} \]

       2. unpow2 N/A

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

       3. lower-*.f64 71.7

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

   11. Simplified 71.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 + \left(\left(x1 + \left(\left(\left(x1 \cdot x1 + 1\right) \cdot \left(\left(\left(x1 \cdot 2\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} \cdot 4 - 6\right)\right) + \left(x1 \cdot \left(x1 \cdot 3\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + x1 \cdot \left(x1 \cdot x1\right)\right)\right) + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \leq 2 \cdot 10^{+283}:\\ \;\;\;\;x2 \cdot -6 - x1\\ \mathbf{else}:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot 9\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)\\ \mathbf{if}\;x1 \leq -165000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{t\_0}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 18000:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{t\_0 - \frac{\mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1} - 3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma 4.0 (fma x2 2.0 -3.0) 9.0)))
   (if (<= x1 -165000.0)
     (+ x1 (* (pow x1 4.0) (+ 6.0 (/ (- (/ t_0 x1) 3.0) x1))))
     (if (<= x1 18000.0)
       (fma x1 (fma 9.0 x1 -1.0) (* x2 (fma x1 (fma x2 8.0 -12.0) -6.0)))
       (+
        x1
        (*
         (pow x1 4.0)
         (+
          6.0
          (/
           (- (/ (- t_0 (/ (* (fma x2 2.0 -3.0) -6.0) x1)) x1) 3.0)
           x1))))))))

C

double code(double x1, double x2) {
	double t_0 = fma(4.0, fma(x2, 2.0, -3.0), 9.0);
	double tmp;
	if (x1 <= -165000.0) {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + (((t_0 / x1) - 3.0) / x1)));
	} else if (x1 <= 18000.0) {
		tmp = fma(x1, fma(9.0, x1, -1.0), (x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	} else {
		tmp = x1 + (pow(x1, 4.0) * (6.0 + ((((t_0 - ((fma(x2, 2.0, -3.0) * -6.0) / x1)) / x1) - 3.0) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(4.0, fma(x2, 2.0, -3.0), 9.0)
	tmp = 0.0
	if (x1 <= -165000.0)
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(t_0 / x1) - 3.0) / x1))));
	elseif (x1 <= 18000.0)
		tmp = fma(x1, fma(9.0, x1, -1.0), Float64(x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	else
		tmp = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(Float64(t_0 - Float64(Float64(fma(x2, 2.0, -3.0) * -6.0) / x1)) / x1) - 3.0) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision]}, If[LessEqual[x1, -165000.0], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(t$95$0 / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 18000.0], N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision] + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(t$95$0 - N[(N[(N[(x2 * 2.0 + -3.0), $MachinePrecision] * -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -165000

   1. Initial program 32.9%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Simplified 94.2%

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

   if -165000 < x1 < 18000

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(-2 + 1\right) \cdot x1} + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-1} \cdot x1 + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot {x1}^{2} + -1 \cdot x1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      7. unpow2 N/A

         \[\leadsto \left(9 \cdot \color{blue}{\left(x1 \cdot x1\right)} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(9 \cdot x1\right) \cdot x1} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{x1 \cdot \left(9 \cdot x1 + -1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x1, 9 \cdot x1 + -1, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -1\right)}, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)}\right) \]

      13. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \color{blue}{\left(-12 \cdot x1 + \left(\left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right) - 6\right)\right)}\right) \]

   10. Simplified 97.3%

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

   11. Taylor expanded in x1 around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 97.3

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

   13. Simplified 97.3%

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

   if 18000 < x1

   1. Initial program 35.7%

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

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Simplified 98.5%

      \[\leadsto x1 + \color{blue}{{x1}^{4} \cdot \left(6 - \frac{3 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{0 + \mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1}}{x1}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -165000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 18000:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right) - \frac{\mathsf{fma}\left(x2, 2, -3\right) \cdot -6}{x1}}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{if}\;x1 \leq -165000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (+
          x1
          (*
           (pow x1 4.0)
           (+ 6.0 (/ (- (/ (fma 4.0 (fma x2 2.0 -3.0) 9.0) x1) 3.0) x1))))))
   (if (<= x1 -165000.0)
     t_0
     (if (<= x1 18500.0)
       (fma x1 (fma 9.0 x1 -1.0) (* x2 (fma x1 (fma x2 8.0 -12.0) -6.0)))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = x1 + (pow(x1, 4.0) * (6.0 + (((fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1)));
	double tmp;
	if (x1 <= -165000.0) {
		tmp = t_0;
	} else if (x1 <= 18500.0) {
		tmp = fma(x1, fma(9.0, x1, -1.0), (x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 + Float64((x1 ^ 4.0) * Float64(6.0 + Float64(Float64(Float64(fma(4.0, fma(x2, 2.0, -3.0), 9.0) / x1) - 3.0) / x1))))
	tmp = 0.0
	if (x1 <= -165000.0)
		tmp = t_0;
	elseif (x1 <= 18500.0)
		tmp = fma(x1, fma(9.0, x1, -1.0), Float64(x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 + N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(N[(N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + 9.0), $MachinePrecision] / x1), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -165000.0], t$95$0, If[LessEqual[x1, 18500.0], N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision] + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -165000 or 18500 < x1

   1. Initial program 34.3%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

   5. Simplified 96.4%

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

   if -165000 < x1 < 18500

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(-2 + 1\right) \cdot x1} + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-1} \cdot x1 + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot {x1}^{2} + -1 \cdot x1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      7. unpow2 N/A

         \[\leadsto \left(9 \cdot \color{blue}{\left(x1 \cdot x1\right)} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(9 \cdot x1\right) \cdot x1} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{x1 \cdot \left(9 \cdot x1 + -1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x1, 9 \cdot x1 + -1, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -1\right)}, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)}\right) \]

      13. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \color{blue}{\left(-12 \cdot x1 + \left(\left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right) - 6\right)\right)}\right) \]

   10. Simplified 97.3%

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

   11. Taylor expanded in x1 around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 97.3

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

   13. Simplified 97.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -165000:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + {x1}^{4} \cdot \left(6 + \frac{\frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), 9\right)}{x1} - 3}{x1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(\left(x1 \cdot x1\right) \cdot \left(\frac{4 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1}}{x1} - 6\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot x1\right) \cdot 9\right)\right)\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -195000.0)
   (* (pow x1 4.0) (+ 6.0 (/ -3.0 x1)))
   (if (<= x1 18500.0)
     (fma x1 (fma 9.0 x1 -1.0) (* x2 (fma x1 (fma x2 8.0 -12.0) -6.0)))
     (+
      x1
      (+
       (+
        x1
        (+
         (* x1 (* x1 x1))
         (+
          (*
           (*
            (* x1 x1)
            (- (/ (- 4.0 (/ (fma 4.0 (fma x2 2.0 -3.0) -6.0) x1)) x1) 6.0))
           (- -1.0 (* x1 x1)))
          (* (* x1 x1) 9.0))))
       (* 3.0 (- (* x2 -2.0) x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -195000.0) {
		tmp = pow(x1, 4.0) * (6.0 + (-3.0 / x1));
	} else if (x1 <= 18500.0) {
		tmp = fma(x1, fma(9.0, x1, -1.0), (x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	} else {
		tmp = x1 + ((x1 + ((x1 * (x1 * x1)) + ((((x1 * x1) * (((4.0 - (fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1)) / x1) - 6.0)) * (-1.0 - (x1 * x1))) + ((x1 * x1) * 9.0)))) + (3.0 * ((x2 * -2.0) - x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -195000.0)
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-3.0 / x1)));
	elseif (x1 <= 18500.0)
		tmp = fma(x1, fma(9.0, x1, -1.0), Float64(x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) + Float64(Float64(Float64(Float64(x1 * x1) * Float64(Float64(Float64(4.0 - Float64(fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1)) / x1) - 6.0)) * Float64(-1.0 - Float64(x1 * x1))) + Float64(Float64(x1 * x1) * 9.0)))) + Float64(3.0 * Float64(Float64(x2 * -2.0) - x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -195000.0], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 18500.0], N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision] + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(N[(N[(4.0 - N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -195000

   1. Initial program 32.9%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 91.5

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

   8. Simplified 91.5%

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

   if -195000 < x1 < 18500

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(-2 + 1\right) \cdot x1} + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-1} \cdot x1 + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot {x1}^{2} + -1 \cdot x1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      7. unpow2 N/A

         \[\leadsto \left(9 \cdot \color{blue}{\left(x1 \cdot x1\right)} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(9 \cdot x1\right) \cdot x1} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{x1 \cdot \left(9 \cdot x1 + -1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x1, 9 \cdot x1 + -1, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -1\right)}, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)}\right) \]

      13. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \color{blue}{\left(-12 \cdot x1 + \left(\left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right) - 6\right)\right)}\right) \]

   10. Simplified 97.3%

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

   11. Taylor expanded in x1 around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 97.3

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

   13. Simplified 97.3%

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

   if 18500 < x1

   1. Initial program 35.7%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

   5. Simplified 34.3%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.3

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 96.8

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

   11. Simplified 96.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + \frac{-3}{x1}\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(\left(x1 \cdot x1\right) \cdot \left(\frac{4 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1}}{x1} - 6\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot x1\right) \cdot 9\right)\right)\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;\mathsf{fma}\left(6 + \frac{-3}{x1}, x1 \cdot t\_0, x1\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(t\_0 + \left(\left(\left(x1 \cdot x1\right) \cdot \left(\frac{4 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1}}{x1} - 6\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot x1\right) \cdot 9\right)\right)\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1))))
   (if (<= x1 -195000.0)
     (fma (+ 6.0 (/ -3.0 x1)) (* x1 t_0) x1)
     (if (<= x1 18500.0)
       (fma x1 (fma 9.0 x1 -1.0) (* x2 (fma x1 (fma x2 8.0 -12.0) -6.0)))
       (+
        x1
        (+
         (+
          x1
          (+
           t_0
           (+
            (*
             (*
              (* x1 x1)
              (- (/ (- 4.0 (/ (fma 4.0 (fma x2 2.0 -3.0) -6.0) x1)) x1) 6.0))
             (- -1.0 (* x1 x1)))
            (* (* x1 x1) 9.0))))
         (* 3.0 (- (* x2 -2.0) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double tmp;
	if (x1 <= -195000.0) {
		tmp = fma((6.0 + (-3.0 / x1)), (x1 * t_0), x1);
	} else if (x1 <= 18500.0) {
		tmp = fma(x1, fma(9.0, x1, -1.0), (x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	} else {
		tmp = x1 + ((x1 + (t_0 + ((((x1 * x1) * (((4.0 - (fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1)) / x1) - 6.0)) * (-1.0 - (x1 * x1))) + ((x1 * x1) * 9.0)))) + (3.0 * ((x2 * -2.0) - x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -195000.0)
		tmp = fma(Float64(6.0 + Float64(-3.0 / x1)), Float64(x1 * t_0), x1);
	elseif (x1 <= 18500.0)
		tmp = fma(x1, fma(9.0, x1, -1.0), Float64(x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 + Float64(t_0 + Float64(Float64(Float64(Float64(x1 * x1) * Float64(Float64(Float64(4.0 - Float64(fma(4.0, fma(x2, 2.0, -3.0), -6.0) / x1)) / x1) - 6.0)) * Float64(-1.0 - Float64(x1 * x1))) + Float64(Float64(x1 * x1) * 9.0)))) + Float64(3.0 * Float64(Float64(x2 * -2.0) - x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -195000.0], N[(N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * t$95$0), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 18500.0], N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision] + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 + N[(t$95$0 + N[(N[(N[(N[(x1 * x1), $MachinePrecision] * N[(N[(N[(4.0 - N[(N[(4.0 * N[(x2 * 2.0 + -3.0), $MachinePrecision] + -6.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(N[(x2 * -2.0), $MachinePrecision] - x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -195000

   1. Initial program 32.9%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 91.5

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

   5. Simplified 91.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 91.5

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 91.5

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 91.5

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

   7. Applied egg-rr 91.5%

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

   if -195000 < x1 < 18500

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(-2 + 1\right) \cdot x1} + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-1} \cdot x1 + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot {x1}^{2} + -1 \cdot x1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      7. unpow2 N/A

         \[\leadsto \left(9 \cdot \color{blue}{\left(x1 \cdot x1\right)} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(9 \cdot x1\right) \cdot x1} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{x1 \cdot \left(9 \cdot x1 + -1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x1, 9 \cdot x1 + -1, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -1\right)}, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)}\right) \]

      13. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \color{blue}{\left(-12 \cdot x1 + \left(\left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right) - 6\right)\right)}\right) \]

   10. Simplified 97.3%

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

   11. Taylor expanded in x1 around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 97.3

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

   13. Simplified 97.3%

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

   if 18500 < x1

   1. Initial program 35.7%

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

   3. Taylor expanded in x1 around -inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

   5. Simplified 34.3%

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

   6. Taylor expanded in x1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 34.3

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x1 around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 96.8

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

   11. Simplified 96.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;\mathsf{fma}\left(6 + \frac{-3}{x1}, x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right), x1\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(\left(x1 + \left(x1 \cdot \left(x1 \cdot x1\right) + \left(\left(\left(x1 \cdot x1\right) \cdot \left(\frac{4 - \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(x2, 2, -3\right), -6\right)}{x1}}{x1} - 6\right)\right) \cdot \left(-1 - x1 \cdot x1\right) + \left(x1 \cdot x1\right) \cdot 9\right)\right)\right) + 3 \cdot \left(x2 \cdot -2 - x1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x1 \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;\mathsf{fma}\left(6 + \frac{-3}{x1}, x1 \cdot t\_0, x1\right)\\ \mathbf{elif}\;x1 \leq 31000:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + t\_0 \cdot \mathsf{fma}\left(x1, 6, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* x1 (* x1 x1))))
   (if (<= x1 -195000.0)
     (fma (+ 6.0 (/ -3.0 x1)) (* x1 t_0) x1)
     (if (<= x1 31000.0)
       (fma x1 (fma 9.0 x1 -1.0) (* x2 (fma x1 (fma x2 8.0 -12.0) -6.0)))
       (+ x1 (* t_0 (fma x1 6.0 -3.0)))))))

C

double code(double x1, double x2) {
	double t_0 = x1 * (x1 * x1);
	double tmp;
	if (x1 <= -195000.0) {
		tmp = fma((6.0 + (-3.0 / x1)), (x1 * t_0), x1);
	} else if (x1 <= 31000.0) {
		tmp = fma(x1, fma(9.0, x1, -1.0), (x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	} else {
		tmp = x1 + (t_0 * fma(x1, 6.0, -3.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(x1 * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -195000.0)
		tmp = fma(Float64(6.0 + Float64(-3.0 / x1)), Float64(x1 * t_0), x1);
	elseif (x1 <= 31000.0)
		tmp = fma(x1, fma(9.0, x1, -1.0), Float64(x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	else
		tmp = Float64(x1 + Float64(t_0 * fma(x1, 6.0, -3.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -195000.0], N[(N[(6.0 + N[(-3.0 / x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * t$95$0), $MachinePrecision] + x1), $MachinePrecision], If[LessEqual[x1, 31000.0], N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision] + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(t$95$0 * N[(x1 * 6.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -195000

   1. Initial program 32.9%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 91.5

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

   5. Simplified 91.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 91.5

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 91.5

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 91.5

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

   7. Applied egg-rr 91.5%

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

   if -195000 < x1 < 31000

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(-2 + 1\right) \cdot x1} + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-1} \cdot x1 + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot {x1}^{2} + -1 \cdot x1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      7. unpow2 N/A

         \[\leadsto \left(9 \cdot \color{blue}{\left(x1 \cdot x1\right)} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(9 \cdot x1\right) \cdot x1} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{x1 \cdot \left(9 \cdot x1 + -1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x1, 9 \cdot x1 + -1, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -1\right)}, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)}\right) \]

      13. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \color{blue}{\left(-12 \cdot x1 + \left(\left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right) - 6\right)\right)}\right) \]

   10. Simplified 97.3%

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

   11. Taylor expanded in x1 around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 97.3

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

   13. Simplified 97.3%

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

   if 31000 < x1

   1. Initial program 35.7%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 94.4

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

   5. Simplified 94.4%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 94.4

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

   8. Simplified 94.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;\mathsf{fma}\left(6 + \frac{-3}{x1}, x1 \cdot \left(x1 \cdot \left(x1 \cdot x1\right)\right), x1\right)\\ \mathbf{elif}\;x1 \leq 31000:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \mathsf{fma}\left(x1, 6, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 94.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{elif}\;x1 \leq 31000:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \mathsf{fma}\left(x1, 6, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -195000.0)
   (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))
   (if (<= x1 31000.0)
     (fma x1 (fma 9.0 x1 -1.0) (* x2 (fma x1 (fma x2 8.0 -12.0) -6.0)))
     (+ x1 (* (* x1 (* x1 x1)) (fma x1 6.0 -3.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -195000.0) {
		tmp = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	} else if (x1 <= 31000.0) {
		tmp = fma(x1, fma(9.0, x1, -1.0), (x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	} else {
		tmp = x1 + ((x1 * (x1 * x1)) * fma(x1, 6.0, -3.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -195000.0)
		tmp = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0));
	elseif (x1 <= 31000.0)
		tmp = fma(x1, fma(9.0, x1, -1.0), Float64(x2 * fma(x1, fma(x2, 8.0, -12.0), -6.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) * fma(x1, 6.0, -3.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -195000.0], N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 31000.0], N[(x1 * N[(9.0 * x1 + -1.0), $MachinePrecision] + N[(x2 * N[(x1 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -195000

   1. Initial program 32.9%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 91.4

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

   8. Simplified 91.4%

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

   if -195000 < x1 < 31000

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x2 around 0

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

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt1-in N/A

         \[\leadsto \left(\color{blue}{\left(-2 + 1\right) \cdot x1} + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{-1} \cdot x1 + 9 \cdot {x1}^{2}\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(9 \cdot {x1}^{2} + -1 \cdot x1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      7. unpow2 N/A

         \[\leadsto \left(9 \cdot \color{blue}{\left(x1 \cdot x1\right)} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(9 \cdot x1\right) \cdot x1} + -1 \cdot x1\right) + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \color{blue}{x1 \cdot \left(9 \cdot x1 + -1\right)} + x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x1, 9 \cdot x1 + -1, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -1\right)}, x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), \color{blue}{x2 \cdot \left(\left(-12 \cdot x1 + \left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right)\right) - 6\right)}\right) \]

      13. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \color{blue}{\left(-12 \cdot x1 + \left(\left(8 \cdot \left(x1 \cdot x2\right) + 12 \cdot {x1}^{2}\right) - 6\right)\right)}\right) \]

   10. Simplified 97.3%

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

   11. Taylor expanded in x1 around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 97.3

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

   13. Simplified 97.3%

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

   if 31000 < x1

   1. Initial program 35.7%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 94.4

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

   5. Simplified 94.4%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 94.4

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

   8. Simplified 94.4%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{elif}\;x1 \leq 31000:\\ \;\;\;\;\mathsf{fma}\left(x1, \mathsf{fma}\left(9, x1, -1\right), x2 \cdot \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, 8, -12\right), -6\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \mathsf{fma}\left(x1, 6, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 88.3% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \mathsf{fma}\left(x1, 6, -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -195000.0)
   (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))
   (if (<= x1 18500.0)
     (fma x2 -6.0 (* x1 (fma x2 (fma x2 8.0 -12.0) -1.0)))
     (+ x1 (* (* x1 (* x1 x1)) (fma x1 6.0 -3.0))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -195000.0) {
		tmp = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	} else if (x1 <= 18500.0) {
		tmp = fma(x2, -6.0, (x1 * fma(x2, fma(x2, 8.0, -12.0), -1.0)));
	} else {
		tmp = x1 + ((x1 * (x1 * x1)) * fma(x1, 6.0, -3.0));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -195000.0)
		tmp = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0));
	elseif (x1 <= 18500.0)
		tmp = fma(x2, -6.0, Float64(x1 * fma(x2, fma(x2, 8.0, -12.0), -1.0)));
	else
		tmp = Float64(x1 + Float64(Float64(x1 * Float64(x1 * x1)) * fma(x1, 6.0, -3.0)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -195000.0], N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 18500.0], N[(x2 * -6.0 + N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(N[(x1 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -195000

   1. Initial program 32.9%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 91.4

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

   8. Simplified 91.4%

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

   if -195000 < x1 < 18500

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 90.4

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

   10. Simplified 90.4%

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

   if 18500 < x1

   1. Initial program 35.7%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 94.4

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

   5. Simplified 94.4%

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

   6. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 94.4

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

   8. Simplified 94.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + \left(x1 \cdot \left(x1 \cdot x1\right)\right) \cdot \mathsf{fma}\left(x1, 6, -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 88.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(6, x1, -3\right), x1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -195000.0)
   (* x1 (fma (* x1 x1) (fma x1 6.0 -3.0) 1.0))
   (if (<= x1 18500.0)
     (fma x2 -6.0 (* x1 (fma x2 (fma x2 8.0 -12.0) -1.0)))
     (fma (* x1 x1) (* x1 (fma 6.0 x1 -3.0)) x1))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -195000.0) {
		tmp = x1 * fma((x1 * x1), fma(x1, 6.0, -3.0), 1.0);
	} else if (x1 <= 18500.0) {
		tmp = fma(x2, -6.0, (x1 * fma(x2, fma(x2, 8.0, -12.0), -1.0)));
	} else {
		tmp = fma((x1 * x1), (x1 * fma(6.0, x1, -3.0)), x1);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -195000.0)
		tmp = Float64(x1 * fma(Float64(x1 * x1), fma(x1, 6.0, -3.0), 1.0));
	elseif (x1 <= 18500.0)
		tmp = fma(x2, -6.0, Float64(x1 * fma(x2, fma(x2, 8.0, -12.0), -1.0)));
	else
		tmp = fma(Float64(x1 * x1), Float64(x1 * fma(6.0, x1, -3.0)), x1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -195000.0], N[(x1 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * 6.0 + -3.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 18500.0], N[(x2 * -6.0 + N[(x1 * N[(x2 * N[(x2 * 8.0 + -12.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * N[(6.0 * x1 + -3.0), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -195000

   1. Initial program 32.9%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 91.5

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

   5. Simplified 91.5%

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

   6. Taylor expanded in x1 around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 91.4

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

   8. Simplified 91.4%

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

   if -195000 < x1 < 18500

   1. Initial program 99.3%

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

   3. Taylor expanded in x1 around 0

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

   5. Simplified 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(\left(-6 + \mathsf{fma}\left(x2, 14, \mathsf{fma}\left(x2, 6, 9\right)\right)\right) + \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(x2, -4, 6\right)\right), x1 \cdot x1, \mathsf{fma}\left(\mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -2\right), x1, x2 \cdot -6\right)\right)} \]

   8. Taylor expanded in x1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 90.4

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

   10. Simplified 90.4%

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

   if 18500 < x1

   1. Initial program 35.7%

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

   3. Taylor expanded in x1 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval 94.4

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

   5. Simplified 94.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. associate-+l+ N/A

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

      10. lower-+.f64 N/A

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

   7. Applied egg-rr 19.3%

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

   8. Taylor expanded in x1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 94.4

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

   10. Simplified 94.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq -195000:\\ \;\;\;\;x1 \cdot \mathsf{fma}\left(x1 \cdot x1, \mathsf{fma}\left(x1, 6, -3\right), 1\right)\\ \mathbf{elif}\;x1 \leq 18500:\\ \;\;\;\;\mathsf{fma}\left(x2, -6, x1 \cdot \mathsf{fma}\left(x2, \mathsf{fma}\left(x2, 8, -12\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x1 \cdot x1, x1 \cdot \mathsf{fma}\left(6, x1, -3\right), x1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 39.3% accurate, 33.1× speedup

Math

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

FPCore

(FPCore (x1 x2) :precision binary64 (- (* x2 -6.0) x1))

C

double code(double x1, double x2) {
	return (x2 * -6.0) - x1;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = (x2 * (-6.0d0)) - x1
end function

Java

public static double code(double x1, double x2) {
	return (x2 * -6.0) - x1;
}

Python

def code(x1, x2):
	return (x2 * -6.0) - x1

Julia

function code(x1, x2)
	return Float64(Float64(x2 * -6.0) - x1)
end

MATLAB

function tmp = code(x1, x2)
	tmp = (x2 * -6.0) - x1;
end

Wolfram

code[x1_, x2_] := N[(N[(x2 * -6.0), $MachinePrecision] - x1), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{\left(-6 \cdot x2 + x1 \cdot \left(\left(4 \cdot \left(x2 \cdot \left(2 \cdot x2 - 3\right)\right) + x1 \cdot \left(\left(2 \cdot \left(-2 \cdot x2 + -1 \cdot \left(2 \cdot x2 - 3\right)\right) + \left(3 \cdot \left(3 - -2 \cdot x2\right) + \left(6 \cdot x2 + 8 \cdot x2\right)\right)\right) - 6\right)\right) - 2\right)\right)} \]

5. Simplified 73.6%

   \[\leadsto x1 + \color{blue}{\mathsf{fma}\left(x1, \mathsf{fma}\left(x1, \mathsf{fma}\left(x2, -4, \mathsf{fma}\left(2, \mathsf{fma}\left(x2, -2, 3\right), \mathsf{fma}\left(3, \mathsf{fma}\left(x2, 2, 3\right), \mathsf{fma}\left(x2, 14, -6\right)\right)\right)\right), \mathsf{fma}\left(4, x2 \cdot \mathsf{fma}\left(x2, 2, -3\right), -2\right)\right), x2 \cdot -6\right)} \]

6. Taylor expanded in x2 around 0

   \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 - 2}, x2 \cdot -6\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{9 \cdot x1 + \left(\mathsf{neg}\left(2\right)\right)}, x2 \cdot -6\right) \]

   2. metadata-eval N/A

      \[\leadsto x1 + \mathsf{fma}\left(x1, 9 \cdot x1 + \color{blue}{-2}, x2 \cdot -6\right) \]

   3. lower-fma.f64 65.7

      \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

8. Simplified 65.7%

   \[\leadsto x1 + \mathsf{fma}\left(x1, \color{blue}{\mathsf{fma}\left(9, x1, -2\right)}, x2 \cdot -6\right) \]

9. Taylor expanded in x1 around 0

   \[\leadsto \color{blue}{-6 \cdot x2 + -1 \cdot x1} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto -6 \cdot x2 + \color{blue}{\left(\mathsf{neg}\left(x1\right)\right)} \]

    2. unsub-neg N/A

       \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]

    3. lower--.f64 N/A

       \[\leadsto \color{blue}{-6 \cdot x2 - x1} \]

    4. *-commutative N/A

       \[\leadsto \color{blue}{x2 \cdot -6} - x1 \]

    5. lower-*.f64 37.2

       \[\leadsto \color{blue}{x2 \cdot -6} - x1 \]

11. Simplified 37.2%

    \[\leadsto \color{blue}{x2 \cdot -6 - x1} \]
12. Add Preprocessing

Alternative 19: 27.0% accurate, 33.1× speedup

Math

\[\begin{array}{l} \\ x1 + x2 \cdot -6 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (+ x1 (* x2 -6.0)))

C

double code(double x1, double x2) {
	return x1 + (x2 * -6.0);
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x1 + (x2 * (-6.0d0))
end function

Java

public static double code(double x1, double x2) {
	return x1 + (x2 * -6.0);
}

Python

def code(x1, x2):
	return x1 + (x2 * -6.0)

Julia

function code(x1, x2)
	return Float64(x1 + Float64(x2 * -6.0))
end

MATLAB

function tmp = code(x1, x2)
	tmp = x1 + (x2 * -6.0);
end

Wolfram

code[x1_, x2_] := N[(x1 + N[(x2 * -6.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   2. lower-*.f64 26.5

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Simplified 26.5%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
6. Add Preprocessing

Alternative 20: 27.0% accurate, 42.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x2, -6, x1\right) \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (fma x2 -6.0 x1))

C

double code(double x1, double x2) {
	return fma(x2, -6.0, x1);
}

Julia

function code(x1, x2)
	return fma(x2, -6.0, x1)
end

Wolfram

code[x1_, x2_] := N[(x2 * -6.0 + x1), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   2. lower-*.f64 26.5

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Simplified 26.5%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{x2 \cdot -6 + x1} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{x2 \cdot -6} + x1 \]

   4. lower-fma.f64 26.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1\right)} \]

7. Applied egg-rr 26.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x2, -6, x1\right)} \]
8. Add Preprocessing

Alternative 21: 26.9% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ x2 \cdot -6 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (* x2 -6.0))

C

double code(double x1, double x2) {
	return x2 * -6.0;
}

Fortran

real(8) function code(x1, x2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = x2 * (-6.0d0)
end function

Java

public static double code(double x1, double x2) {
	return x2 * -6.0;
}

Python

def code(x1, x2):
	return x2 * -6.0

Julia

function code(x1, x2)
	return Float64(x2 * -6.0)
end

MATLAB

function tmp = code(x1, x2)
	tmp = x2 * -6.0;
end

Wolfram

code[x1_, x2_] := N[(x2 * -6.0), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x1 around 0

   \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

   2. lower-*.f64 26.5

      \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

5. Simplified 26.5%

   \[\leadsto x1 + \color{blue}{x2 \cdot -6} \]

6. Taylor expanded in x1 around 0

   \[\leadsto \color{blue}{-6 \cdot x2} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{x2 \cdot -6} \]

   2. lower-*.f64 26.3

      \[\leadsto \color{blue}{x2 \cdot -6} \]

8. Simplified 26.3%

   \[\leadsto \color{blue}{x2 \cdot -6} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))))